Cooperation between Top-Down and Bottom-Up Theorem Provers

Top-down and bottom-up theorem proving approaches each have specific advantages and disadvantages. Bottom-up provers profit from strong redundancy control but suffer from the lack of goal-orientation, whereas top-down provers are goal-oriented but often have weak calculi when their proof lengths are considered. In order to integrate both approaches, we try to achieve cooperation between a top-down and a bottom-up prover in two different ways: The first technique aims at supporting a bottom-up with a top-down prover. A top-down prover generates subgoal clauses, they are then processed by a bottom-up prover. The second technique deals with the use of bottom-up generated lemmas in a top-down prover. We apply our concept to the areas of model elimination and superposition. We discuss the ability of our techniques to shorten proofs as well as to reorder the search space in an appropriate manner. Furthermore, in order to identify subgoal clauses and lemmas which are actually relevant for the proof task, we develop methods for a relevancy-based filtering. Experiments with the provers SETHEO and SPASS performed in the problem library TPTP reveal the high potential of our cooperation approaches

leanCoP: lean connection-based theorem proving

AbstractThe Prolog programimplements a theorem prover for classical first-order (clausal) logic which is based on the connection calculus. It is sound and complete (provided that an arbitrarily large I is iteratively given), and demonstrates a comparatively strong performance

On SGGS and Horn clauses

SGGS (Semantically-Guided Goal-Sensitive reasoning) is a refutationally complete theorem-proving method that offers first-order conflict-driven reasoning and is model complete in the limit. This paper investigates the behavior of SGGS on Horn clauses, which are widely used in declarative programming, knowledge representation, and verification. We show that SGGS generates the least Herbrand model of a set of definite clauses, and that SGGS terminates on Horn clauses if and only if hyperresolution does, with the advantage that SGGS builds a model. We report on experiments applying the SGGS prototype prover Koala to Horn problems, with promising performances especially on satisfiable inputs

OTTER 3.3 Reference Manual

OTTER is a resolution-style theorem-proving program for first-order logic with equality. OTTER includes the inference rules binary resolution, hyperresolution, UR-resolution, and binary paramodulation. Some of its other abilities and features are conversion from first-order formulas to clauses, forward and back subsumption, factoring, weighting, answer literals, term ordering, forward and back demodulation, evaluable functions and predicates, Knuth-Bendix completion, and the hints strategy. OTTER is coded in ANSI C, is free, and is portable to many different kinds of computer.Comment: 66 page

How To Efficiently Implement An OSHL-Based Automatic Theorem Prover

Ordered Semantic Hyper-linking (OSHL) is a general-purpose instance-based first-order automated theorem proving algorithm. Although OSHL has many useful properties, previous implementations of OSHL were not very efficient. The implementation of such a theorem prover differs from other more traditional programs in that a lot of its subroutines are more mathematical than procedural. The low performance of previous implementations prevents us from evaluating how the proof strategy used in OSHL matches up against other theorem proving strategies. This dissertation addresses this problem on three levels. First, an abstract, generalized version genOSHL is defined which captures the essential features of OSHL and for which the soundness and completeness are proved. This gives genOSHL the flexibility to be tweaked while still preserving soundness and completeness. A type inference algorithm is introduced which allows genOSHL to possibly reduce its search space while still preserving the soundness and completeness. Second, incOSHL, a specialized version of genOSHL, which differs from the original OSHL algorithm, is defined by specializing genOSHL. Its soundness of completeness follows from that of genOSHL. Third, an embedded programming language called STACK EL, which allows managing program states and their dependencies on global mutable data, is designed and implemented. STACK EL allows our prover to generate instances incrementally. We also study the performance of our incremental theorem prover that implements incOSHL.Doctor of Philosoph