The Space Efficiency of OSHL Swaha Miller David A. Plaisted UNC Chapel Hill.

The Space The Space Efficiency of OSHLEfficiency of OSHL

Swaha MillerSwaha Miller

David A. PlaistedDavid A. Plaisted

UNC Chapel HillUNC Chapel Hill

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

SemanticsSemantics

Case analysisCase analysis

Sequential search through space Sequential search through space of possible structuresof possible structures

Focus 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.

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.

Strategy Selection in Strategy Selection in EE

Strategy SelectionStrategy Selection

Schulz, Stephan, E-A Brainiac Theorem Schulz, Stephan, E-A Brainiac Theorem Prover, Journal of AI Communications Prover, Journal of AI Communications 15(2/3):111-126, 2002. 15(2/3):111-126, 2002.

Strategy SelectionStrategy SelectionThe Vampire kernel provides a fairly large The Vampire kernel provides a fairly large

number of features for strategy selection. number of features for strategy selection. The most important ones are: The most important ones are:

Choice of the main saturation procedure : (i) Choice of the main saturation procedure : (i) OTTER loop, with or without the Limited OTTER loop, with or without the Limited Resource Strategy, (ii) DISCOUNT loop. Resource Strategy, (ii) DISCOUNT loop.

A variety of optional simplifications. A variety of optional simplifications. Parameterised reduction orderings. Parameterised reduction orderings. A number of built-in literal selection A number of built-in literal selection

functions and different modes of functions and different modes of comparing literals. comparing literals.

Age-weight ratio that specifies how strongly Age-weight ratio that specifies how strongly lighter clauses are preferred for inference lighter clauses are preferred for inference selection. selection.

Set-of-support strategy. Set-of-support strategy.

Strategy SelectionStrategy Selection

The automatic mode of Vampire 7.0 is The automatic mode of Vampire 7.0 is derived from extensive experimental data derived from extensive experimental data obtained on problems from TPTP v2.6.0. obtained on problems from TPTP v2.6.0. Input problems are classified taking into Input problems are classified taking into account simple syntactic properties, such account simple syntactic properties, such as being Horn or non-Horn, presence of as being Horn or non-Horn, presence of equality, etc. Additionally, we take into equality, etc. Additionally, we take into account the presence of some important account the presence of some important kinds of axioms, such as set theory kinds of axioms, such as set theory axioms, associativity and commutativity. axioms, associativity and commutativity. Every class of problems is assigned a Every class of problems is assigned a fixed schedule consisting of a number of fixed schedule consisting of a number of kernel strategies called one by one with kernel strategies called one by one with different time limits. different time limits.

DCTP Strategy SelectionDCTP Strategy SelectionDCTP 1.31 has been implemented as a DCTP 1.31 has been implemented as a

monolithic system in the Bigloo dialect monolithic system in the Bigloo dialect of the Scheme language.of the Scheme language.

DCTP 1.31 is a single strategy prover. DCTP 1.31 is a single strategy prover. Individual strategies are started by Individual strategies are started by DCTP 10.21p using the schedule based DCTP 10.21p using the schedule based resource allocation scheme known from resource allocation scheme known from the E-SETHEO system. Of course, the E-SETHEO system. Of course, different schedules have been different schedules have been precomputed for the syntactic problem precomputed for the syntactic problem classes. The problem classes are more classes. The problem classes are more or less identical with the sub-classes of or less identical with the sub-classes of the competition organisers.the competition organisers.

In CASC-J2 DCTP 10.21p performed In CASC-J2 DCTP 10.21p performed substantially better.substantially better.

Goal of OSHLGoal of OSHL

First-order logicFirst-order logic

Clause formClause form

Propositional efficiencyPropositional efficiency

SemanticsSemanticsRequires ground decidabilityRequires ground decidability

Structure of OSHLStructure of OSHL

Goal 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

groupgroup

Sequential 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 ground instances Di Progresses by generating ground instances Di

of input clauses and refining interpretationsof input clauses and refining interpretations

unsatisfiable

I0 I1 I2 I3 …

D0 D1 D2 T

SemanticsSemantics

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

atoms, first D is all positive. atoms, first D is all positive. Forward chaining.Forward chaining.

Negative: Choose INegative: Choose I00 to satisfy all to satisfy all atoms, first D is all negative. atoms, first D is all negative. Backward chaining.Backward chaining.

Natural semantics: INatural semantics: I00 chosen by chosen by useruser

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

extended to literalsextended to literals

Extend <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 samesame

Suppose ISuppose I00 is given is given

I <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

Rules of OSHL

Start with empty sequence

(C1,C2, …, Cn), D minimal ground instance of an input clause that contradicts I, I minimal model of sequence

(C1,C2, …, Cn,D)

(C1,C2, …, Cn, D), Cn “out of order”

(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]

╨

U RulesU RulesChoose clauses instances to match Choose clauses instances to match

existing literals. Look for a existing literals. Look for a contradiction.contradiction.

Basic clauses and U clausesBasic clauses and U clausesBasic clauses are used in three rules givenBasic clauses are used in three rules givenSequence can also have U clauses on the Sequence can also have U clauses on the

endendU clauses have a selected literalU clauses have a selected literalIn basic clauses the max. lit. is selectedIn basic clauses the max. lit. is selectedIn U clauses other literals can be selected.In U clauses other literals can be selected.Significant performance enhancement.Significant performance enhancement.

UR Resolution ExampleUR Resolution Example

Given the sequenceGiven the sequence ({s(a), ({s(a), p(b) p(b) }, {t(a),}, {t(a), q(b) q(b)})})

and the clauseand the clause {{p(X), p(X), q(X), r(X)}q(X), r(X)}

create the sequencecreate the sequence ({s(a), ({s(a), p(b)p(b)}, {t(a), }, {t(a), q(b)q(b)}, {}, {p(b), p(b), q(b), q(b), r(b)r(b)} )} )

X b

Filtering ExampleFiltering Example

Given the sequence ({s(a), Given the sequence ({s(a), p(b)p(b)}, {t(a), }, {t(a), q(b)q(b)}) })

and the clause {and the clause {p(X), p(X), q(X)} q(X)}

create the sequence create the sequence

({s(a), ({s(a), p(b)p(b)}, {t(a), }, {t(a), q(b)q(b)}, {}, {p(b), p(b), q(b)q(b)} )} )

X b

Case Analysis ExampleCase Analysis Example

Given the sequence ({s(a), Given the sequence ({s(a), p(b)p(b)}, {t(a), }, {t(a), q(b)q(b)}) })

and the clause {and the clause {q(X), r(X), s(X)} q(X), r(X), s(X)}

create the sequence create the sequence

({s(a), ({s(a), p(b)p(b)}, {t(a), }, {t(a), q(b)q(b)}, {}, {q(b), r(b), q(b), r(b), s(b)s(b)} )} )

X b

Example Proof Using U Example Proof Using U RulesRules

All positive semanticsAll positive semanticsClauses:Clauses:A1. A1. XXY, Y, YYX, X=YX, X=YA2. A2. ZZX, X, XXY, ZY, ZYYA3. g(X,Y)A3. g(X,Y)X, XX, XYYA4. A4. g(X,Y)g(X,Y)Y, XY, XYYA5. A5. ZZX, ZX, ZX X Y Y A6. A6. ZZY, ZY, ZX X Y YA7. A7. ZZX X Y, Z Y, ZX, ZX, ZYYT. T. A A B = B B = B A A


1. {1. {A A B = B B = B A} A} (T)(T)2. {2. {A A B B B B A A, , B B A A A A B, A B, A B = B =

B B A} (Case Analysis, A1) A} (Case Analysis, A1)3. {3. {g(A g(A B, B B, B A) A) B B A A, A , A B B B B A} A}

(UR resolution, A4)(UR resolution, A4)4. {g(A 4. {g(A B, B B, B A) A) B B A, A, g(…) g(…) B B} (UR } (UR

resolution, A5)resolution, A5)5. {g(A 5. {g(A B, B B, B A) A) B B A, A, g(…) g(…) A A} (UR } (UR

resolution, A6)resolution, A6)6. {g(…) 6. {g(…) B, g(…) B, g(…) A, A, g(…) g(…) A A B B} }

(UR resolution, A7)(UR resolution, A7)7. {A 7. {A B B B B A, A, g(…) g(…) A A B B} (Filtering, } (Filtering,

A3)A3)


1. {1. {A A B = B B = B A} A}2. {2. {A A B B B B A A, , B B A A A A B, A B, A B = B =

B B A} (Case Analysis) A} (Case Analysis)3. {3. {g(A g(A B, B B, B A) A) B B A A, A , A B B B B

A} (UR resolution)A} (UR resolution)4. {g(A 4. {g(A B, B B, B A) A) B B A, A, g(…) g(…) B B} }

(UR resolution)(UR resolution)5. {g(A 5. {g(A B, B B, B A) A) B B A, A, g(…) g(…) A A} }

(UR resolution)(UR resolution)8. {g(…) 8. {g(…) B, B, g(…) g(…) A A, A , A B B B B A,} A,}

(Resolution of 6. and 7.)(Resolution of 6. and 7.)




A} (UR resolution)A} (UR resolution)4. {g(A 4. {g(A B, B B, B A) A) B B A, A, g(…) g(…) B B} }

(UR resolution)(UR resolution)9. {g(A 9. {g(A B, B B, B A) A) B B A, A, g(…) g(…) B B, A , A

B B B B A} (Resolution of 8. and 5.) A} (Resolution of 8. and 5.)




A} (UR resolution)A} (UR resolution)10. {10. {g(A g(A B, B B, B A) A) B B A A} (Resolution } (Resolution

of 9. and 4.)of 9. and 4.)


1. {1. {A A B = B B = B A} A}2. {2. {A A B B B B A A, , B B A A A A B, A B, A B = B B = B

A} (Case Analysis)A} (Case Analysis)11. {11. {A A B B B B A A} (Resolution of 10. and 3.)} (Resolution of 10. and 3.)


1. {1. {A A B = B B = B A} A}12. {12. {B B A A A A B B, A , A B = B B = B A} A}

(Resolution of 11 and 2)(Resolution of 11 and 2)

Now the other half of the proof will be done. Note that there is only one ascending sequence of clauses constructed by OSHL and we are only indicating part of it.

Implementation ResultsImplementation ResultsSlower implementation speed of OSHLSlower implementation speed of OSHL

Uniform strategy versus strategy Uniform strategy versus strategy selectionselection

The choice of OtterThe choice of Otter

Influence of U rules on an earlier Influence of U rules on an earlier version:version:None: 233 proofs in 30 seconds on TPTP None: 233 proofs in 30 seconds on TPTP

problemsproblems

Using them: 900 proofs in 30 secondsUsing them: 900 proofs in 30 seconds

All results for trivial semanticsAll results for trivial semantics

Implementation ResultsImplementation Results

OSHL has no special data OSHL has no special data structures.structures.

Implemented in OCaMLImplemented in OCaML

No special equality methodsNo special equality methods

Semantics was implemented but Semantics was implemented but frequently only trivial semantics frequently only trivial semantics was used.was used.

Thus significant performance Thus significant performance improvements are possible.improvements are possible.

Various ProversVarious Provers

PTTP solved 999 of 2200 tested problems.PTTP solved 999 of 2200 tested problems.

Otter proved 1595.Otter proved 1595.

leanCoP proved 745.leanCoP proved 745.

Source:Source:

Jens Otten and Wolfgang Bibel.Jens Otten and Wolfgang Bibel.leanCoP: Lean Connection-Based leanCoP: Lean Connection-Based Theorem Proving. Theorem Proving. Journal of Symbolic Journal of Symbolic Computation, Volume 36, pages 139-161. Computation, Volume 36, pages 139-161. Elsevier Science, 2003.Elsevier Science, 2003.

Vampire 6.0: 3286 refutations of 7267 Vampire 6.0: 3286 refutations of 7267 problems, more solvedproblems, more solved

Total Number of ProofsTotal Number of Proofs##

PP

RR

OO

BB

SS

# Otter Proofs# Otter Proofs # OSHL-U # OSHL-U ProofsProofs

AllAll HH

OO

RR

NN

Non-Non-HornHorn

AllAll HH

OO

RR

NN

Non-Non-HornHorn

AlAlll

RR

==

00

RR

>>

00

AlAlll

RR

==

00

RR

>>

00

AllAll 44144177

16916977

764764 939333

636636 297297 10210277

311311 717166

451451 265265

FLDFLD 143143 2828 00 2828 1717 1111 6868 00 6868 2121 4747

SETSET 604604 168168 22 161666

126126 4040 211211 22 202099

114114 9797

R denotes the TPTP difficulty rating

30 second time limit on each problem with each prover


Shows that a prover working entirely Shows that a prover working entirely at the ground level can come into at the ground level can come into the range of performance of a the range of performance of a respectable resolution theorem respectable resolution theorem prover.prover.

DCTP and FDPLL probably perform DCTP and FDPLL probably perform better than OSHL.better than OSHL.

DCTP and FDPLL do not work entirely DCTP and FDPLL do not work entirely at the ground level and do not use at the ground level and do not use natural semantics.natural semantics.

Search spaceSearch space

AllAll HornHorn Non-Non-HornHorn

R=0R=0 R>0R>0 Non-Horn, Non-Horn, R>0R>0

OtterOtter 708708 9090 618618 357357 351351 348348

OSHL-UOSHL-U 104104 3939 6565 7878 2626 2626

Number of clauses generated (in 1,000s) computed on 827 problems that were proved by both provers

RatioRatio AllAll HornHorn Non-Non-HornHorn


OSHL-UOSHL-U

OtterOtter0.1470.147 0.4330.433 0.1050.105 0.210.21

880.070.07

550.0750.075

Ratio of number of clauses generated

Storage spaceStorage space

AllAll HornHorn Non-Non-HornHorn


OtterOtter 423423 8181 342342 230230

193193 192192

OSHL-UOSHL-U 9191 3737 5555 6767 2525 2525Max. number of clauses stored (in 1,000s) computed on

827 problems that were proved by both provers

RatioRatio AllAll HornHorn Non-Non-HornHorn


OSHL-UOSHL-U

OtterOtter0.2150.215 0.4570.457 0.1610.161 0.290.29

110.130.13

000.1300.130

Ratio of number of clauses stored


In a given number of inferences In a given number of inferences OSHL finds more proofs than OSHL finds more proofs than Otter for non Horn problemsOtter for non Horn problems

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The Space Efficiency of OSHL Swaha Miller David A. Plaisted UNC Chapel Hill.

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