A. Darwiche Searching while Keeping a Trace The Evolution from SAT to Knowledge Compilation Adnan Darwiche Computer Science Department UCLA.

A. Darwiche

Searching while Keeping a Trace

The Evolution from SAT to Knowledge Compilation

Adnan DarwicheComputer Science Department

UCLA

A. Darwiche

Searching while Keeping a Trace The evolution from SAT to Knowledge

Compilation

Satisfiability (SAT) Knowledge compilation The connection: The trace of

search Implications Open questions

A. Darwiche

Is a set of Boolean constraints satisfiable?

Input to SAT is typically a CNF SAT is mostly solved by DPLL

search

Satisfiability (SAT)

A & okX => BA & okX => B

B & okY => CB & okY => C

A. Darwiche

SAT Solvers: Significant growth in last decade; many solvers publicly available (source code); millions of variables & constraints not uncommon.

Applications: Verification, planning, diagnosis, CAD, non-propositional reasoning (e.g., SMTs), …

Satisfiability (SAT)

A. Darwiche

A & okX => BA & okX => B

B & okY => CB & okY => C

CompiledStructureCompiler

Evaluator(Polytime)

Queries

Knowledge Compilation

A. Darwiche

A & okX => BA & okX => B

B & okY => CB & okY => C ?Compiler

Evaluator(Polytime)

Queries

Knowledge Compilation

A. Darwiche

A & okX => BA & okX => B

B & okY => CB & okY => C

.....Prime Implicates

OBDD…

Compiler

Evaluator(Polytime)

Queries

Knowledge Compilation

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Knowledge Compilation Map What’s the space of possible target

compilation languages? Can it be synthesized in a

semantically systematic way?

How do the languages compare? Succinctness (relative size) Operations they support in polytime

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Diagnosis Is this a normal behavior? What are the possible faults?

Planning Can this goal be achieved? Generate a set of plans

Probabilistic reasoning What is the probability of X given Y

Non-monotonic reasoning (penalty logics) Does X follow preferentially from Y

Formal verification / CAD: Is it possible that the design will exhibit behavior X? Are two designs equivalent?

Applications

A. Darwiche

Knowledge Compilation Map

For a given application: identify needed operations

Choose most succinct language that supports desired operations

Compile knowledge base into chosen language

A. Darwiche

Succinctness

Polytime OperationsConsistency (CO)Validity (VA)Clausal entailment (CE)Sentential entailment (SE)Implicant testing (IP)Equivalence testing (EQ)Model Counting (CT)Model enumeration (ME)

Projection (exist. quantification)ConditioningConjoin, Disjoin, Negate

DecomposabilityDeterminismSmoothness

FlatnessDecision

Ordering

Negation Normal Form

A B B A C D D C

and and and and and and and and

or or or or

and and

or

A Knowledge Compilation MAP

A. Darwiche

A B B A C D D C

and and and and and and and and

or or or or

and and

or

rooted DAG (Circuit)

Negation Normal Form

A. DarwicheA B B A C D D C

and and and and and and and and

or or or or

and and

orDecomposability

DeterminismSmoothness

FlatnessDecision

Ordering

Negation Normal Form

A. DarwicheA B B A C D D C

and and and and and and and and

or or or or

and and

or

A,B C,D

Decomposability

A. Darwiche

A B B A C D D C

and and and and and and and and

or or or or

and and

or

Determinism

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X X

and

or

and

X

Decision

A. Darwiche

X1

X2 X2

X3X3

1 0

or

and and

X1 X1or or

and and andand

X2 X2 X2 X2

and and andand

X3 X3 X3 X3

or or

true false

Binary Decision Diagrams(BDDs)

Decision + decomposability = FBDD

A. Darwiche

X1

X2 X2

X3X3

1 0

or

and and

X1 X1or or

and and andand

X2 X2 X2 X2

and and andand

X3 X3 X3 X3

or or

true false

Binary Decision Diagrams(BDDs)

Decision + decomposability + ordering = OBDD

A. Darwiche

NNF

d-NNF s-NNF f-NNF

sd-DNNF

DNNF

CO, CE, ME

d-DNNF

VA,IP,CT

EQ?

CNFDNF

IP PI

CO,CE,MEVA,IP,SE,EQVA,IP, SE,EQ

BDD

FBDD EQ?

OBDD

SE,EQ

MODSSE,EQ

NNF Subsets (2002)

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OBDD

FBDD

d-DNNF

DNNF

Space Efficiency (succinctness)

Tractable OperationsNNF

decomposability

determinism

decision

ordering

Diagnosis,Non-mon

Probabilisticreasoning

Tractability & Succinctness

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Inference by Compiling to d-DNNF

Deterministic Conformant Planning Blai Bonet and Hector Geffner (KR 2006)

Probabilistic Conformant Planning Jinbo Huang (AIPS 2006)

Model-based diagnosis Paul Elliott and Brian Williams (AAAI 2006) Anthony Barrett (IJCAI 2005)

Databases (query re-write) Yolife Arvelo, Blai Bonet and Maria Esther Vidal (AAAI 2006)

Inference in Bayesian Networks (2006 competition) Mark Chavira, Adnan Darwiche (IJCAI 2005)

Inference in Probabilistic Relational Models Mark Chavira, Adnan Darwiche and Manfred Jaeger (IJAR 2006)

c2d compiler: http://reasoning.cs.ucla.edu/c2d

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x y ¬x ¬z

¬w z ¬vv w z

Terminating condition for recursion:

empty set (satisfied), or empty clause (contradiction)

SAT?

x y ¬x ¬z

w z

v = false

SAT?x y

¬x ¬z¬w z

v = true

SAT?

SAT by DPLL Search

•Unit resolution•Conflict-directed backtracking•Clause learning•Branching heuristics•Restarts

A. Darwiche

Recent Trend: Exhaustive DPLL

Count number of models: Model counters, e.g., relsat, cachet

Generate all/subset of models: Image computation in model checking SMTs (non-propositional reasoning)

Variations on DPLL Search

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KnowledgeCompiler

ExhaustiveDPLL

Record Trace

Variations Languages

The Language of Search

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Trace of DPLL

X

Y

Z

unsat

unsat

sat

X Y

X Y Z

X Y Z

0

0 1

01

A. Darwiche

X

YY

Z Z

unsat

sat

unsat

unsat

satsat

0 1

0 1

01

0

01

1

X Y

X Y Z

X Y Z

Run to Exhaustion

Exhaustive DPLL

A. Darwiche

X

or

X

and

Y Y0

and

andand

or

Y Y

andand

or

1Z Z

X

YY

Z Z

unsat

sat

unsat

unsat

satsat

Trace of DPLL:a Formula

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X

or

X

and

Y Y0

and

andand

or

Y Y

andand

or

1Z Z

Equivalent to original CNF

Tractable (e.g., count models)

Trace of DPLL: a Formula

A. Darwiche

X

YY

Z Z

unsatunsat

unsat sat

satsat

Level One: Do not record redundant portions of trace

Level Two: Try not to solve equivalent subproblems

Dealing with Redundancy

A. Darwiche

X

YY

Z Z

unsatunsat

unsat

sat

Dealing with Redundancy

A. Darwiche

X

YY

Z Z

unsat sat

Do not createSimply point to existing node

Dealing with Redundancy

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X

YY

Z

0 1

This is an OBDD!

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X

YY

Z

0 1

X

or

X

and

0

and

andand

or

YY

andand

or

1Z

This is an OBDD!

NNF + decision, decomposability, ordering

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X Y

X Y Z

X Y Z

Compile

0

X

Y Y

Z

1

A Non-traditional OBDD Compiler

Exhaustive DPLL,Fixed variable order,Unique nodes

New complexity guarantees

A. Darwiche

0

X

Y Z

Z

1

Y

FBDD

Compile

Exhaustive DPLL,Dynamic variable order,Unique nodes

X Y

X Y Z

X Y Z

NNF + decision, decomposability

A. Darwiche

FBDD more succinct than OBDD (dynamic var ordering in sat)

Top-down vs bottom-up algorithms

OBDD: equivalence test (canonical) FBDD: probabilistic equivalence test Both allow model counting

FBDD vs OBDD

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Level One: Unique nodes (done)

Level Two: Avoid redundant compilation (searches)

Dealing with Redundancy

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Redundant Compilation

x5 x6

x4 x5 x6

x1 x3 x4 x5

x2 x3

x1 x2 x3

X1

X2X2

X3 X3

0 1

1

1

0

1

x5 x6

x4 x5 x6

x5 x6

x4 x5 x6

Formula Caching: complexity guarantees

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Formula Caching

Majercik and Litmman, 1998 Darwiche, 2002 Bacchus et al, 2003, 2004 Huang, 2004 Sang, Kautz, Beam, 2004, 2005 Thurley, 2006

A. Darwiche

Plain DPLL FBDD

Fixed Variable Ordering OBDD

Beyond BDDs…

A. Darwiche

Combine as AND node

d-DNNF

Decomposition (Component Analysis)

Solve disjoint subproblems independently

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A B C A B CA D E A D E

B CD E

D EB C

and

0

A

B

1

and

D B

C

D

E

Deterministic Decomposable Negation Norm Form

(d-DNNF)

A. Darwiche

or

and

A

and

Aand and

or

and

B

C

or

and

D

E

or or

B D

and and

Deterministic Decomposable Negation Norm Form

(d-DNNF)

A. Darwiche

d-DNNF more succinct than FBDD (effectiveness of decomposition)

Deterministic equivalence test open

Probabilistic equivalence test apply Other queries same…

FBDD vs d-DNNF

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Plain DPLL FBDD

Fixed Variable Ordering OBDD

Allowing Decomposition d-DNNF

The Language of Search

Other languages: deterministic DNF

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Relation to AND/OR Search (CP)

AND/OR graphs are deterministic and decomposable

AND/OR search algorithms are doing enough work to compile networks into (multi-valued equivalent of) d-DNNF

Capable of more than answering a single query (model counting, belief revision, etc)

A. Darwiche

Implications SAT techniques harnessed for

knowledge compilation c2d compiler based on

Rsat Solver (SAT-race 06)

Language properties (succinctness/tractability) help characterize power and limitations of search

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Understanding DPLL

Take any program X that runs exhaustive DPLL-style search

Examine traces, if traces L, then

X can answer all queries tractable for L

X is hopeless on any input having no polynomial-size representation in L

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Power of DPLL

Traces of several model counters (Relsat, Cachet, e.g.) are in d-DNNF

Are doing enough work to compile formulas into d-DNNF solve tasks beyond model counting

(e.g., minimum cardinality, probabilistic equivalent testing)

A. Darwiche

or

and

X

and

X

Decision nodes(d-DNNF’)

Deterministic nodes(d-DNNF)

or

A B C

and and

andand

or

A B A B

Limitation of DPLL:General determinism

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Beyond DPLL:Decomposability (D) without determinism (d)

or

or

and

and

X1 X2

X3

DNNF:CO, CE, ME, exist quant

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Summary

Overview of recent results in knowledge compilation

Overview of recent trends in exhaustive DPLL search

A connection between SAT and knowledge compilation (search trace): SAT techniques harnessed for compilation

into various languages Language properties (succinctness,

tractability) characterize power and limitations of search algorithms

A. Darwichehttp://reasoning.cs.ucla.edu

A. Darwiche

•Knowledge Compilation Map, with Pierre Marquis•DPLL with a Trace, •Language of Search, with Jinbo Huang

A. Darwiche

Multi-Linear Functions Arithmetic Circuits

ababaababaababaababaf ||||

A B

**

* *

+

+ +

* * * *

a ab ba aab| ab | ab| ab |

Factoring

A. Darwiche

a c + a b c + cMulti-linear function:Propositional theory:

c ^ (a b) Encode

c

b 1

a 1Arithmetic Circuit

Decode

c

b b

a aSmooth d-DNNF

Compile

MLFsACsCNFsd-DNNF