compact propositional encoding of first order theories

Compact Encoding of FOLEyal Amir, Cambridge Present., May 2006

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Compact Propositional Compact Propositional Encoding ofEncoding of

First Order TheoriesFirst Order Theories

Eyal AmirEyal AmirUniversity of Illinois at Urbana-ChampaignUniversity of Illinois at Urbana-Champaign

(Joint with (Joint with Deepak RamachandranDeepak Ramachandran and and Igor GammerIgor Gammer))


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Example: Pigeonhole PrincipleExample: Pigeonhole Principle

p1

p2

p3

p4

H1

H2

H3

H4


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Pigeonhole: Classical EncodingPigeonhole: Classical Encoding

HH11(p(pii) ) H H22(p(pii) ) H H33(p(pii) ) H H44(p(pii) )

[H[H11(p(pii) ) H H22(p(pii)])]

p,q (pp,q (pq q H Hii(p) (p) H Hii(q))(q))

O(n2) props

Every i:Every i:


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A ReformulationA ReformulationHHii

++(p) : pigeon p is not in a hole in the subtree rooted at i.(p) : pigeon p is not in a hole in the subtree rooted at i.

H1+

H3+H2

+

p1

p2

p3

p411 22 33 44


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A ReformulationA Reformulation

H1+p[H1

+(p) (H2+(p) H3

+(p))

p[H2+(p) (-H1(p) -H2(p)) p[H1

+(p) (-H3(p)-H4(p))

p1

p2

p3

p4

HHii++(p) : pigeon p is not in a hole in the subtree rooted at i.(p) : pigeon p is not in a hole in the subtree rooted at i.

11 22 33 44


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p[H1(p) H2(p)] p[H3(p)H4(p)]

p[H1+(p) H2

+(p)]

p1

p2

p3

p4

Hi+(p) : pigeon p is not in a hole in the subtree rooted at i.Hi+(p) : pigeon p is not in a hole in the subtree rooted at i.

11 22 33 44


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H1+

H3+H2

+

p,q[Hi(p)Hi(q) (p=q)]

p1

p2

p3

p4



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H3+H2

+

p[H1+(p)]

p1

p2

p3

p4


O(n·logn) props


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Propositional Encodings: Propositional Encodings: MotivationMotivation

• How do we do this in general? – scientific How do we do this in general? – scientific curiositycuriosity

• Applications:Applications:– Formal verification, BMC – specification in FOLFormal verification, BMC – specification in FOL– AI: planning, diagnosis, spatial reasoning, AI: planning, diagnosis, spatial reasoning,

question answering from knowledge – specified question answering from knowledge – specified in FOLin FOL

– Useful with many objects, predicates, time Useful with many objects, predicates, time steps, etc.steps, etc.

• Can use efficient SAT solversCan use efficient SAT solvers


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OutlineOutline1.1. Algorithm for compact prop. encodingAlgorithm for compact prop. encoding

2.2. AnalysisAnalysis1.1. Interpolants – correctness of algorithmInterpolants – correctness of algorithm

2.2. A different Treewidth – number of props.A different Treewidth – number of props.

3.3. Towards automatic partitioningTowards automatic partitioning

3.3. Related work & open questionsRelated work & open questions


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Naïve Propositional EncodingNaïve Propositional Encoding• Tasks converted into propositional logicTasks converted into propositional logic

– P(a) P(a) P Pa a x P(x) x P(x) P Pa a PPb b PPc c

((Naive Naive Propositionalization)Propositionalization)

• Problem: Very many propositions:Problem: Very many propositions:

|P||C||P||C|kk for |P| predicates of arity k, for |P| predicates of arity k, and |C| constant symbolsand |C| constant symbols

• Note: Assumes Domain Closure (DCA)Note: Assumes Domain Closure (DCA)– constant symbols name all domain objectsconstant symbols name all domain objects


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Our ResultsOur Results

• New method for propositioal encodingNew method for propositioal encoding

• Size depends on optimal tree Size depends on optimal tree decomposition (e.g., decomposition (e.g., O(|P|+|C|) sometimes; O(|P|+|C|) sometimes; O(n logn) instead of O(nO(n logn) instead of O(n22) for Pigeonhole ) for Pigeonhole PrinciplePrinciple))

• Applications: Applications: – Machine scheduling problemMachine scheduling problem– Board-Level Routing Problem for FPGA’sBoard-Level Routing Problem for FPGA’s


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Our New AlgorithmOur New AlgorithmRough algorithm (given theory T):Rough algorithm (given theory T):

1.1. Partition theory T into TPartition theory T into T11…T…Tr r (*1)(*1)

2.2. Encode every TEncode every Tii in propositional logic in propositional logic

TTii’ separately (create proposition P’ separately (create proposition Paa for for

constant a and predicate P in L(Tconstant a and predicate P in L(Tii)) ))

(*2)(*2)

3.3. Return UReturn Uii≤≤rrTTii’’


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Partitioned TheoryPartitioned Theory

…

T

))()()(.( xPxRxQx

))()()(.( xTxRxQx

))()()(.( xPxWxVx

))()()(.( xWxYxZx


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Partitioned TheoryPartitioned Theory

… …

T1 T2

))()(

)(.(

xPxR

xQx

))()()(.( xTxRxQx P

))()(

)(.(

xPxW

xVx

))()()(.( xWxYxZx


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Propositional EncodingPropositional Encoding

… …

T1 T2

))()(

)(.(

xPxR

xQx

))()()(.( xTxRxQx P

))()(

)(.(

xPxW

xVx

))()()(.( xWxYxZx


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… …

T1 T2

))()(

)(.(

xPxR

xQx

))()()(.( xTxRxQx P

))()(

)(.(

xPxW

xVx

))()()(.( xWxYxZx


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… …

T1 T2

))()(

)(.(

xPxR

xQx

TRQE ,,

P))()(

)(.(

xPxW

xVx

))()()(.( xWxYxZx


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… …

T1 T2

PRQE ,,

TRQE ,,

P))()(

)(.(

xPxW

xVx

))()()(.( xWxYxZx


20


… …

T1 T2

PRQE ,,

TRQE ,,

P

PWVE ,,

WYZE ,,


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… …

T1 T2

PRQE ,,

TRQE ,, P

PWVE ,,

WYZE ,,

...1

,,, ...

PC

RQTRQ

EP

EE


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Partitioned Propositional Partitioned Propositional EncodingEncoding

Algorithm:Algorithm:

1.1. Partition theory T into TPartition theory T into T11…T…Trr

2.2. Propositionalize each TPropositionalize each Tii separately separately

3.3. Return the propositional encoding UReturn the propositional encoding Uii≤≤rrTTii

How to make this sound+complete?How to make this sound+complete?

# props. in partitioned encoding 1# props. in partitioned encoding 1# props. in naïve encoding # props. in naïve encoding #partitions#partitions

an exponential speed-up for SAT solver

|P|Pii||C||Cii| props.| props.


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Experiments: Board-Level Experiments: Board-Level RoutingRouting

#prop. vars per problem SAT Running time (sec.) per problem


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Experiments: Machine Experiments: Machine SchedulingScheduling

#prop. vars per problem SAT Running time (sec.) per problem


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Key to Exact EncodingKey to Exact Encoding

• Craig’s interpolation theoremCraig’s interpolation theorem (First-Order Logic): (First-Order Logic):– If T1 T2, then there is a formula C including only If T1 T2, then there is a formula C including only

symbols from L(T1) symbols from L(T1) L(T2) such that T1 C and L(T2) such that T1 C and C T2C T2

P(x)

T1 T2


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• Craig’s interpolation theoremCraig’s interpolation theorem (First-Order Logic): (First-Order Logic):– If T1 If T1 T2, then there is a formula C including only T2, then there is a formula C including only

symbols from L(T1) symbols from L(T1) L(T2) such that T1 C and L(T2) such that T1 C and C C T2T2

P(x)

T1 T2T1T1T2 has no model, iff T2 has no model, iff

T1 T1 T2T2

The only interpolants (the C’s) The only interpolants (the C’s) can be can be xP(x),xP(x),xP(x),… xP(x),…

(+6 others)(+6 others)


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• Key idea: If our prop. encoding of T1 [T2] Key idea: If our prop. encoding of T1 [T2] implies all possible interpolants in implies all possible interpolants in L(T1)L(T1)L(T2), then the combined encoding L(T2), then the combined encoding is SAT iff T (in FOL) is SAT.is SAT iff T (in FOL) is SAT.

P(x)

T1 T2T1T1T2 has no model, iff T2 has no model, iff

T1 T1 T2T2

The only interpolants (the C’s) The only interpolants (the C’s) can be can be xP(x),xP(x),xP(x),… xP(x),…



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… …

T1 T2

PRQE ,,

TRQE ,, P

PWVE ,,

WYZE ,,

...1

,,, ...

PC

RQTRQ

EP

EE

The only interpolants (the C’s)The only interpolants (the C’s) can be can be xP(x),xP(x),xP(x),…xP(x),…



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Correctness of Encoding Alg.Correctness of Encoding Alg.• Encoding of each partition needs to Encoding of each partition needs to

include include encodings of all possible encodings of all possible interpolants (interpolants (detailed nextdetailed next))

• When there are more than two When there are more than two partitions, the partitions are arranged in partitions, the partitions are arranged in a a tree decomposition (in next section)tree decomposition (in next section)


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Propositional Encoding SizePropositional Encoding Size

… …

T1 T2

PRQE ,,

TRQE ,, P

PWVE ,,

WYZE ,,

...1

,,, ...

PC

RQTRQ

EP

EE

The only interpolants (the C’s)The only interpolants (the C’s) can be can be xP(x),xP(x),xP(x),…xP(x),…



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Propositional encoding of Propositional encoding of Monadic FOLMonadic FOL

)()(())()(())()(())()(())()((

yyQxPxyQxPxyyQxxPyyQxPxyyQxPyx

FactorFactor = single-variable existentially = single-variable existentiallyquantified formulaquantified formula

TheoremTheorem : : Can convert any monadic FOL Can convert any monadic FOL formula to a conjunction of disjunctions of formula to a conjunction of disjunctions of factorsfactors

Example: Example:

QP EE~


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Propositionalizing a Partition in Propositionalizing a Partition in Monadic FOLMonadic FOL

1.1. Convert TConvert Tii to to factorized formfactorized form

2.2. Replace Replace factorsfactors in in ΤΤii with prop. symbols with prop. symbols

3.3. Add meaning of new propositionsAdd meaning of new propositions

...},{...},{

,,

,

QPQPQPQP

QPccPa

EEEEEEEQPEP

QPExQxPxaP

,))()(()(

εε(T(Tii) =) =

aP

This approach creates |PThis approach creates |P ii||C||Cii|+3|+3|Pi||Pi| propositional symbols. propositional symbols.


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Automatic Partitioning and Automatic Partitioning and Propositional EncodingPropositional Encoding

• Optimization criteriaOptimization criteria– Tree of partitions of axioms that satisfies Tree of partitions of axioms that satisfies

the running intersection propertythe running intersection property

– Minimize |PMinimize |Pii||C||Cii|+3|+3|Pi||Pi| (contrast with (contrast with

treewidth, where we minimize |Pi|)treewidth, where we minimize |Pi|)

• Can use current algorithms for treewidth Can use current algorithms for treewidth and decomposition as a starting pointand decomposition as a starting point

• Sometime need reformulation of axiomsSometime need reformulation of axioms


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Automatic PartitioningAutomatic Partitioning• Find a Find a tree decompositiontree decomposition G of G of

minimum width:minimum width:– Every node in G is a set of symbols from TEvery node in G is a set of symbols from T– G satisfies the G satisfies the running intersection running intersection

propertyproperty: if P appears in both v,u in G, then : if P appears in both v,u in G, then P appears in all the nodes on the path P appears in all the nodes on the path connecting themconnecting them

– Choose G with minimum Choose G with minimum widthwidth (width(G) is (width(G) is the size of its largest node v)the size of its largest node v)


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Related WorkRelated Work• [McMillan 2005] – some ground (QF) [McMillan 2005] – some ground (QF)

FOL theories have ground interpolantsFOL theories have ground interpolants– May use his results for similar results on May use his results for similar results on

compact prop. for those theoriescompact prop. for those theories

• McMillan’s talk yesterday – may use our McMillan’s talk yesterday – may use our compact prop. method to generate all compact prop. method to generate all interpolantsinterpolants


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Related WorkRelated Work• [Pnueli, Rodeh, Strichman, Siegel ’99, [Pnueli, Rodeh, Strichman, Siegel ’99,

’02] – equational theories (no ’02] – equational theories (no predicates), and weak(er?) boundpredicates), and weak(er?) bound

where R = resulting universe size, and where R = resulting universe size, and others are problem dependent (worst others are problem dependent (worst case n!, for n variables/constants)case n!, for n variables/constants)


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ConclusionConclusion• Created efficient propositional encodings:Created efficient propositional encodings:

– Monadic FOL – Sound + CompleteMonadic FOL – Sound + Complete– Ramsey class – Sound, Ramsey class – Sound, IncompleteIncomplete without DCA without DCA– Applies to general FOL (with DCA)Applies to general FOL (with DCA)

• Speedup in SAT solvingSpeedup in SAT solving• Exploits the structure of real world problemsExploits the structure of real world problems• Future workFuture work: Classes beyond Monadic FOL: Classes beyond Monadic FOL

– Right now: ground FOL with equalityRight now: ground FOL with equality– 2-var fragment2-var fragment

• Some software on Some software on http://reason.cs.uiuc.edu/eyalhttp://reason.cs.uiuc.edu/eyal


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THE ENDTHE END


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Problems with Naïve Prop.Problems with Naïve Prop.• Problem 1: Very large number of Problem 1: Very large number of

propositions:propositions:

|P||C||P||C|kk for |P| predicates of arity k, for |P| predicates of arity k, and |C| constant symbolsand |C| constant symbols

• Problem 2: Assumes Domain Closure Problem 2: Assumes Domain Closure Assumption (DCA) – only elements are Assumption (DCA) – only elements are those with constant symbolsthose with constant symbols


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General FOL (with DCA)General FOL (with DCA)

Use DCA repeatedly until we get a Use DCA repeatedly until we get a formula in Monadic FOLformula in Monadic FOL

– Replace Replace xx11QxQx22…Qx…Qxnn(x(x11,…, x,…, xnn) with) with

ccCC QxQx22…Qx…Qxnn(x(x11,…, x,…, xnn))

– Replace Replace xx11QxQx22…Qx…Qxnn(x(x11,…, x,…, xnn) with ) with

ccCC QxQx22…Qx…Qxnn(x(x11,…, x,…, xnn))


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Automatic PartitioningAutomatic Partitioning

• Begin with logical Knowledge Base in FOLBegin with logical Knowledge Base in FOL

• Construct symbol graphConstruct symbol graph– vertex = symbolvertex = symbol– edge = axiom uses both symbolsedge = axiom uses both symbols

• Find a Find a tree decompositiontree decomposition

• Partition axioms using clusters in tree Partition axioms using clusters in tree decompositiondecomposition


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key locked can_opencan_open try_open openopen fetch broomkey opentry_open open broom fetch

broom dry can_cleancan_clean cleanbroom let_dry drytime let_drydrier let_drytime drier



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key locked

can_open try_open

open fetch

broom

dry

clean

let_dry

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drier

can_clean


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key locked

can_open try_open

open fetch

broom

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let_dry

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drier

can_clean


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key locked

can_open try_open

open fetch

broom

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clean

let_dry

time

drier

can_clean

broom


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broom

key locked

can_open try_open

open fetch

broom

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let_dry

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can_clean

broom


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broom

compact propositional encoding of first order theories

Documents

fpgasapproximating vertexcuts

pigeon p

planar graphspigeonhole

planar graphsexample

planar graphsoutlinealgorithm

qhiphiq p

predicate p

p predicates of arity