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

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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

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T2

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Constraint GraphConstraint GraphBinary constraints

T

WA

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

T1

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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 example

Backtracking example

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?

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

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

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

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 1

Conflict-directed A*

consistent

inconsistent

IncreasingCost

Conflict 2

Conflict 1

Conflict-directed A*

consistent

inconsistent

IncreasingCost

Conflict 2

Conflict 1

Conflict-directed A*

consistent

inconsistent

IncreasingCost

Conflict 3

Conflict 2

Conflict 1

Conflict-directed A*

consistent

inconsistent

IncreasingCost

Conflict 3

Conflict 2

Conflict 1

Conflict-directed A*

• 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

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

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

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

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Extract Conflict and Constituent Diagnoses:

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

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Extract Conflict and Constituent Diagnoses:

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

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

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

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

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

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

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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

Third Iteration3

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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)

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

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

{ }

> >

Conflict-directed A*: Expand Best Child & Sibling

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

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

{ }

Conflict-directed A*: Expand Best Child & Sibling

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

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

Conflict-directed A*: Expand Best Child & Sibling

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

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

Terminateswhen . . .

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

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*