Topics in artificial intelligence 1/1 Dr hab. inż. Joanna Józefowska, prof. PP Reasoning and...
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Dr hab. inż. Joanna Józefowska, prof. PP pics in artificial intelligence 1/1 Reasoning and search techniques
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Transcript of Topics in artificial intelligence 1/1 Dr hab. inż. Joanna Józefowska, prof. PP Reasoning and...
Dr hab. inż. Joanna Józefowska, prof. PP
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Reasoning and search techniques
Dr hab. inż. Joanna Józefowska, prof. PP
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Plan
• Reasoning in Description logics– Subsumption– Classification– Satisfiability– Tableau algorithms
• Reasoning and search– Search space– MIN-MAX algorithm– Alpha-beta algorithm
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Lecturer attends.Course T Student
Reasoning task: sumsumption
C is subsumed by D with respect to T
iff
CI DI holds for all models I of T
C T D
C T D
Intuition
If then D is more general than C.
Lecturer = Person teaches.Course
Student = Person attends.Course
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PhDStudent = teaches.Course Student
Reasoning task: classification
Arrange all defined objects from TBox in a hierarchy with respect to generality.
Lecturer = Person teaches.CourseStudent = Person attends.Course
Student
Person
Lecturer
PhDStudent
Can be computed using multiple subsumption tests.
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Then sibling.Woman sibling.Man is unsatisfiable w.r.t. T.
Reasoning task: satisfiability
C is satisfiable w.r.t. T iff T has a model with CI .
Woman = Person FemaleMan = Person Female
Subsumption can be reduced to (un)satisfiability and vice versa.
Intuition: If unsatisfiable the concept contains a contradiction.
iff C D is not satisfiable w.r.t. TC T D
C T C is satisfiable w.r.t. T iff not
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Description logics are more than concept language
Knowledge base
TBoxterminological knowledgebackground knowledge
ABoxknowledge about individuals
Use concept
language
DL Reasoner
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Definitorial TBoxes
A primitive interpretation for TBox T interprets
• the primitive concept names
• all role names
A TBox is called definitorial if every primitive interpretation for T can be uniquely extended to a model of T.
i.e. primitive concepts (and roles) uniquely determine defined concepts.
Not all TBoxes are definitorial Person = parent.Person
Non-definitorial TBoxes describe constraints, e.g. from background knowledge.
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Acyclic TBoxes
TBox is acyclic if there are no definitorial cycles.
Lecturer = Person teaches.Course
Course = hastitle.Title tought-by.Lecturer
Expansion of acyclic TBox T
exhaustively replace defined concept name with their definition (terminates due to acyclicity)
Acyclic TBoxes are always definitorial
first expand then set AI := CI for all A = C T
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Acyclic TBoxes II
For reasoning acyclic TBoxes can be eliminated
• to decide with T acyclic• expand T• replace defined concept names in C, D with their
definition• decide
• analogously for satisfiability
C T D
C D
May yield an exponential blow-up.
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General concept inclusions
General Tbox: finite set of general concept implications (GCIs)
with both C and D allowed to be complex.
C D
Course attended-by.Sleeping Boring
Note: C D equivalent to T = C D
(in terms of model I)
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Tableau algorithms
Goal: an algorithm which takes an ALC concept C0 and
1. Returns „satisfiable” iff C0 is satisfiable
2. Terminates on every input
i.e. decides satisfiability of ALC concepts
Recall: such an algorithm cannot exist for FOL since satisfiability of FOL is not decidable!
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Negation normal form (NNF)
Negation occurs only in front of concept names
C
C D
C D
R. C
R. C
C
(C D)
(C D)
R.C
R.C
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IntuitionIs A R.B R. B satisfiable?
The tableau algorithm works on a complete tree which
• represents a model I:
• nodes represent elements of I
each node x is labeled with concepts L(x) sub(C0), C L(x) is read as „x should be an instance of C”
• edges represent role successorship
each edge x,y is labelled with a role name from C0, R L(x,y) is read as „(x,y) should be in RI”
• is initialized with a single root node x0 with L(x0) = {C0}
• is expanded using completion rules
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Completion rules
rule: if (C1 C2) L(x) and {C1, C2} L(x)
then set L(x) = L(x) {C1, C2}
rule: if (C1 C2) L(x) and {C1, C2} =
then set L(x) = L(x) C for some C {C1, C2}
rule: if S.CL(x) and x has no S-successor y with CL(x) then create a new node y with L(x,y)={S} and L(y)={C}
rule: if S.CL(x) and there is an S-successor y of x with CL(y)
then set L(y) = L(y) {C}
We only apply rules if their application does „something new”
The rule is non-deterministic
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Clash
A c-tree contains a clash if it has a node x with L(x) or {A, A} L(x) – otherwise it is clash-free
C0 is satisfiable iff the completion rules can be applied in such a way that it results in a complete and clash-free c-tree.
Careful: this is non-deterministic
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Properties of the tableau algorithm
Let C0 be an ALC concept in NNF. Then:
1. the algorithm terminates when applied to C0 and
2. the rules can be applied such that they generate a clash-free and complete completion tree iff C0 is satisfiable.
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Example L(x) = {A, R.B, R.B}x
w L(w) = {B,R.B}
R
CLASH!
y
R
L(y) = {B, B}
xAI xAI
x(R.B)I d: (x,d)RI, dBI
x (R.B)I d(B)I
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ABoxes
An ABox is a finite set of assertions
a : C (a – individual name, C – concept)
(a,b) : R (a, b – individual names, R – role name)
E.g. {peter : Student, (ai-course, joanna) : tought-by}
Interpretations I map each individual name a to an element of I.
I satisfies an assertion
a : C iff aI CI
(a,b) : R iff (aI,bI ) RI
I is a model for an Abox A if I satisfies all assertions in A.
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ABoxes
• Interpretations describe the state of the world in a complete way
• ABoxes describe the state of the world in an incomplete way
• An ABox has many models
• An ABox constraints the set of admissible models similar to a TBox
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Reasoning with ABoxes
Given an ABox A and a TBox T do they have a common model?
ABox consistency
Given an ABox A, a TBox T , an individual name a, and a concept C does aI CI hold in all models of A and T ?
Instance checking
A, T = a : C
The two tasks are interreducible:
• A consistent w.r.t T iff A, T |= a :
• A, T = a : C iff A {a : C} is not consistent
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Example
ABox
TBox
dumbo : Mammalt14 : Trunk(dumbo, t14) : bodypartg23 : Darkgrey(dumbo, g23) : color
Elephant = Mammal bodypart.Trunk color.GreyGrey = Lightgrey Darkgrey = Lightgrey Darkgrey
• ABox is inconsistent w.r.t. TBox.
• dumbo is an instance of Elephant.
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Reasoning and search
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cbac
bcac cabc
bacc acbc
abcc abcc
(3) (1)
(2) (2)
(3)(1)
baab (1)caac (2)cbbc (3)
Production rules
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State space is an ordered 4-tuple [N, A, S, GD], where:
N is a set of nodes corresponding to the states of the problem in the solution process
A is a set of arcs corresponding to the steps in the solution process
S is a non-empty subset of N containing the initial states of the problem
GD is a non-empty subset of N containing the goal states of the problem.
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cbac
bcac cabc
bacc acbc
abcc abcc
(3) (1)
(2) (2)
(3)(1)
Systemy produkcyjneN – state set
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cbac
bcac cabc
bacc acbc
abcc abcc
(3) (1)
(2) (2)
(3)(1)
N – state set
A – step setS – set of initial states
GD – set of goal states
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The states in GD are defined:
1. by properties of states occurring during search
2. by properties of the path created during search
Solution path is the path from a node in S to a node in GD.
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cbac
bcac cabc
bacc acbc
abcc abcc
(3) (1)
(2) (2)
(3)(1)
Two solution paths
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NIM
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NIM
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NIM
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NIM
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NIM
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NIM
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NIM
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NIM
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NIM
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Algorytm MIN-MAX
Players are denoted MIN and MAX
The value of the game is the score of MAX.
The score of MAX plus the score of MIN equals zero.
MAX attempts to maximize the value of the game.
MIN attempts to minimize the value of the game.
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NIM
MIN
MIN
MIN
MAX
MAX
MAX
+1
+1
-1+1
+1
+1
-1
+1 -1-1
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NIM
MIN
MIN
MIN
MAX
MAX
MAX
+1
+1
-1+1
+1
+1
-1
+1 -1-1
-1-1-1
-1
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Algorithm MINiMAX
If the father is MIN, assign it the minimum value of all its children.
If the father is MIN, assign it the maximum value of all its children.
Both players have the same information about the game and want to win.
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Algorithm alpha-beta
Assumptions:
1. The rules prohibit infinite path.
2. Only finite number of successors can be generated from any node.
3. The length of any game is finite.
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Algorithm alpha beta
MAX
MAX
MIN
MIN
alfa=-
beta=+ beta=+beta=+
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Algorithm alpha beta
MAX
MAX
MIN
MIN
alfa=-
beta=+ beta=+beta=+
9
Ł 9
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Algorithm alpha beta
MAX
MAX
MIN
MIN
alfa=-
beta=+ beta=+beta=9
9
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Algorithm alpha beta
MAX
MAX
MIN
MIN
alfa=-
beta=+ beta=+beta=7
9 7
Ł 7
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Algorithm alpha beta
MAX
MAX
MIN
MIN
alfa=-
beta=+ beta=+beta=7
9 78
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Algorithm alpha beta
MAX
MAX
MIN
MIN
alfa=-
beta=+ beta=+beta=7
9 78
ł7
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Alpha cuts
MAX
MAX
MIN
MIN
alfa=7
beta=+ Ł6beta=7
9 78
6
Any value found in this branch can
not increase beta.
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Alpha cuts
Search can complete below any MIN node with value less than or equal from value alpha of any of its predecessors (of type MAX).
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Beta cuts
MAX
MAX
MIN
MIN
alfa=7
beta=+beta=7
9 78
MAX
beta=8
8
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Beta cuts
MAX
MAX
MIN
MIN
alfa=7
beta=+beta=7
9 78
MAX
9
ł9
beta=8
8
Alfa cannot decrease
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Beta cuts
Search can complete below any MAX node with value greater than or equal from value beta of any of its predecessors (of type MIN).
