Foundations of Constraint Processing, Fall 2005 Sep 22, 2005Consistency: Properties & Algorithms 1...

Sep 22 2005 ConsistencyProperties amp Algorithms

1

Foundations of Constraint Processing Fall 2005

Foundations of Constraint ProcessingCSCE421821 Fall 2005

wwwcseunledu~choueiryF05-421-821

Berthe Y Choueiry (Shu-we-ri)Avery Hall Room 123Bchoueirycseunledu

Tel +1(402)472-5444

Consistency Algorithms


2


Lecture SourcesRequired reading1 Algorithms for Constraint Satisfaction Problems Mackworth and

Freuder AIJ852 Sections 31 32 33 Chapter 3 Constraint Processing Dechter

Recommended1 Sections 34mdash310 Chapter 3 Constraint Processing Dechter2 Networks of Constraints Fundamental Properties and Application

to Picture Processing Montanari Information Sciences 743 Consistency in Networks of Relations Mackworth AIJ77 4 Constraint Propagation with Interval Labels Davis AIJ875 Path Consistency on Triangulated Constraint Graphs Bliek amp Sam-

Haroud IJCAI99


3


Outline1 Motivation and material

2 Node and arc consistency and their complexity

3 Criteria for performance comparison and CSP parameters

4 Path consistency and its complexity

5 Consistency properties of a CSP

6 Other related results box consistency constraint synthesis look-ahead non-binary constraints etc


4


Consistency checkingMotivation bull CSP are solved with search (conditioning)bull Search performance is affected by

- problem size- amount of backtracking

bull Backtracking may yield

ThrashingExploring non-promising sub-trees and rediscovering the same inconsistencies over and over again

Malady of backtrack search


5


(Some) causes of thrashingSearch order V1 V2 V3 V4 V5

What happens in search if webull do not check the constraint on V3 (node inconsistency)bull do not check constraint between V3 and V5 (arc

inconsistency)bull do not check constraints between V3mdashV4 V3mdashV5 and

V4mdashV5 (path inconsistency)

1 2V4

1 2 3

0 1 2

1 2V5

1 2 3V1 V2

V3ltV2

V3gt0

V3ltV4

V5ltV4

V3ltV5

V3ltV1

V3


6


Consistency checkingGoal eliminate inconsistent combinations

Algorithms (for binary constraints)

bull Node consistencybull Arc consistency (AC-1 AC-2 AC-3 AC-7 AC-31 etc)bull Path consistency (PC-1 PC-2 DPC PPC etc)bull Constraints of arbitrary arity

ndash Waltz algorithm ancestor of ACsndash Generalized arc-consistency (Dechter Section 351 )ndash Relational m-consistency (Dechter Section 81)


7


Overview of Recommended Reading

bull Davis AIJ77

ndash focuses on the Waltz algorithm ndash studies its performance and quiescence for given

bull constraint types (order relations bounded diff algebraic etc)bull domains types (continuous or finite)

bull Mackworth AIJ77ndash presents NC AC-1 AC-2 PC-1 PC-2

bull Mackworth and Freuder AIJ85

ndash studies their complexity Mackworth and Freuder concentrate on finite domains More people work on finite domains than on continuous ones Continuous domains can be quite tough


8


Constraint Propagation with Interval Labels Ernest Davis

bull `Old paper (1987) terminology slightly differentbull Interval labels continuous domains

ndash Section 8 sign labels (discrete)bull Concerned with Waltz algorithm (eg

quiescence completeness)bull Constraint types vs domain types (Table 3)bull Addresses applications from reasoning

ndash about Physical Systems (circuit SPAM)ndash about time relations (TMM)

Advice read Section 3 more if you wish


9


Waltz algorithm for label inference

bull Refine(C(Vi Vk Vm Vn) Vi) - finds a new label for Vi consistent with C

bull Revise(C (Vi Vk Vm Vn)) refines the domains of Vi Vk Vm Vn by iterating over these variables until quiescence (no domains is further refined)

bull Waltz Algorithm revises all constraints by iterating over each constraint connected to a variable whose domain has been revised

bull Waltz Algorithm ancestor of arc-consistency

C

Vk Vm Vn

Vi


10


WARNINGrarr Completeness Running the Waltz Algorithm does not solve

the problem

A=2 and B=3 is still not a solution

rarr Quiescence The Waltz algorithm may go into infinite loops even if problem is solvable

x [0 100] x = y

y [0 100] x = 2y

rarr Davis provides a classification of problems in terms of completeness and quiescence of the Waltz algorithm

f(constraint types domain types) see Table 3

1 2 3 2 3 4

BA BA

2 3 2 3


11


Importance of this paper Establishes that constraints of bounded differences

(temporal reasoning) can be efficiently solved by the Waltz algorithm (O(n3) n number of variables)

Early paper that attracts attention on the difficulty of label propagation in interval labels (continuous domains)

This work has been continued by Faltings (AIJ 92) who studied the early-quiescence problem of the Waltz algorithm


12


Basic consistency algorithms

Examining finite binary CSPs and their complexity

Notation

Given variables i j k with values x y z the predicate

Pijk(x y z) is true iff the 3-tuple x y z Cijk

Node consistency checking Pi(x)

Arc consistency checking Pij(xy)

Path consistency bit-matrix manipulation


13


Worst-case complexity as a function of the input parameters

Upper bound f(n) is O(g(n)) means that f(n) cg(n)

f grows as g or slower

Lower bound f(n) is (h(n)) means that f(n) ch(n)

f grows as g or faster

Input parameters for a CSP

n = number of variables

a = (max) size of a domain

dk = degree of Vk ( n-1)

e = number of edges (or constraints) [(n-1) n(n-1)2]

Worst-case asymptotic complexitytime

space


14


Outline1 Motivation and material2 Node amp arc consistency and their

complexity NC AC-1 AC-3 AC-4 3 Criteria for performance comparison and CSP

parameters 4 Path consistency and its complexity5 Consistency properties of a CSP 6 Other related results box consistency

constraint synthesis look-ahead non-binary constraints etc


15


Node consistency (NC)Procedure NC(i)

Di Di x | Pi(x)

Begin

for i 1 until n do NC(i)

end


16


Complexity of NC Procedure of NC(i)

Di Di x | Pi(x) Begin for i 1 until n do NC(i)end

bull For each variable we check a valuesbull We have n variables we do na checksbull NC is O(an)


17


Arc-consistency Adapted from Dechter

Definition Given a constraint graph G

bull A variable Vi is arc-consistent relative to Vj iff for every value aDVi there exists a value bDVj | (a b)CViVj

bull The constraint CViVj is arc-consistent iff ndash Vi is arc-consistent relative to Vj and ndash Vj is arc-consistent relative to Vi

bull A binary CSP is arc-consistent iff every constraint (or sub-graph of size 2) is arc-consistent

1

22

3

1

2

3

1

2

3

Vi VjVjVi


18


Procedure ReviseRevises the domains of a variable i

Procedure Revise(ij) Begin

DELETE false for each x Di do

if there is no y Dj such that Pij(x y) then begin

delete x from Di

DELETE true end return DELETE end

bull Revise is directionalbull What is the complexity of Revise

a2


19


Revise example R Dechter

Apply the Revise procedure to the following example

5 6 15 1 2 10

YX+Y = 10X


20


Solution DVi DVi Vi (CViVj DVi)

This is actually equivalent to DVi Vi (CViVj DVi)

Effect of Revise Adapted from Dechter

Question Given

bull two variables Vi and Vj

bull their domains DVi and DVj and

bull the constraint CViVj

write the effect of the Revise procedure as a sequence of operations in relational algebra

Hintbull Think about the domain DVi as a unary constraint CVi and

bull consider the composition of this unary constraint and the binary one

1

22

3

1

2

3

1

2

3

Vi VjVjVi


21


Arc consistency (AC-1)Procedure AC-11 begin2 for i 1 until n do NC(i)3 Q (i j) | (ij) arcs(G) i j 4 repeat5 begin6 CHANGE false7 for each (i j) Q do CHANGE ( REVISE(i j) or CHANGE )8 end9 until not CHANGE10 end

bull AC-1 does not update Q the queue of arcsbull No algorithm can have time complexity below O(ea2)


22


Arc consistency

1 AC may discover the solution Example borrowed from Dechter

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency

2 AC may discover inconsistency Example borrowed from Dechter

1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24


NC amp AC Example courtesy of M Fromherz

In the temporal problem below

1 AC propagates bound

2 Unary constraint xgt3 is imposed

3 AC propagates bounds again

[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25


Complexity of AC-1Procedure AC-11 begin2 for i 1 until n do NC(i)3 Q (i j) | (ij) arcs(G) i j4 repeat5 begin6 CHANGE false7 for each (i j) Q do CHANGE (REVISE(i j) or CHANGE)8 end9 until not CHANGE10 end

Note Q is not modified and |Q| = 2ebull 4 9 repeats at most na timesbull Each iteration has |Q| = 2e calls to REVISEbull Revise requires at most a2 checks of Pij

AC-1 is O(a3 n e)


26


AC versions

bull AC-1 does not update Q the queue of arcs

bull AC-2 iterates over arcs connected to at least one node whose domain has been modified Nodes are ordered

bull AC-3 same as AC-2 nodes are not ordered


27


Arc consistency (AC-3)AC-3 iterates over arcs connected to at least one

node whose domain has been modified

Procedure AC-31 begin2 for i 1 until n do NC(i)3 Q (i j) | (i j) arcs(G) i j 4 While Q is not empty do5 begin6 select and delete any arc (k m) from Q7 If Revise(k m) then Q Q (i k) | (i k) arcs(G) i k i m 8 end9 end


28


Complexity of AC-3Procedure AC-3

1 begin

2 for i 1 until n do NC(i)

3 Q (i j) | (i j) arcs(G) i j

4 While Q is not empty do

5 begin

6 select and delete any arc (k m) from Q

7 If Revise(k m) then Q Q (i k) | (i k) arcs(G) i k i m

8 end

9 end

bull First |Q| = 2e then it grows and shrinks 48

bull Worst case ndash 1 element is deleted from DVk per iteration

ndash none of the arcs added is in Q

bull Line 7 a(dk - 1)

bull Lines 4 - 8

bull Revise a2 checks of Pij

AC-3 is O(a2(2e + a(2e-n)))

Connected graph (e n ndash 1) AC-3 is O(a3e)

If AC-p AC-3 is O(a2e) AC-3 is (a2e) Complete graph O(a3n2)

)2(2121

neaedaen

k k


29


Example Apply AC-3Example Apply AC-3 Thanks to Xu Lin

DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30


Applying AC-3 Thanks to Xu Lin

bull Queue = CV2 V4 CV4V2 CV1V3 CV2V3 CV3 V1 CV3V2

bull Revise(V2V4) DV2 DV2 5 = 1 2 3 4

bull Queue = CV4V2 CV1V3 CV2V3 CV3 V1 CV3 V2

bull Revise(V4 V2) DV4 DV4 4 = 1 2 3 5

bull Queue = CV1V3 CV2V3 CV3 V1 CV3 V2

bull Revise(V1 V3) DV1 1 2 3 4 5

bull Queue = CV2V3 CV3 V1 CV3 V2

bull Revise(V2 V3) DV2 1 2 3 4


31


Applying AC-3 (cont) Thanks to Xu Lin

bull Queue = CV3 V1 CV3 V2



bull Revise(V2 V3) DV2 DV2

bull Queue = CV3 V2

bull Revise(V3 V2) DV3 1 3 5

bull Queue = CV1 V3

bull Revise(V1 V3) DV1 1 3 5 END


32


Main ImprovementsMohr amp Henderson (AIJ 86) bull AC-3 O(a3e) AC-4 O(a2e)bull AC-4 is optimalbull Trade repetition of consistency-check operations with heavy

bookkeeping on which and how many values support a given value for a given variable

bull Data structures it usesndash m values that are active not filteredndash s lists all vvps that support a given vvpndash counter given a vvp provides the number of support provided by a

given variable

bull How it proceeds1 Generates data structures2 Prepares data structures3 Iterates over constraints while updating support in data structures


33


Step 1 Generate 3 data structures

bull m and s have as many rows as there are vvprsquos in the problembull counter has as many rows as there are tuples in the constraints

TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34


Step 2 Prepare data structuresData structures s and counter

bull Checks for every constraint CViVj all tuples Vi=ai Vj=bj_)bull When the tuple is allowed then update

ndash s(Vjbj) s(Vjbj) (Vi ai) andndash counter(Vjbj) (Vjbj) + 1

Update counter ((V2 V3) 2) to value 1Update counter ((V3 V2) 2) to value 1Update s-htable (V2 2) to value ((V3 2))Update s-htable (V3 2) to value ((V2 2))



35


Constraints CV2V3 and CV3V2 S

(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m

Note that (V3 V2)4 0 thus

we remove 4 from the domain of V3

and update (V3 4) nil in m

Updating counter

Since 4 is removed from DV3 then

for every (Vk l) | (V3 4) s[(Vk l)]

we decrement counter[(Vk V3) l] by 1


36


Summary of arc-consistency algorithmsbull AC-4 is optimal (worst-case) [Mohr amp Henderson AIJ 86]

bull Warning worst-case complexity is pessimistic Better worst-case complexity AC-4

Better average behavior AC-3 [Wallace IJCAI 93]

bull AC-5 special constraints [Van Hentenryck Deville Teng 92]

functional anti-functional and monotonic constraints

bull AC-6 AC-7 general but rely heavily on data structures for bookkeeping[Bessiegravere amp Reacutegin]

bull Now back to AC-3 AC-2000 AC-2001 AC-31 AC33 etc

bull Non-binary constraints ndash GAC (general) [Mohr amp Masini 1988] ndash all-different (dedicated) [Reacutegin 94]


37


Outline1 Motivation and material2 Node and arc consistency and their complexity 3 Criteria for performance comparison amp CSP

parameters bull Which AC algorithm to choose Project results

may tellhellip 4 Path consistency and its complexity5 Consistency properties of a CSP6 Other related results box consistency

constraint synthesis look-ahead non-binary CSPs etc


38


CSP parameters lsaquon a t drsaquobull n is number of variablesbull a is maximum domain sizebull t is constraint tightness

bull d is constraint density

where e is the constraints emin=(n-1) and emax = n(n-1)2

Lately we use constraint ratio p = eemax

rarr Constraints in random problems often generated uniform

rarr Use only connected graphs (throw the unconnected ones away)

tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39


Criteria for performance comparison

1 Bounding time and space complexity (theoretical)ndash worst-casendash average-case ndash best- case

2 Counting the number of constraint checks (theoretical empirical)

3 Measuring CPU time (empirical)


40


Performance comparison Courtesy of Lin XU

AC-7

AC-4

AC-3


41


Wisdom () Free adaptation from Bessiegravere

bull When a single constraint check is very expensive to make use AC-7

bull When there is a lot of propagation use AC-4bull When little propagation and constraint checks

are cheap use AC-3x

Advice Instructors personal opinion

Use AC-3


42






5 Consistency properties of CSPs

6 Other related results box consistency constraint synthesis look-ahead non-binary CSPs etc


43


AC is not enough Example borrowed from Dechter

Arc-consistentSatisfiable

seek higher levels of consistency

V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44


Consistency of a pathA path (V0 V1 V2 hellip Vm) of length m is consistent iff bull for any value xDV0 and for any value yDVm that are consistent (ie PV0 Vm(x y))

a sequence of values z1 z2 hellip zm-1 in the domains of variables V1 V2 hellip Vm-1 such that all constraints between them (along the path not across it) are satisfied

(ie PV0 V1(x z1) PV1 V2(z1 z2) hellip PVm-1 Vm(zm-1 zm) )

for all x DV0 for all y DVm

V0 Vm

Vm-1

V2

V1


45


Note The same variable can appear more than once in the path

Every time it may have a different value

Constraints considered PV0Vm and those along the path

All other constraints are neglected


V0 Vm

Vm-1

V2

V1


46


Example consistency of a path Check path length = 2 3 4 5 6

a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6

All mutex constraints


47


Path consistency definition

A path of length m is path consistent A CSP is path consistent Property of a

CSP

Definition A CSP is path consistent (PC) iff every path is

consistent (ie any length of path)

Question should we enumerate every path of any length

Answer No only length 2 thanks to [Mackworth AIJ77]


48


Making a CSP Path Consistent (PC)

Special case Complete graphTheorem In a complete graph if every path of length 2 is

consistent the network is path consistent [Mackworth AIJ77]

PC-1 two operations composition and intersection

Proof by induction

Special case Triangulated graphTheorem In a triangulated graph if every path of length 2 is

consistent the network is path consistent [Bliek amp Sam-Haroud lsquo99]

PPC (partially path consistent) PC


49


Tools for PC-1

Two operators

1 Constraint composition ( bull )

R13 = R12 bull R23

2 Constraint intersection ( )

R13 R13 old R13 induced


50


Path consistency (PC-1)Achieved by composition and intersection (of binary relations

expressed as matrices) over all paths of length twoProcedure PC-11 Begin2 Yn R3 repeat4 begin5 Y0 Yn

6 For k 1 until n do7 For i 1 until n do8 For j 1 until n do

9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj

10 end11 until Yn = Y0

12 Y Yn

10 end


51


Properties of PC-1Discrete CSPs [Montanari74]

1 PC-1 terminates2 PC-1 results in a path consistent CSP

bull PC-1 terminates It is complete sound (for finding PC network)

bull PC-2 Improves to PC-1similarly to (AC-1 AC-3)

Complexity of PC-1


52


Complexity of PC-1Procedure PC-11 Begin2 Yn R3 repeat4 begin5 Y0 Yn


9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3

Lines 6ndash10 n3 a3

Line 3 at most n2 relations x a2 elements

PC-1 is O(a5n5)

PC-2 is O(a5n3) and (a3n3)


53


Some improvementsbull Mohr amp Henderson (AIJ 86)

ndash PC-2 O(a5n3) PC-3 O(a3n3)ndash Open question PC-3 optimal

bull Han amp Lee (AIJ 88)ndash PC-3 is incorrectndash PC-4 O(a3n3) space and time

bull Singh (ICTAI 95)ndash PC-5 uses ideas of AC-6 (support bookkeeping)

Note PC is seldom used in practical applications unless in presence of special type of constraints (eg bounded difference)


54


Path consistency as inference of binary constraints

Path consistency corresponds to inferring a new constraint

(alternatively tightening an existing constraint) between every two

variables given the constraints that link them to a third variable

Considers all subgraphs of 3 variables

3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56


Question Adapted from Dechter

Given three variables Vi Vk and Vj and the constraints CViVk CViVj and CVkVj write the effect of PC as a sequence of operations in relational algebra

B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C

Solution CViVj CViVj ij(CViVk CVkVj)


57


Constraint propagation courtesy of Dechter

After Arc-consistency

After Path-consistency

bull Are these CSPs the samendash Which one is more explicitndash Are they equivalent

bull The more propagationndash the more explicit the constraintsndash the more search is directed towards a solution

1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent

PC can detect unsatisfiability

V3

a b

V4

a bV2

V1 a b

a b

a b


59


Outline1 Motivation and material2 Node and arc consistency and their complexity 3 Criteria for performance comparison and CSP

parameters bull Which AC algorithm to choose Project results may

tell4 Path consistency and its complexity5 Consistency properties of a CSP6 Other related results box consistency



60


Consistency properties of a CSP

bull (Arc Consistency AC)bull (Path Consistency PC)bull Minimalitybull Decomposability (ref lossless join decomposition)bull When PC approximates minimality and

decomposability ndash Example temporal reasoning

bull (strong) k-consistencybull (ij)-consistency


61


MinimalityPC tightens the binary constraints

The tightest possible binary constraints yield the minimal network

Minimal network aka central problem

Given two values for two variables if they are consistent then they appear in at least one solution

Notebull Minimal path consistentbull The definition of minimal CSP is concerned with binary CSPs


62


Minimal CSP



Informallybull In a minimal CSP the remainder of the CSP does not add any

further constraint to the direct constraint CVi Vj between the two variables Vi and Vj [Mackworth

AIJ77]

bull A minimal CSP is perfectly explicit as far as the pair Vi and Vj is concerned the rest of the network does not add any further constraint to the direct constraint CVi Vj [Montanari74]

bull The binary constraints are explicit as possible [Montanari74]


63


Decomposabilitybull Any combination of values for k variables that satisfy the constraints

between them can be extended to a solution

bull Decomposability generalizes minimality

Minimality any consistent combination of values for

any 2 variables is extendable to a solution

Decomposability any consistent combination of values for

any k variables is extendable to a solution

Decomposable Minimal Path Consistent

Strong n-consistent n-consistent Solvable


64


Relations to (theory of) DB

CSP DatabaseMinimal Pair-wise consistent

Decomposable Complete join


65


PC approximates In generalbull Decomposability minimality path consistentbull PC is used to approximate minimality (which is the central problem)

When is the approximation the real thingSpecial casesbull When composition distributes over intersection [Montanari74]

PC-1 on the completed graph guarantees minimality and decomposability

bull When constraints are convex [Bliek amp Sam-Haroud 99]

PPC on the triangulated graph guarantees minimality and decomposability (and the existing edges are as tight as possible)


66


PC algorithms and propertyCSP0 Complete

the graph PC-1

CSP0 Triangulate the graph PPC

Result

bull CSP0 is not path consistent or

bull PC-1(CSP0) is path consistent

1 It is tight but could be tighter

2 The graph is complete

Result


bull PPC(CSP0) is path consistent

1 It is generally less tight than PC-1(CSP0)

2 The graph is triangulatedWatch for PPC


67


PC algorithms special casesCSPa Complete

the graph PC-124Result [Montanari 74]

bull CSPa is not path consistent or

bull PC-1(CSPa) is

1 Path consistent minimal amp decomposable

2 Graph is complete

With a(bc)=(ab)(bc)

CSPb Triangulatethe graph PPC

Constraint are convex

Result [BampS-H 99]

bull CSPb is not path consistent or

bull PPC(CSPb) is

1 Path consistent minimal decomposable

2 Existing edges are as tight as PC-1(CSPb)

3 Graph is triangulated


68


Special conditions examples

bull Distributivity propertyndash 1 loop in PC-1 (PC-3) is sufficient

bull Exploiting special conditions in temporal reasoning ndash Temporal constraints in the Simple Temporal Problem

(STP) composition amp intersectionndash Composition distributes over intersection

bull PC-1 is a generalization of the Floyd-Warshall algorithm (all pairs shortest path)

ndash Convex constraintsbull PPC


69


Distributivity propertyIn PC-1 two operations

RAB bull (RBC RBC) = (RAB bull RBC) (RAB bull RrsquoBC)

When ( bull ) distributes over ( ) then [Montanari74]

1 PC-1 guarantees that CSP is minimal and decomposable

2 1 loop in PC-1 is sufficient

Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70


Temporal reasoning constraints of bounded difference

Variables X Y Z etc

Constraints a Y-X b ie Y-X = [a b] = I

Composition I 1 bull I2 = [a1 b1] bull [a2 b2] = [a1+ a2 b1+b2] Interpretation ndash intervals indicate distancesndash composition is triangle inequality

Intersection I1 I2 = [max(a1 a2) min(b1 b2)]

Distributivity I1 bull (I2 I3) = (I1 bull I2) (I1 bull I3)

Proof left as an exercise


71


Composition of intervals + Rrsquo13 = R12 + R23 = [4 12]

R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]

Intersection of intervals R13 R13 = [4 12] [3 5] = [4 5]

R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]

Path consistency guarantees minimality and decomposability

Example constraints of bounded difference

Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP

bull PC-1 generalizes Floyd-Warshall algorithm (all-pairs shortest path) where ndash composition is lsquoscalar additionrsquo andndash intersection is lsquoscalar minimalrsquo


73


Convex constraints temporal reasoning (again)

Thanks to Xu Lin (2002)

bull Constraints of bounded difference are convex

bull We triangulate the graph (good heuristics exist)

bull Apply PPC restrict propagations in PC to triangles of the graph (and not in the complete graph)

bull According to [Bliek amp Sam-Haroud 99] PPC becomes equivalent to PC thus it guarantees minimality and decomposability


74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable

we should seek (even) higher levels of consistency

k-consistency k = 1 2 3 hellip

a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75


k-consistencyA CSP is k-consistent iff

ndash given any consistent instantiation of any (k-1) distinct variables

ndash there exists an instantiation of any kth variable such that the k values taken together satisfy all the constraints among the k variables

Example

(courtesy of Dechter) Q

Q

Q

Q

Q

3-consistent 4-consistent


76


bull Questionndash Is this CSP 3-consistent ndash is it 2-consistent

bull Lesson ndash 3-consistency does not guarantee 2-consistency

Warning Does 3-consistency guarantee 2-consistency

red blue

A

B

red

red blue

red

C


77


(Strong) k-consistency definition

Definitions

bull k-consistency vs strong k-consistency

bull k-consistency ndash any consistent assignment for (k-1)-variable

can be extended to any kth variable

bull strong k-consistency ndash has to be j-consistent for all j k


78


(ij)-consistency definition

bull (ij) consistency ndash every consistent instantiation to i variablesndash can be extended to every j variables

bull Consequently ndash 2-consistency i=1 j=1ndash 3-consistency i=2 j=1ndash k-consistency i=k-1 j=1ndash Minimality i=2 j=n-2ndash Decomposability i=k j=n-k


79


Consistency properties amp algorithms summary

bull New terms AC PC minimal CSP decomposable CSP complete graph triangulated graph Revise AC-1 AC-3 PC-1 PC-2 PPC

bull Complexity of AC PC algorithmsbull AC tightens domains PC tightens binary constraintsbull Consistency operations in terms of composition

intersection in relational algebrabull Path consistency approximates minimality which

approximates decomposabilitybull Situations when path consistency guarantees minimality

and decomposabilitybull Strong k-consistency and (ij)-consistency


80




3 Criteria for performance comparison and CSP parameters bull Which AC algorithm to choose Project results may tell



6 Other related results bull constraint synthesisbull box consistency bull look-ahead bull non-binary CSPs etc


81


Constraint synthesisbull k-consistency an alternative view

ndash Constraint of arity (k-1) are used to synthesize a constraint of arity k which is then used to filter constraints of arity (k-1)

bull Examplesndash 2-consistency DVi DVi Vi (CViVj DVj)

Projection of binary constraint is used to tighten unary constraint

ndash 3-consistency CViVj CViVj ViVj(CViVk CVkVj)Binary constraints are used to synthesize a ternary constraintProjection of the ternary constraint is used to tighten binary constraint

ndash k-consistency

[Freuder 78 82]

Constraints of arity (k-1) are used to synthesize constraints of arity kProjection of k-ary constraint is used to tighten (k-1)-ary constraints


82


Solving CSPs by Constraint synthesis [Freuder 78]

bull From k=2 to k=n ndash achieve k-consistency by using (k-1)-arity

constraints to synthesize k-arity constraints ndash then use the k-ary constraints to filter

constraints of arity k-1 k-2 etc

bull Process ends with a unique n-ary constraint whose tuples are all the solutions to the CSP


83


Interval constraints

bull Domains are (continuous) intervals ndash Historically also called continuous CSPs continuous

domains

bull Domains are infinite ndash We cannot enumerate consistent valuestuplesndash [Davis AIJ 87] (see recommended reading) showed

that even AC may be incomplete or not terminate

bull We apply consistency (usually arc-consistency) on the boundaries of the interval


84


Look-ahead (already discussed)

bull Integration of consistency algorithm with searchbull AC

ndash Forward Checking (FC) [Haralick amp Elliott]

ndash Directional Arc-Consistency (DAC) [Dechter]

ndash Maintaining arc consistency (MAC) [Sabin amp Freuder]

bull PCndash Directed path consistency (DPC) [Dechter]


85


Non-binary CSPs

bull (Almost) all algorithms and properties discussed so far were restricted to binary CSPs

bull Consistency properties for non-binary CSPs are the topic of current research Mainly properties and algorithms forndash GAC generalized arc-consistency [Mohr amp Masini]

ndash Relational m-consistency

[Dechter Chap 8]


86


Generalized Arc-Consistencybull Conceptually project the constraint on each of

the variables in its scope to tighten the domain of the variable

bull When constraint is not defined in extension GAC may be problematic (eg NP-hard in TCSP)


87


Relational consistency

bull Dechter (see Section 811) generalizes consistency properties to non-binary constraints as relational m-consistencyndash Relational 1-consistency = relational arc-

consistency = GAC ndash Relational 2-consistency = relational path-

consistency


88


Look-ahead in non-binary CSPsbull In binary CSPs when a variable is instantiated

Revise (Vf Vc) updates the domain of a future variable Vf that is a neighbor to the current variable Vc

bull In non-binary CSPs a constraint that applies to a current variable may involve one or more past variables and one or more future variables The question is when to trigger Revise

bull There are 5 strategies nFC0 nFC1 hellip nFC5 in increasing lsquoaggressivityrsquo [Bessiere et al AIJ 02 ]

bull For example nFC0 triggers Revise only when all but one variable has been instantiated


89


Related results summarybull Interval constraints box consistencybull Solving CSPs by Constraint synthesis [Freuder]

ndash From k=2 to k=n-1 achieve k-consistency by using (k-1)-arity constraints to synthesize k-arity constraints then filter constraints of arity k-1 k-2 etc

bull Integration with searchndash AC

bull Forward Checking (FC) [Haralick amp Elliott] bull Directional Arc-Consistency (DAC) [Dechter]

bull Maintaining arc consistency (MAC) [Sabin amp Freuder]ndash PC Directed path consistency (DPC) [Dechter]

bull Non-binary constraintsndash Generalized arc consistency (GAC) [Mohr amp Masini]

ndash Relational consistency [Dechter]

bull Non-binary constraints and integration with search nFC0 nFC1 hellip nFC5

bull etc


2


Lecture SourcesRequired reading1 Algorithms for Constraint Satisfaction Problems Mackworth and

Freuder AIJ852 Sections 31 32 33 Chapter 3 Constraint Processing Dechter

Recommended1 Sections 34mdash310 Chapter 3 Constraint Processing Dechter2 Networks of Constraints Fundamental Properties and Application

to Picture Processing Montanari Information Sciences 743 Consistency in Networks of Relations Mackworth AIJ77 4 Constraint Propagation with Interval Labels Davis AIJ875 Path Consistency on Triangulated Constraint Graphs Bliek amp Sam-

Haroud IJCAI99


3









4








5






1 2V4

1 2 3

0 1 2

1 2V5

1 2 3V1 V2

V3ltV2

V3gt0

V3ltV4

V5ltV4

V3ltV5

V3ltV1

V3


6







7



bull Davis AIJ77







8









9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


3









4








5






1 2V4

1 2 3

0 1 2

1 2V5

1 2 3V1 V2

V3ltV2

V3gt0

V3ltV4

V5ltV4

V3ltV5

V3ltV1

V3


6







7



bull Davis AIJ77







8









9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


4








5






1 2V4

1 2 3

0 1 2

1 2V5

1 2 3V1 V2

V3ltV2

V3gt0

V3ltV4

V5ltV4

V3ltV5

V3ltV1

V3


6







7



bull Davis AIJ77







8









9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


5






1 2V4

1 2 3

0 1 2

1 2V5

1 2 3V1 V2

V3ltV2

V3gt0

V3ltV4

V5ltV4

V3ltV5

V3ltV1

V3


6







7



bull Davis AIJ77







8









9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


6







7



bull Davis AIJ77







8









9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


7



bull Davis AIJ77







8









9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


8









9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


9







C

Vk Vm Vn

Vi


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


10



the problem



x [0 100] x = y

y [0 100] x = 2y



1 2 3 2 3 4

BA BA

2 3 2 3


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


11







12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


12




Notation







13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


13













space


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


14







15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


15



Di Di x | Pi(x)

Begin


end


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


16






17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


17







1

22

3

1

2

3

1

2

3

Vi VjVjVi


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


18






delete x from Di



a2


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


19




5 6 15 1 2 10

YX+Y = 10X


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


20





Question Given







1

22

3

1

2

3

1

2

3

Vi VjVjVi


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


21





22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


22


Arc consistency


V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1

V2 V3

V1


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


23


Arc consistency


1 2 3

1 2 3 1 2 3 YltZ

ZltX

ZY

X

XltY


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


24







[0 10]

[0 10]

x y-3

x

y

[0 7]

[3 10]

x y-3

x

y

[4 7]

[7 10]

x y-3

x

y


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


25




AC-1 is O(a3 n e)


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


26


AC versions





27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


27






28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


28



1 begin




5 begin



8 end

9 end





bull Lines 4 - 8


AC-3 is O(a2(2e + a(2e-n)))



)2(2121

neaedaen

k k


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


29



DV1 = 1 2 3 4 5

DV2 = 1 2 3 4 5

DV3 = 1 2 3 4 5

DV4 = 1 2 3 4 5

CV2V3 = (2 2) (4 5) (2 5) (3 5) (2 3) (5 1) (1 2) (5 3) (2 1) (1 1)

CV1V3 = (5 5) (2 4) (3 5) (3 3) (5 3) (4 4) (5 4) (3 4) (1 1) (3 1)

CV2V4 = (1 2) (3 2) (3 1) (4 5) (2 3) (4 1) (1 1) (4 3) (2 2) (1 5)

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

V1

V3

V2

V4

V1

V3

V2

V4 1 3 5 1 2 3 5

1 2 3 4 1 3 5


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


30












31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


31







bull Queue = CV3 V2


bull Queue = CV1 V3



32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


32





given variable



33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


33




TV

TV

TV

TV

TV

m

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

NILV

NILV

NILV

NILV

NILV

s

)4(

)2(

)3(

)2(

)5(

4

2

3

3

4

04)(

01)(

01)(

02)(

03)(

31

32

24

24

24

VV

VV

VV

VV

VV

counter


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


34








35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


35



(V4 5) Nil

(V3 2) ((V2 1) (V2 2))

(V3 3) ((V2 5) (V2 2))

(V2 2) ((V31) (V33) (V3 5) (V32))

(V3 1) ((V2 1) (V2 2) (V2 5))

(V2 3) ((V3 5))

(V1 2) Nil

(V2 1) ((V3 1) (V3 2))

(V1 3) Nil

(V3 4) Nil

(V4 2) Nil

(V3 5) ((V2 3) (V2 2) (V2 4))

(v4 3) Nil

(V1 1) Nil

(V2 4) ((V3 5))

(V2 5) ((V3 3) (V3 1))

(V4 1) Nil

(V1 4) Nil

(V1 5) Nil

(V4 4) Nil

Counter(V4 V2) 3 0

(V4 V2) 2 0

(V4 V2) 1 0

(V2 V3) 1 2

(V2 V3) 3 4

(V4 V2) 5 1

(V2 V4) 1 0

(V2 V3) 4 0

(V4 V2) 4 1

(V2 V4) 2 0

(V2 V3) 5 0

(V2 V4) 3 2

(V2 V4) 4 0

(V2 V4) 5 0

(V3 V1) 1 0

(V3 V1) 2 0

(V3 V1) 3 0

hellip hellip

(V3 V2) 4 0

Etchellip Etc

Updating m




Updating counter





36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


36











37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


37







38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


38








tuplesall

tuplesforbiddent

)(

)(

minmax

min

ee

eed


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


39







40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


40



AC-7

AC-4

AC-3


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


41





are cheap use AC-3x


Use AC-3


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


42









43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


43





V1

a b

a b a bV2 V3

=

V1

b a

a b b aV2 V3

=

=


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


44






V0 Vm

Vm-1

V2

V1


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


45







V0 Vm

Vm-1

V2

V1


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


46



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


47




CSP






48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


48






Proof by induction





49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


49


Tools for PC-1

Two operators


R13 = R12 bull R23




50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


50





9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


51






Complexity of PC-1


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


52




9 Ylij Yl-1

ij Yl-1ik bull Yl-1

kk bull Yl-1kj


12 Y Yn

10 end

Line 9 a3



PC-1 is O(a5n5)



53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


53








54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


54







3-consistency

B

C

A

B lt C

A lt B

B

C

AA lt B

A lt C

B lt C


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


55



Another example

V4

V3

a b

V4

V2

V1

a b

V3

a b

a b

V2

a b

V1

a b

a b

=

=a b a b


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


56




B

C

A

B lt C

A lt B

A + 3 gt C

B

C

A

A lt B

A + 3 gt C

A lt C

B lt C

B

C

A

A lt B

-3 lt A ndashC lt 0

B lt C



57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


57







1

2

3

1

2

3

1

2 2

3

( 0 1 )

( 0 1 )( 0 1 )

( 0 1 )

( 0 1 )( 0 1 )


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


58


Arc-consistent

Path-consistent


V3

a b

V4

a bV2

V1 a b

a b

a b


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


59







60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


60







61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


61








62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


62


Minimal CSP





AIJ77]




63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


63












64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


64






65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


65








66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


66



the graph PC-1


Result





Result






67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


67





bull PC-1(CSPa) is


2 Graph is complete

With a(bc)=(ab)(bc)



Result [BampS-H 99]


bull PPC(CSPb) is





68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


68









69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


69







Intersection

Composition bull

RAB RrsquoBCRBC

A

B

C


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


70



Variables X Y Z etc







71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


71



R01 + R13 = [25] + [3 5] = [5 10]

R01 + R13 = [25] + [4 12] = [6 17]


R01 + (R13 R13) = (R01 + R13) (R01 + R13)

R01 + (R13 R13) = [2 5] + [4 5] = [6 10]

(R01 + R13) (R01 + R13) = [5 10] [617] = [6 10]



Rrsquo13

R13 =[35]

R23=[18]R12=[34]

V1 V3

V2

V0 R01=[25] [


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


72


PC-1 on the STP



73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


73









74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


74


PC is not enough

Arc-consistent

Path-consistent

Satisfiable



a b c

V2

a b c

V3

a b c

V1

a b c

V4

a b cV7

a b c V5

a b c

V6



75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


75





Example


Q

Q

Q

Q



76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


76





red blue

A

B

red

red blue

red

C


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


77



Definitions






78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


78






79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


79









80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


80









81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


81







ndash k-consistency

[Freuder 78 82]



82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


82








83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


83




domains





84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


84









85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


85


Non-binary CSPs




[Dechter Chap 8]


86






87





consistency


88








89










bull etc


86






87





consistency


88








89










bull etc


87





consistency


88








89










bull etc


88








89










bull etc


89










bull etc

Foundations of Constraint Processing, Fall 2005 Sep 22, 2005Consistency: Properties & Algorithms 1...

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Transcript of Foundations of Constraint Processing, Fall 2005 Sep 22, 2005Consistency: Properties & Algorithms 1...