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

How do humans prove theorems? Semantics Case analysis Sequential search through space of possible structures Focus on the theorem

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

DPLL Example {p,r},{  p,  q,r},{p,  r} {T,r},{  T,  q,r},{T,  r}{F,r},{  F,  q,r},{F,  r} p=Tp=F {  q,r}{r},{  r} {} SIMPLIFY

Hyper Linking

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

Later propositional strategies Billon’s disconnection calculus, derived from hyper-linking Disconnection calculus theorem prover (DCTP), derived from Billon’s work FDPLL

Performance of DCTP on TPTP, 2003 DCTP 1.3 first in EPS and EPR (largely propositional) DCTP 10.2p third in FNE (first-order, no equality) solving same number as best provers DCTP 10.2p fourth in FOF and FEQ (all first- order formulae, and formulae with equality) DCTP 1.3 is a single strategy prover.

Strategy Selection in E

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

Strategy Selection The Vampire kernel provides a fairly large number of features for strategy selection. The most important ones are: Choice of the main saturation procedure : (i) OTTER loop, with or without the Limited Resource Strategy, (ii) DISCOUNT loop. A variety of optional simplifications. Parameterised reduction orderings. A number of built-in literal selection functions and different modes of comparing literals. Age-weight ratio that specifies how strongly lighter clauses are preferred for inference selection. Set-of-support strategy.

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

DCTP Strategy Selection DCTP 1.31 has been implemented as a monolithic system in the Bigloo dialect of the Scheme language. DCTP 1.31 is a single strategy prover. Individual strategies are started by DCTP 10.21p using the schedule based resource allocation scheme known from the E-SETHEO system. Of course, different schedules have been precomputed for the syntactic problem classes. The problem classes are more or less identical with the sub-classes of the competition organisers. In CASC-J2 DCTP 10.21p performed substantially better.

Goal of OSHL First-order logic Clause form Propositional efficiency Semantics Requires ground decidability

Structure of OSHL Goal sensitivity if semantics chosen properly Choose initial semantics to satisfy axioms Use of natural semantics For group theory problems, can specify a group Sequential search through possible interpretations Thus similar to Davis and Putnam’s method Propositional Efficiency Constructs a semantic tree

Ordered Semantic Hyperlinking (Oshl) Reduce first-order logic problem to propositional problem Imports propositional efficiency into first-order logic The algorithm Imposes an ordering on clauses Progresses by generating ground instances Di of input clauses and refining interpretations unsatisfiable I0 I1 I2 I3 … D0 D1 D2 T

Semantics Trivial semantics: Positive: Choose I 0 to falsify all atoms, first D is all positive. Forward chaining. Negative: Choose I 0 to satisfy all atoms, first D is all negative. Backward chaining. Natural semantics: I 0 chosen by user

Semantics Ordering < t a well founded ordering on atoms, extended to literals Extend < t to interpretations as follows: I and J agree on L if they interpret L the same Suppose I 0 is given I < t J if I and J are not identical, A is the minimal atom on which they disagree, and I agrees with I 0 on A

Rules of OSHL Start with empty sequence (C 1,C 2, …, C n ), D minimal ground instance of an input clause that contradicts I, I minimal model of sequence (C 1,C 2, …, C n,D) (C 1,C 2, …, C n, D), C n “out of order” (C 1,C 2, …, C n-1,D) (C 1,C 2, …, C n,D), max resolution possible (C 1,C 2, …, C n-1,res(C n,D,L)) Proof if empty clause derived

Propositional Example (  p I 0 p) () ({-p1, -p2, -p3}) I 0 [-p3] ({-p1, -p2, -p3}, {-p4, -p5, -p6}) I 0 [-p3,-p6] ({…}, {…}, {-p7}) I 0 [-p3,-p6,-p7] ({…}, {…}, {-p7}, {p3, p7}) ({…}, {-p4, -p5, -p6}, {p3}) ({-p1, -p2, -p3},{p3}) ({-p1, -p2 }) I 0 [-p2] ╨

U Rules Choose clauses instances to match existing literals. Look for a contradiction. Basic clauses and U clauses Basic clauses are used in three rules given Sequence can also have U clauses on the end U clauses have a selected literal In basic clauses the max. lit. is selected In U clauses other literals can be selected. Significant performance enhancement.

UR Resolution Example Given the sequence ({s(a), p(b) }, {t(a), q(b)}) and the clause {  p(X),  q(X), r(X)} create the sequence ({s(a), p(b)}, {t(a), q(b)}, {  p(b),  q(b), r(b)} ) X  b

Filtering Example Given the sequence ({s(a), p(b)}, {t(a), q(b)}) and the clause {  p(X),  q(X)} create the sequence ({s(a), p(b)}, {t(a), q(b)}, {  p(b),  q(b)} ) X  b

Case Analysis Example Given the sequence ({s(a), p(b)}, {t(a), q(b)}) and the clause {  q(X), r(X), s(X)} create the sequence ({s(a), p(b)}, {t(a), q(b)}, {  q(b), r(b), s(b)} ) X  b

Example Proof Using U Rules All positive semantics Clauses: A1.  X  Y,  Y  X, X=Y A2.  Z  X,  X  Y, Z  Y A3. g(X,Y)  X, X  Y A4.  g(X,Y)  Y, X  Y A5.  Z  X, Z  X  Y A6.  Z  Y, Z  X  Y A7.  Z  X  Y, Z  X, Z  Y T.  A  B = B  A

Example Proof Using U Rules 1. {  A  B = B  A} (T) 2. {  A  B  B  A,  B  A  A  B, A  B = B  A} (Case Analysis, A1) 3. {  g(A  B, B  A)  B  A, A  B  B  A} (UR resolution, A4) 4. {g(A  B, B  A)  B  A,  g(…)  B} (UR resolution, A5) 5. {g(A  B, B  A)  B  A,  g(…)  A} (UR resolution, A6) 6. {g(…)  B, g(…)  A,  g(…)  A  B} (UR resolution, A7) 7. {A  B  B  A, g(…)  A  B} (Filtering, A3)

Example Proof Using U Rules 1. {  A  B = B  A} 2. {  A  B  B  A,  B  A  A  B, A  B = B  A} (Case Analysis) 3. {  g(A  B, B  A)  B  A, A  B  B  A} (UR resolution) 4. {g(A  B, B  A)  B  A,  g(…)  B} (UR resolution) 5. {g(A  B, B  A)  B  A,  g(…)  A} (UR resolution) 8. {g(…)  B, g(…)  A, A  B  B  A,} (Resolution of 6. and 7.)

Example Proof Using U Rules 1. {  A  B = B  A} 2. {  A  B  B  A,  B  A  A  B, A  B = B  A} (Case Analysis) 3. {  g(A  B, B  A)  B  A, A  B  B  A} (UR resolution) 4. {g(A  B, B  A)  B  A,  g(…)  B} (UR resolution) 9. {g(A  B, B  A)  B  A, g(…)  B, A  B  B  A} (Resolution of 8. and 5.)

Example Proof Using U Rules 1. {  A  B = B  A} 2. {  A  B  B  A,  B  A  A  B, A  B = B  A} (Case Analysis) 3. {  g(A  B, B  A)  B  A, A  B  B  A} (UR resolution) 10. {g(A  B, B  A)  B  A} (Resolution of 9. and 4.)

Example Proof Using U Rules 1. {  A  B = B  A} 2. {  A  B  B  A,  B  A  A  B, A  B = B  A} (Case Analysis) 11. {A  B  B  A} (Resolution of 10. and 3.)

Example Proof Using U Rules 1. {  A  B = B  A} 12. {  B  A  A  B, A  B = B  A} (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 Results Slower implementation speed of OSHL Uniform strategy versus strategy selection The choice of Otter Influence of U rules on an earlier version: None: 233 proofs in 30 seconds on TPTP problems Using them: 900 proofs in 30 seconds All results for trivial semantics

Implementation Results OSHL has no special data structures. Implemented in OCaML No special equality methods Semantics was implemented but frequently only trivial semantics was used. Thus significant performance improvements are possible.

Various Provers PTTP solved 999 of 2200 tested problems. Otter proved leanCoP proved 745. Source: Jens Otten and Wolfgang Bibel. leanCoP: Lean Connection-Based Theorem Proving. Journal of Symbolic Computation, Volume 36, pages Elsevier Science, Vampire 6.0: 3286 refutations of 7267 problems, more solved

Total Number of Proofs #PROBS # Otter Proofs # OSHL-U Proofs AllHORNNon-HornAllHORNNon-Horn AllR=0R>0AllR=0R>0 All FLD SET R denotes the TPTP difficulty rating 30 second time limit on each problem with each prover

Implementation Results Shows that a prover working entirely at the ground level can come into the range of performance of a respectable resolution theorem prover. DCTP and FDPLL probably perform better than OSHL. DCTP and FDPLL do not work entirely at the ground level and do not use natural semantics.

Search space AllHornNon-HornR=0R>0 Non-Horn, R>0 Otter OSHL-U Number of clauses generated (in 1,000s) computed on 827 problems that were proved by both provers Ratio RatioAllHornNon-HornR=0R>0 Non-Horn, R>0 OSHL-U Otter Otter Ratio of number of clauses generated

Storage space AllHornNon-HornR=0R>0 Non-Horn, R>0 Otter OSHL-U Max. number of clauses stored (in 1,000s) computed on 827 problems that were proved by both provers Ratio RatioAllHornNon-HornR=0R>0 Non-Horn, R>0 OSHL-U Otter Otter Ratio of number of clauses stored

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