The Relative Power The Relative Power of Semantics and of Semantics and

UnificationUnificationDavid A. PlaistedDavid A. PlaistedUNC Chapel HillUNC Chapel Hill

GoalGoal

Attempt to understand in a Attempt to understand in a formal way the influence of formal way the influence of semantics on OSHLsemantics on OSHL

Relate the benefit to be Relate the benefit to be obtained from semantics to obtained from semantics to that of unificationthat of unification

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Current theorem proversCurrent theorem proversLargely syntacticLargely syntacticResolution or ME (tableau) basedResolution or ME (tableau) basedFirst-order provers are often poor First-order provers are often poor

on non-Horn clauseson non-Horn clausesRarely can solve hard problemsRarely can solve hard problemsHuman interaction needed for Human interaction needed for

hard problemshard problems

How do humans prove How do humans prove theorems?theorems?

SemanticsSemanticsCase analysisCase analysisSequential search through space Sequential search through space

of possible structuresof possible structuresFocus on the theoremFocus on the theorem

““Systematic methods can Systematic methods can now routinely solve now routinely solve verification problems with verification problems with thousands or tens of thousands or tens of thousands of variables, thousands of variables, while local search methods while local search methods can solve hard random can solve hard random 3SAT problems with 3SAT problems with millions of variables.”millions of variables.”(from a conference (from a conference announcement)announcement)

DPLL ExampleDPLL Example

{p,r},{p,q,r},{p,r}

{T,r},{T,q,r},{T,r}

{F,r},{F,q,r},{F,r}

p=T p=F

{q,r} {r},{r}

{}

SIMPLIFY

SIMPLIFY

SIMPLIFY

Hyper LinkingHyper Linking

Problem Input Clauses

OTTER (sec)

Hyper Linking

Ph5 45 38606.76 1.8 Ph9 297 >24 hrs 2266.6 Latinsq 16 >24 hrs 56.4 Salt 44 1523.82 28.0 Zebra 128 >24 hrs 866.2

Eliminating Duplication with the Eliminating Duplication with the Hyper-Linking Strategy, Shie-Jue Hyper-Linking Strategy, Shie-Jue Lee and David A. Plaisted, Lee and David A. Plaisted, Journal of Automated Reasoning Journal of Automated Reasoning 9 (1992) 25-42.9 (1992) 25-42.

DefinitionDefinition DetectionDetection

Problem OSHL Time

Otter Time

Otter Clauses

P1 0.3 0.03 51 P2 2.3 1000+ 41867 P3 11.25 1000+ 27656 P4 1.35 1000+ 105244 P5 2.0 1000+ 54660

Replacement Rules with Definition Replacement Rules with Definition

Detection, David A. Plaisted and Detection, David A. Plaisted and Yunshan Zhu, in Caferra and Yunshan Zhu, in Caferra and Salzer, eds., Automated Salzer, eds., Automated Deduction in Classical and Non-Deduction in Classical and Non-Classical Logics, LNAI 1761 Classical Logics, LNAI 1761 (1998) 80-94.(1998) 80-94.

More DefinitionsMore DefinitionsSS1 1 S S2 2 … … S Snn=S=Sn n S Sn-1 n-1 … …

SS11

Left AssociativeLeft Associativenn OSHLOSHL OtterOtter VampireVampire E-E-SethSetheoeo

DCTDCTPP

timtimee

GeGenn

KepKeptt

timtimee

GeGenn

KepKeptt

timtimee

GenGen KepKeptt

timtimee

timtimee

22 0.10.17575

4141 3636 600600++

100310030303

2471247122

0.00.000

103103 9090 0.00.0 0.010.01

33 0.60.67878

8585 8080 600600++

6675667533

3149314966

70.70.11

36067360674242

5035038282

0.30.3 300300++

44 2.12.10707

141141 136136 600600++

4721472199

2211221199

300300++

2589825898955955

6836838585

0.30.3 300300++

55 5.35.31717

207207 202202 600600++

4605460544

2094209411

300300++

2529825298293293

6786786464

2.62.6 300300++

66 12.12.0202

283283 278278 600600++

6024602477

2292229233

300300++

2561225612105105

6846845757

300300++

300300++

77 38.38.9797

77 33 600600++

5629562999

1966196600

300300++

2564125641650650

6796797777

300300++

300300++

88 77.77.9494

77 33 600600++

5635563522

1893189322

300300++

2586325863117117

6856854242

300300++

300300++

More DefinitionsMore DefinitionsSimilar results for other definitions:Similar results for other definitions:SS1 1 S S2 2 … … S Snn=S=Sn n S Sn-1 n-1 … … S S11, left side left , left side left

associated, right side right associatedassociated, right side right associatedSS1 1 S S2 2 … … S Snn== SS1 1 S S2 2 … … S Sn n SS1 1 S S2 2 … …

SSnn, both sides associated to the left, both sides associated to the leftSS1 1 S S2 2 … … S Snn== SS1 1 S S2 2 … … S Sn n SS1 1 S S2 2 … …

SSnn, left side left associated, right side , left side left associated, right side right associatedright associated

Similar results for ∩Similar results for ∩

Later propositional Later propositional strategiesstrategies

Billon’s disconnection calculus, Billon’s disconnection calculus, derived from hyper-linkingderived from hyper-linking

Disconnection calculus theorem Disconnection calculus theorem prover (DCTP), derived from prover (DCTP), derived from Billon’s workBillon’s work

FDPLLFDPLL

Performance of DCTP on Performance of DCTP on TPTP, 2003TPTP, 2003

DCTP 1.3 first in EPS and EPR DCTP 1.3 first in EPS and EPR (largely propositional)(largely propositional)

DCTP 10.2p third in FNE (first-order, DCTP 10.2p third in FNE (first-order, no equality) solving same number no equality) solving same number as best proversas best provers

DCTP 10.2p fourth in FOF and FEQ DCTP 10.2p fourth in FOF and FEQ (all first-order formulae, and (all first-order formulae, and formulae with equality)formulae with equality)

DCTP 1.3 is a single strategy prover.DCTP 1.3 is a single strategy prover.

SemanticsSemanticsGelernter 1959 Geometry Theorem Gelernter 1959 Geometry Theorem

ProverProverAdapt semantics to clause form:Adapt semantics to clause form:An interpretation (semantics) An interpretation (semantics) II is an is an

assignment of truth values to assignment of truth values to literals so that literals so that I I assigns opposite assigns opposite truth values to truth values to LL and and LL for atoms for atoms LL..

The literals The literals LL and and LL are said to be are said to be complementarycomplementary..

SemanticsSemanticsWe write We write I CI C ( (II satisfiessatisfies CC) to ) to

indicate that semantics indicate that semantics I I makes the makes the clause clause CC true. true.

If If CC is a ground clause then is a ground clause then II satisfies satisfies C C if I satisfies at least one of its literals.if I satisfies at least one of its literals.

Otherwise Otherwise II satisfies satisfies CC if if I I satisfies all satisfies all ground instances ground instances DD of of CC. (Herbrand . (Herbrand interpretations.)interpretations.)

If If II does not satisfy does not satisfy CC then we say then we say II falsifiesfalsifies CC..

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Example SemanticsExample SemanticsSpecify I by interpreting symbolsSpecify I by interpreting symbolsInterpret predicate Interpret predicate p(x,y)p(x,y) as as x = yx = yInterpret function Interpret function f(x,y)f(x,y) as as x + yx + yInterpret a as 1, b as 2, c as 3Interpret a as 1, b as 2, c as 3Then Then p(f(a,b),c)p(f(a,b),c) interprets to TRUE interprets to TRUE

but but p(a,b)p(a,b) interprets to FALSE interprets to FALSEThus I satisfies Thus I satisfies p(f(a,b),c)p(f(a,b),c) but I but I

falsifies falsifies p(a,b)p(a,b)

Obtaining SemanticsObtaining Semantics

Humans using mathematical Humans using mathematical knowledgeknowledge

Automatic methods (finite models)Automatic methods (finite models)Trivial semanticsTrivial semantics

Goal of OSHLGoal of OSHL

First-order logicFirst-order logicClause formClause formPropositional efficiencyPropositional efficiencySemanticsSemantics

Requires ground decidabilityRequires ground decidability

Structure of OSHLStructure of OSHLGoal sensitivity if semantics chosen Goal sensitivity if semantics chosen

properlyproperlyChoose initial semantics to satisfy axiomsChoose initial semantics to satisfy axioms

Use of natural semanticsUse of natural semanticsFor group theory problems, can specify a For group theory problems, can specify a

groupgroupSequential search through possible Sequential search through possible

interpretationsinterpretationsThus similar to Davis and Putnam’s methodThus similar to Davis and Putnam’s methodPropositional EfficiencyPropositional Efficiency

Constructs a semantic treeConstructs a semantic tree

Ordered Semantic Hyperlinking (Oshl)Ordered Semantic Hyperlinking (Oshl)

Reduce first-order logic problem to Reduce first-order logic problem to propositional problem propositional problem

Imports propositional efficiency into first-Imports propositional efficiency into first-order logicorder logic

The algorithmThe algorithmImposes an ordering on clausesImposes an ordering on clausesProgresses by generating instances and refining Progresses by generating instances and refining

interpretationsinterpretations

unsatisfiable

I0 I1 I2 I3 …

D0 D1 D2 T

OSHLOSHLII00 is specified by the user is specified by the userDDii is chosen is chosen minimal minimal so that Iso that Iii falsifies falsifies

DDii

DDii is an instance of a clause in S is an instance of a clause in SIIii is chosen is chosen minimalminimal so that I so that Iii satisfies satisfies

DDjj for all j < i for all j < iLet TLet Tii be {D be {D00,D,D11, …, D, …, Di-1i-1}.}.

IIii falsifies D falsifies Di i but satisfies Tbut satisfies Tii

When TWhen Tii is unsatisfiable OSHL stops and is unsatisfiable OSHL stops and reports that S is unsatisfiable.reports that S is unsatisfiable.

Clause OrderingClause Ordering||L||||L||linlin

||P(f(x),g(x,c))||||P(f(x),g(x,c))||linlin = 6 = 6||L||||L||dagdag

||P(f(x),f(x))||||P(f(x),f(x))||dag dag = 4= 4Extend to clauses additively, ignoring Extend to clauses additively, ignoring

negationsnegationsOSHL chooses DOSHL chooses Dii minimal in such an minimal in such an

orderingordering

Alternate version of Alternate version of OSHLOSHL

Want to keep the size of T smallWant to keep the size of T smallDo this by throwing away clauses of T Do this by throwing away clauses of T

subject to the condition:subject to the condition:The minimal model of TThe minimal model of Ti+1i+1 is larger than is larger than

the minimal model of Tthe minimal model of Tii for all i. for all i.This guarantees completeness.This guarantees completeness.Leads to a formulation using Leads to a formulation using

sequences of clauses and sequences of clauses and resolutions between clauses.resolutions between clauses.

Rules of OSHL

Start with empty sequence(C1,C2, …, Cn), D minimal contradict I, I minimal model

(C1,C2, …, Cn,D)

(C1,C2, …, Cn, D), Cn not needed

(C1,C2, …, Cn-1,D)

(C1,C2, …, Cn,D), max resolution possible

(C1,C2, …, Cn-1,res(Cn,D,L))

Proof if empty clause derived

Propositional Example (p I0 p)

()

({-p1, -p2, -p3}) I0[-p3]

({-p1, -p2, -p3}, {-p4, -p5, -p6}) I0 [-p3,-p6]

({…}, {…}, {-p7}) I0 [-p3,-p6,-p7]

({…}, {…}, {-p7}, {p3, p7})

({…}, {-p4, -p5, -p6}, {p3})

({-p1, -p2, -p3},{p3})

({-p1, -p2 }) I0 [-p2]

╨

Semantics OrderingSemantics Ordering<<t t a well founded ordering on atoms, a well founded ordering on atoms,

extended to literalsextended to literalsExtend <Extend <t t to interpretations as follows:to interpretations as follows:I and J agree on L if they interpret L the I and J agree on L if they interpret L the

samesameSuppose ISuppose I00 is given is givenI <I <tt J if I and J are not identical, A is the J if I and J are not identical, A is the

minimal atom on which they disagree, minimal atom on which they disagree, and I agrees with Iand I agrees with I00 on A on A

Semantics OrderingSemantics Ordering

<<t t is not a well founded ordering on is not a well founded ordering on interpretations. But <interpretations. But <t t minimal minimal models of T always exist.models of T always exist.

IIii is always chosen as the < is always chosen as the <tt minimal minimal model of T.model of T.

Theorem: Such ITheorem: Such Iii always has the form always has the form II00[L[L11 … L … Lmm] where L] where Lii are literals of are literals of clauses of T.clauses of T.

II00[L[L11 … L … Lmm] L iff at(L) ] L iff at(L) {at(L {at(L11 … L … Lnn)} )} and Iand I0 0 L, or for some i L = LL, or for some i L = Lii..

╨

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Instantiation ExampleInstantiation ExampleSuppose ISuppose I00 interprets arithmetic in interprets arithmetic in

the standard way.the standard way.Suppose S contains axioms of Suppose S contains axioms of

arithmetic and the clause X+3arithmetic and the clause X+35.5.Then the first instance chosen could Then the first instance chosen could

be 2+3be 2+35, (1+1)+35, (1+1)+35, (3-1)+35, (3-1)+35 5 et cetera but it could not be et cetera but it could not be 3+33+35, nor could it be an instance 5, nor could it be an instance of an axiom.of an axiom.

Instantiation ExampleInstantiation ExampleSuppose the first instance chosen is Suppose the first instance chosen is

2+32+35.5.Then IThen I11 is I is I00[2+3[2+35], which interprets all 5], which interprets all

atoms as in standard arithmetic except atoms as in standard arithmetic except that the statement 2+3that the statement 2+35 is true.5 is true.

The next instance chosen might be 2+3-1 The next instance chosen might be 2+3-1 = 5-1 = 5-1 2+3 = 5. This contradicts I 2+3 = 5. This contradicts I11. It . It is an instance of the clause X-1 = Y-1 is an instance of the clause X-1 = Y-1 X X = Y and corresponds to generating the = Y and corresponds to generating the subgoal 2+3-1 = 5-1.subgoal 2+3-1 = 5-1.

SemanticsSemantics

Trivial semantics:Trivial semantics:Positive: Choose IPositive: Choose I00 to falsify all to falsify all

atoms, first D is all positiveatoms, first D is all positiveNegative: Choose INegative: Choose I00 to satisfy all to satisfy all

atoms, first D is all negativeatoms, first D is all negativeNatural semantics: INatural semantics: I00 chosen by chosen by

useruser

Another Semantics Another Semantics OrderingOrdering

I ≤I ≤pos pos J if for all atoms A, I A J if for all atoms A, I A implies J A.implies J A.

J is ≤J is ≤pos pos minimal model of S if J is a minimal model of S if J is a model of S and there is no model model of S and there is no model I of S such that I ≤I of S such that I ≤pos pos JJ

╨╨

Theoretical Results: Benefit Theoretical Results: Benefit of Semanticsof Semantics

Two complementary unifiable Two complementary unifiable literalsliterals

Horn ClausesHorn ClausesUnique interpretationUnique interpretationArbitrary set of first-order clausesArbitrary set of first-order clausesA number of other results in the A number of other results in the

paperpaper

Complementary unifiable Complementary unifiable literalsliterals

P(f(x),y), P(f(x),y), P(w,g(w)) P(w,g(w))Resolution: Linear timeResolution: Linear timeOSHL with trivial semantics, linear OSHL with trivial semantics, linear

ordering: Double exponential timeordering: Double exponential timeOSHL with trivial semantics, dag OSHL with trivial semantics, dag

ordering: Single exponentialordering: Single exponentialOSHL with semantics as ≤OSHL with semantics as ≤pos pos minimal minimal

model of positive literal : Polynomial model of positive literal : Polynomial timetime

UnifiabilityUnifiabilitySuppose C = {LSuppose C = {L11 … L … Lmm} is a clause in S. } is a clause in S.

Let Unif(C) be {CLet Unif(C) be {C : : CC11 … C … Cnn S, S, literals Mliterals M1 1 CC11, …, M, …, Mn n C Cnn s.t. s.t. is a is a most general simultaneous unifier of most general simultaneous unifier of LLii and and MMiiii for all i where for all i where ii are are renamings of variables of Mrenamings of variables of Mii so that L so that Lii and Mand Mii have no common variables} have no common variables}

UnifUnifnegneg(C): {L(C): {L11 … L … Lmm} are the } are the negativenegative literals in C.literals in C.

Summary of resultsSummary of resultsSeveral results show that OSHL with an Several results show that OSHL with an

appropriate semantics is implicitly appropriate semantics is implicitly performing unifications. Thus the performing unifications. Thus the choice of semantics has a profound choice of semantics has a profound effect on the operation of OSHL.effect on the operation of OSHL.

OSHL has some features of OSHL has some features of propositional methods and some propositional methods and some features of unification-based features of unification-based methods.methods.

Horn ClausesHorn ClausesTheorem. Suppose that S is an Theorem. Suppose that S is an

unsatisfiable set of Horn clauses and unsatisfiable set of Horn clauses and II00 is a ≤ is a ≤pos pos minimal model of the minimal model of the axioms of S. Then for all instances D axioms of S. Then for all instances D generated by OSHL there is a clause generated by OSHL there is a clause D' in Unif(S) such that D is an D' in Unif(S) such that D is an instance of D'.instance of D'.

This shows that OSHL is implicitly This shows that OSHL is implicitly performing unifications with this performing unifications with this semantics.semantics.

Unique ModelUnique ModelTheorem. Suppose that S is an Theorem. Suppose that S is an

unsatisfiable set of clauses and T is a unsatisfiable set of clauses and T is a subset of S. Suppose that S – T has a subset of S. Suppose that S – T has a unique Herbrand model. If Iunique Herbrand model. If I00 is is chosen as this unique Herbrand chosen as this unique Herbrand model then for all instances D model then for all instances D generated by OSHL there is a clause generated by OSHL there is a clause D' in Unif(S) such that D is an D' in Unif(S) such that D is an instance of D'.instance of D'.

Thus OSHL is implicitly unifying with Thus OSHL is implicitly unifying with this semantics.this semantics.

General CaseGeneral CaseTheorem. Suppose that S is an Theorem. Suppose that S is an

unsatisfiable set of clauses and T unsatisfiable set of clauses and T is a subset of S. Suppose that S – is a subset of S. Suppose that S – T is satisfiable. Let IT is satisfiable. Let I00 be a ≤ be a ≤pos pos minimal model of S – T. Then for minimal model of S – T. Then for all instances D generated by OSHL all instances D generated by OSHL there is a clause D' in Unifthere is a clause D' in Unifnegneg(S) (S) such that D is an instance of D'.such that D is an instance of D'.

Thus OSHL is implicitly performing Thus OSHL is implicitly performing unifications with this semantics.unifications with this semantics.

Lifting SemanticsLifting Semantics

Implementation IssuesImplementation Issues