MBD and CSP

MBD and CSP

Meir Kalech

Partially based on slides of Jia You and Brian Williams

Outline Last lecture:

1. Autonomous systems

2. Model-based programming

3. Livingstone

Today’s lecture:

1. Constraints satisfaction problems (CSP)

2. Optimal CSP

3. Conflict-directed A*

4. Expand best child

Intro Example: 8-QueensIntro Example: 8-Queens

Generate-and-test, 88 combinations

Intro Example: 8-QueensIntro Example: 8-Queens

Constraint Satisfaction Constraint Satisfaction ProblemProblem

Set of variables {X1, X2, …, Xn} Each variable Xi has a domain Di of possible

values Usually Di is discrete and finite

Set of constraints {C1, C2, …, Cp} Each constraint Ck involves a subset of variables

and specifies the allowable combinations of values of these variables

Goal: Assign a value to every variable such that all constraints are satisfied

Example: 8-Queens ProblemExample: 8-Queens Problem 8 variables Xi, i = 1 to 8 Domain for each variable {1,2,…,8} Constraints are of the forms:

Xi = k Xj k for all j = 1 to 8, ji

Example: Map ColoringExample: Map Coloring

• 7 variables {WA,NT,SA,Q,NSW,V,T}

• Each variable has the same domain {red, green, blue}• No two adjacent variables have the same value: WANT, WASA, NTSA, NTQ, SAQ, SANSW, SAV,QNSW, NSWV

WA

NT

SA

Q

NSWV

T

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NT

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Q

NSWV

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Example: Task SchedulingExample: Task Scheduling

T1 must be done during T3T2 must be achieved before T1 startsT2 must overlap with T3T4 must start after T1 is complete

T1

T2

T3

T4

Constraint GraphConstraint GraphBinary constraints

T

WA

NT

SA

Q

NSW

V

Two variables are adjacent or neighbors if theyare connected by an edge or an arc

T1

T2

T3

T4

CSP as a Search ProblemCSP as a Search Problem Initial state: empty assignment Successor function: a value is assigned to

any unassigned variable, which does not conflict with the currently assigned variables

Goal test: the assignment is complete Path cost: irrelevant

Backtracking example

Backtracking AlgorithmBacktracking Algorithm

CSP-BACKTRACKING(PartialAssignment a) If a is complete then return a X select an unassigned variable D select an ordering for the domain of X For each value v in D do

If v is consistent with a then Add (X= v) to a result CSP-BACKTRACKING(a) If result failure then return result Remove (X= v) from a

Return failure

Start with CSP-BACKTRACKING({})

Improving backtracking efficiency

Which variable should be assigned next?

In what order should its values be tried?

Can we detect obvious failure early?




3. Livingstone

Today’s lecture:


2. Optimal CSP



What is an Optimal CSP (OCSP)? Set of decision variables y, A utility function g on y, A set of constraints C that y must satisfy.

The solution is an assignment to y, maximizing g and satisfying C.

What is an OCSP formally? OCSP is: Where CSP =

x – set of variables Dx – Domain over variables

Cx – set of constraints over variables

- set of decision variables Cost function –

We call the elements of Dy decision states. A solution y* to an OCSP is a minimum cost decision

state that is consistent with the CSP.

Methods for solving OCSP

Traditional method: A* Good at finding optimal solutions. Visits every state that whose estimated cost is less

than the true optimum*. Computationally unacceptable for model-based executives.* A* heuristic is admissible, that is, it never overestimates cost.

Solution: CONFLICT-DIRECTED A* Also searches in best-first order Eliminates subspaces around inconsistent states




3. Livingstone

Today’s lecture:


2. Optimal CSP



CONFLICT-DIRECTED A*Williams&Rango 03

Step 1: Best candidate S generated

Step 2: S is tested for consistency

Step 3: If S is inconsistent, the inconsistency

is generalized to conflicts, removing

a subspace of the solution space

Step 4: The program jumps to the next-best

candidate, resolving all conflicts.

Enumerating Probable Candidates

GenerateCandidate

TestCandidate

Consistent?Keep

ComputePosterior p

BelowThreshold?

ExtractConflict

DoneYes No

Yes No

Leading CandidateBased on Priors

IncreasingCost

consistent

inconsistent

A* - must visit all states with cost lower than optimum

IncreasingCost

Conflict-directed A* - found inconsistent state eliminates all states entail the same symptom

consistent

inconsistent

IncreasingCost Conflict 1

Conflict-directed A*

consistent

inconsistent

IncreasingCost

Conflict 2

Conflict 1


consistent

inconsistent

IncreasingCost

Conflict 3

Conflict 2

Conflict 1


consistent

inconsistent

IncreasingCost

Conflict 3

Conflict 2

Conflict 1


• Feasible regions described by the implicates of known conflicts (Kernel Assignments)

•Want kernel assignment containing the best cost candidate

consistent

inconsistent

Conflict-directed A*Function Conflict-directed-A*(OCSP) returns the leading minimal cost solutions. Conflicts[OCSP] {} OCSP Initialize-Best-Kernels(OCSP) Solutions[OCSP] {} loop do decision-state Next-Best-State-Resolving-Conflicts(OCSP) if no decision-state returned or Terminate?(OCSP) then return Solutions[OCSP] if Consistent?(CSP[OCSP ], decision-state) then add decision-state to Solutions[OCSP] else

new-conflicts Extract-Conflicts(CSP[OCSP], decision-state) Conflicts[OCSP] Eliminate-Redundant-Conflicts(Conflicts[OCSP] new-conflicts)end

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Assume Independent failures: PG(mi) >> PU(mi) //G-good, U-unknown

Psingle >> Pdouble

PU(M2) > PU(M1) > PU(M3) > PU(A1) > PU(A2)

Example: Diagnosis as an OCSP

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OBS = observation.

COMPONENTS = variable set y.

System Description = constraints - Cy.

M1=G X=A+B

Candidate = a mode assignments to y.

Diagnosis = a candidate that is consistent with Cy and OBS.

Utility = candidate probability P(y).

Cost = 1/P(y).

Example: Diagnosis

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Conflicts / Constituent Diagnoses none

Best Kernel: {}

Best Candidate: ?

First Iteration

{ }

M1=? M2=? M3=? A1=? A2=?

M1=G M2=G M3=G A1=G A2=G

Select most likely value for unassigned modes

Test: M1=G M2=G M3=G A1=G A2=G

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Extract Conflict and Constituent Diagnoses:


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Extract Conflict and Constituent Diagnoses: ¬ [M1=G M2=G A1=G]


Extract Conflict and Constituent Diagnoses: ¬ [M1=G M2=G A1=G]

M1=U M2=U A1=U

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This process has been done through propositional satisfiability algorithm

DPLL algorithm*

Conflicts / Constituent Diagnoses M1=U M2=U A1=U

Best Kernel: M2=U

Best Candidate: M1=G M2=U M3=G A1=G A2=G

Second Iteration

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Later we will describe how to determine the best Kernel

Test: M1=G M2=U M3=G A1=G A2=G

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A1

Extract Conflict:

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Extract Conflict:Not [G(M1) & G(M3) & G(A1) & G(A2)]

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Extract Conflict:¬ [M1=G M3=G A1=G A2=G]

M1=U M3=U A1=U A2=U

Conflicts / Constituent Diagnoses M1=U M2=U A1=U

M1=U M3=U A1=U A2=U

Best Kernel: M1=U

Best Candidate: M1=U M2=G M3=G A1=G A2=G

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Test: M1=U M2=G M3=G A1=G A2=G

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Consistent!

A2=U M1=U

M3=UA1=U

A1=U, A2=U, M1=U, M3=UA1=U M1=U M2=U

A1=U M1=U M1=U A2=U M2=U M3=U

Conflict-Directed A*: Generating Best Kernels

A1=U, M1=U , M2=U

Constituent Diagnoses

• Minimal set covering is an instance of breadth first search.• To find the best kernel, expand tree in best first order.

Insight:• Kernels found by minimal set covering

Superset of M1

A1=U, A2=U, M1=U, M3=UA1=U M1=U M2=U

A2=U M1=U

M3=UA1=U

M2=U A2=U M2=U M3=UM1=U

Conflict-Directed A*: Generating Best Kernels

A1=U, M1=U , M2=U

Constituent Diagnoses

A1=U

• Minimal set covering is an instance of breadth first search.• To find the best kernel, expand tree in best first order.

Insight:• Kernels found by minimal set covering

• Continue expansion to find best candidate.

PU(M2) > PU(M1)




3. Livingstone

Today’s lecture:


2. Optimal CSP



OCSP Admissible Estimate:

M1=? M3=? A1=? A2=?

x PM1=G x PM3=G x PA1=G x PA2=G

Select most likely value for unassigned modes

F = G + H (H is admissible)

PM2=u

M2=U

MPI: mutual, preferential independenceTo find the best candidate we assign each variable its best utility value, independent of the values assigned to the other variables.

M2=U A1=UM1=U

M2=U M1=U A1=U

{ }

For any Node N: The child of N containing the best cost candidate is

the child with the best estimated cost f = g+h (by MPI). Only need to expand the best child.

Conflict-directed A*: Expand Best Child & Sibling

Constituent kernels

M2=U A1=UM1=U

Order constituents by decreasing likelihood

{ }

> >



the child with the best estimated cost f = g+h. Only need to expand the best child.

M2=U M1=U A1=U

Constituent kernels

M2=U

Order constituents by decreasing likelihood

{ }



the child with the best estimated cost f = g+h. Only need to expand the best child.

M2=U M1=U A1=U

Constituent kernels

M1=U M3=U A1=U A2=U

M1=U

M2=U M1=U


M2=U M1=U A1=U

Constituent kernels

When a best child loses any candidate, expand child’s next best sibling: If unresolved conflicts:

expand as soon as next conflict of child is resolved. If conflicts resolved:

expand as soon as child is expanded to a full candidate.

{ }

M2=G

M3=G

A1=G

A2=G

M2=U

M3=U

A1=U

A2=U

Appendix: A* Search: PreliminariesProblem: State Space Search Problem Initial State Expand(node) Children of Search Node = next states Goal-Test(node) True if search node at a goal-state

h Admissible Heuristic -Optimistic cost to go

Search Node: Node in the search tree State State the search is at Parent Parent in search tree

A* Search: PreliminariesProblem: State Space Search Problem Initial State Expand(node) Children of Search Node = adjacent states Goal-Test(node) True if search node at a goal-state Nodes Search Nodes to be expanded Expanded Search Nodes already expanded Initialize Search starts at , with no expanded nodes

h Admissible Heuristic -Optimistic cost to go

Search Node: Node in the search tree State State the search is at Parent Parent in search tree

Nodes[Problem]: Remove-Best(f) Removes best cost node according to f Enqueue(new-node, f ) Adds search node to those to be expanded

A* SearchFunction A*(problem, h) returns the best solution or failure. Problem pre-initialized. f(x) g[problem](x) + h(x) loop do if Nodes[problem] is empty then return failure node Remove-Best(Nodes[problem], f) state State(node) remove any n from Nodes[problem] such that State(n) = state Expanded[problem] Expanded[problem] {state} new-nodes Expand(node, problem) for each new-node in new-nodes unless State(new-node) is in Expanded[problem] then Nodes[problem] Enqueue(Nodes[problem], new-node, f ) if Goal-Test[problem] applied to State(node) succeeds then return nodeend

Expandbest first


Terminateswhen . . .


DynamicProgrammingPrinciple . . .

Next Best State Resolving ConflictsFunction Next-Best-State-Resolving-Conflicts (OCSP) returns the best cost state consistent with Conflicts[OCSP]. f(x) G[problem] (g[problem](x), h(x)) loop do if Nodes[OCSP] is empty then return failure node Remove-Best(Nodes[OCSP], f) state State[node] add state to Visited[OCSP] new-nodes Expand-State-Resolving-Conflicts(node, OCSP) for each new-node in new-nodes unless Exists n in Nodes[OCSP] such that State[new-node] = state OR State[new-node] is in Visited[problem] then Nodes[OCSP] Enqueue(Nodes[OCSP], new-node, f ) if Goal-Test-State-Resolving-Conflicts[OCSP] applied to state succeeds then return nodeend

An instanceof A*

MBD and CSP

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