OSHL: A Propositional Prover with Semantics for First-Order Logic David A. Plaisted UNC Chapel Hill

Current theorem provers Largely syntactic Resolution or ME (tableau) based First-order provers are often poor on non-Horn clauses Rarely can solve hard problems Human interaction needed for hard problems 11/9/2018

Unit Resolution and General Resolution Resolution is efficient for Horn and renameable Horn problems. Resolution is efficient if the proof can be found by UR resolution. Hard problems tend not to be Horn, renameable Horn, or UR resolvable. Of 1697 TPTP problems provable by Otter in 30 seconds, 1042 can be proved by UR resolution.

Unit Resolution and General Resolution Of the 1697 problems provable by Otter, only 297 were both non Horn and had rating greater than zero. Of these 297, at most 215 are not UR resolvable. Otter can do hundreds of thousands of resolutions in 30 seconds on this machine. Resolution is inefficient on hard, non UR resolvable problems. Need for new approaches.

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} p=T p=F {T,r},{T,q,r},{T,r} {F,r},{F,q,r},{F,r} SIMPLIFY SIMPLIFY {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) 25-42.

Definition Detection

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

More Definitions S1  S2  …  Sn=Sn  Sn-1  …  S1 Left Associative OSHL Otter Vampire E-Setheo DCTP time Gen Kept 2 0.175 41 36 600+ 100303 24712 0.00 103 90 0.0 0.01 3 0.678 85 80 66753 31496 70.1 3606742 50382 0.3 300+ 4 2.107 141 136 47219 22119 25898955 68385 5 5.317 207 202 46054 20941 25298293 67864 2.6 6 12.02 283 278 60247 22923 25612105 68457 7 38.97 56299 19660 25641650 67977 8 77.94 56352 18932 25863117 68542

More Definitions Similar results for other definitions: S1  S2  …  Sn=Sn  Sn-1  …  S1, left side left associated, right side right associated S1  S2  …  Sn= S1  S2  …  Sn  S1  S2  …  Sn, both sides associated to the left S1  S2  …  Sn= S1  S2  …  Sn  S1  S2  …  Sn, left side left associated, right side right associated Similar results for ∩

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):111-126, 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 v2.6.0. 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.

Various Provers PTTP solved 999 of 2200 tested problems. Otter proved 1595. leanCoP proved 745. Source: Jens Otten and Wolfgang Bibel. leanCoP: Lean Connection-Based Theorem Proving. Journal of Symbolic Computation, Volume 36, pages 139-161. Elsevier Science, 2003. Vampire 6.0: 3286 refutations of 7267 problems, more solved

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.

Semantics Gelernter 1959 Geometry Theorem Prover Adapt semantics to clause form: An interpretation (semantics) I is an assignment of truth values to literals so that I assigns opposite truth values to L and L for atoms L. The literals L and L are said to be complementary.

Semantics ╨ We write I C (I satisfies C) to indicate that semantics I makes the clause C true. If C is a ground clause then I satisfies C if I satisfies at least one of its literals. Otherwise I satisfies C if I satisfies all ground instances D of C. (Herbrand interpretations.) If I does not satisfy C then we say I falsifies C.

Example Semantics Specify I by interpreting symbols Interpret predicate p(x,y) as x = y Interpret function f(x,y) as x + y Interpret a as 1, b as 2, c as 3 Then p(f(a,b),c) interprets to TRUE but p(a,b) interprets to FALSE Thus I satisfies p(f(a,b),c) but I falsifies p(a,b)

Obtaining Semantics Humans using mathematical knowledge Automatic methods (finite models) Trivial semantics

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 instances and refining interpretations unsatisfiable I0 I1 I2 I3 … D0 D1 D2 T

OSHL I0 is specified by the user Di is chosen minimal so that Ii falsifies Di Di is an instance of a clause in S Ii is chosen minimal so that Ii satisfies Dj for all j < i Let Ti be {D0,D1, …, Di-1}. Ii falsifies Di but satisfies Ti When Ti is unsatisfiable OSHL stops and reports that S is unsatisfiable.

Clause Ordering ||L||lin ||P(f(x),g(x,c))||lin = 6 ||L||dag ||P(f(x),f(x))||dag = 4 Extend to clauses additively, ignoring negations OSHL chooses Di minimal in such an ordering

Alternate version of OSHL Want to keep the size of T small Do this by throwing away clauses of T subject to the condition: The minimal model of Ti+1 is larger than the minimal model of Ti for all i. This guarantees completeness. Leads to a formulation using sequences of clauses and resolutions between clauses.

Start with empty sequence 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 Trivial semantics: Natural semantics: I0 chosen by user Positive: Choose I0 to falsify all atoms, first D is all positive. Forward chaining. Negative: Choose I0 to satisfy all atoms, first D is all negative. Backward chaining. Natural semantics: I0 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 I0 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 I0 on A

Semantics Ordering <t is not a well founded ordering on interpretations. But <t minimal models of T always exist. Ii is always chosen as the <t minimal model of T. Theorem: Such Ii always has the form I0[L1 … Lm] where Li are literals of clauses of T. I0[L1 … Lm] L iff at(L)  {at(L1 … Ln)} and I0 L, or for some i L = Li. ╨ ╨

Instantiation Example Suppose I0 interprets arithmetic in the standard way. Suppose S contains axioms of arithmetic and the clause X+35. Then the first instance chosen could be 2+35, (1+1)+35, (3-1)+35 et cetera but it could not be 3+35, nor could it be an instance of an axiom.

Instantiation Example Suppose the first instance chosen is 2+35. Then I1 is I0[2+35], which interprets all atoms as in standard arithmetic except that the statement 2+35 is true. The next instance chosen might be 2+3-1 = 5-1  2+3 = 5. This contradicts I1. It is an instance of the clause X-1 = Y-1  X = Y and corresponds to generating the subgoal 2+3-1 = 5-1.

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.

U Rules UR resolution: Find C in S having a ground UR resolvent with selected literals. Let C' be the corresponding instance of C. Add C' to the end of the sequence of clauses and select the UR resolvent from it. Filtering: Find C in S such that NIL is derivable by unit resolution from selected literals and C. Let C' be the corresponding instance of C. Add C' to the end of the sequence of clauses. Select a literal from it.

U Rules Case Analysis: Find C in S and L in C such that L has all the variables of C. Find instance L' of L that is complementary to a selected literal of some clause in the sequence. Let C' be the corresponding instance of C. Add C' to the end of the sequence and select a literal from it. This rule expands definitions.

Examples of U Rules UR resolution: Given the sequence ({s(a), p(b)}, {t(a), q(b)}) and the clause {not p(X), not q(X), r(X)} create the sequence ({s(a), p(b)}, {t(a), q(b)}, {not p(b), not q(b), r(b)} ) Filtering: Given the sequence ({s(a), p(b)}, {t(a), q(b)}) and the clause {not p(X), not q(X)} create the sequence ({s(a), p(b)}, {t(a), q(b)}, {not p(b), not q(b)} )

Examples of U Rules Case analysis: Given the sequence ({s(a), p(b)}, {t(a), q(b)}) and the clause {not q(X), r(X), s(X)} create the sequence ({s(a), p(b)}, {t(a), q(b)}, {not q(b), r(b), s(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.

Implementation Results # P R O B S # Otter Proofs # OSHL Proofs All H N Non-Horn = > 4417 1697 764 933 297 636 1027 311 716 265 451 FLD 143 28 11 17 68 47 21 SET 604 168 2 166 40 126 211 209 93 116 Total Number of Proofs, 30 seconds

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.

Implementation Results All Horn Non-Horn R=0 R>0 Non Horn, Clauses, Otter 3483094 215290 3267804 915737 2567357 2460992 Clauses, OSHL 17212 8110 9102 14888 2324 2216 Ratio 202 26.6 359 61.5 1105 1111 For problems for which both provers found proofs in 30 seconds..

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

Summary of theoretical results about semantics Several results show that OSHL with an appropriate semantics is implicitly performing unifications. Thus the choice of semantics has a profound effect on the operation of OSHL. OSHL has some features of propositional methods and some features of unification-based methods. Semantics might significantly improve OSHL.

Number of Clauses Generated Problem #clauses, Otter Oshl+semantics GRP005-1 57 3 GRP006-1 62 7 GRO007-1 85 22 GRP018-1 266 16 GRP019-1 267 15 GRP020-1 265 18 GRP021-1 264 19 GRP023-1 79 22 GRP032-3 83 14 GRP034-3 141 30 GRP034-4 222 6 GRP042-2 21 15 GRP043-2 80 81 GRP136-1 0 8 GRP137-1 0 8

Lifting Semantics