The search efficiency of theorem proving strategies: an analytical comparison (original) (raw)

The search efficiency of theorem proving strategies

Springer eBooks, 1994

We analyze the search e ciency of a number of common refutational theorem proving strategies for rst-order logic. Search e ciency is concerned with the total number of proofs and partial proofs generated, rather than with the sizes of the proofs. We s h o w that most common strategies produce search spaces of exponential size even on simple sets of clauses, or else are not sensitive to the goal. However, clause linking, which uses a reduction to propositional calculus, has behavior that is more favorable in some respects, a property that it shares with methods that cache subgoals. A strategy which is of interest for term-rewriting based theorem proving is the A-ordering strategy, a n d we discuss it in some detail. We show some advantages of A-ordering over other strategies, which m a y help to explain its e ciency in practice. We also point out some of its combinatorial ine ciencies, especially in relation to goal-sensitivity and irrelevant clauses. In addition, SLD-resolution, which is of importance for Prolog implementation, has combinatorial ine ciencies this may suggest basing Prolog implementations on a di erent theorem proving strategy.

SATCHMO: a theorem prover implemented in Prolog

1988

Satchmo is a theorem prover consisting of just a few short and simple Prolog programs. Prolog may be used for representing problem clauses as well. SATCHMO is based on a model-generation paradigm. It is refutation-complete if used in a level-saturation manner. The paper provides a thorough report on experiences with SATCHMO. A considerable amount of problems could be solved with surprising efficiency.

Parthenon: A parallel theorem prover for non-horn clauses

Journal of Automated Reasoning, 1992

We describe a parallel resolution theorem prover, called Parthenon* that handles full first order logic. Although there has been much work on parallel implementations of logic programming languages, our system is apparently the first general purpose theorem prover to be developed for a multiprocessor. The system implements a variant of Loveland's model elimination procedure within the framework of Warren's SRI model for or-parallelism. It has been evaluated on various shared memory multiprocessors including a 16-processor Encore Multimax, producing both impressive absolute run times and near-linear speedup curves.

A Partial Instantiation based First Order Theorem Prover

Tepper School of Business, 1998

Satis ability algorithms for propositional logic have improved enormously in recent years. This increases the attractiveness of satis ability methods for rst order logic that reduce the problem to a series of ground-level satis ability problems. Partial Instantiation for rst order satis ability di ers radically from standard resolution based methods. Two approaches to partial instantiation based rst order theorem provers have been studied by R. Jeroslow 10] and by Plaisted and Zhu 14]. Hooker and Rago 8, 9] have described improvements of Jeroslow's approach by a) extending it to logic with functions, b) accelerating it through use of satis ers, as introduced by Gallo and Rago 6] and c) simplifying it to obtain further speedup. The correctness of the Partial Instantiation algorithms described here for full rst-order logic with functions as well as termination on unsatis able formulas are shown in 9]. This paper describes the implementation of a theorem prover based on the primal algorithm and its application to solving planning problems. We obtained improved e ciency by incorporating incrementality into the primal algorithm (incremental blockage testing). This extended abstract describes the Partial Primal Instantiation algorithm, its implementation and preliminary results on rst order formulation of planning problems.

Proof search in first-order linear logic and other cut-free sequent calculi

We present a general framework for proof search in first-order cut-free sequent calculi and apply it to the specific case of linear logic. In this framework, Herbrand functions are used to encode universal quantification, and unification is used to instantiate existential quantifiers so that the eigenvariable conditions are respected. We present an optimization of this procedure that exploits the permutabilities of the subject logic. We prove the soundness and completeness of several related proof search procedures. This proof search framework is used to show that provability for firstorder MALL is in nexptime, and first-order MLL is in np. Performance comparisons based on Prolog implementations of the procedures are also given. The optimization of the quantifier steps in proof search can be combined effectively with a number of other optimizations that are also based on permutability.

On structuring proof search for first order linear logic

Theoretical Computer Science, 2006

Full first order linear logic can be presented as an abstract logic programming language in Miller's system Forum, which yields a sensible operational interpretation in the 'proof search as computation' paradigm. However, Forum still has to deal with syntactic details that would normally be ignored by a reasonable operational semantics. In this respect, Forum improves on Gentzen systems for linear logic by restricting the language and the form of inference rules. We further improve on Forum by restricting the class of formulae allowed, in a system we call G-Forum, which is still equivalent to full first order linear logic. The only formulae allowed in G-Forum have the same shape as Forum sequents: the restriction does not diminish expressiveness and makes G-Forum amenable to proof theoretic analysis. G-Forum consists of two (big) inference rules, for which we show a cut elimination procedure. This does not need to appeal to finer detail in formulae and sequents than is provided by G-Forum, thus successfully testing the internal symmetries of our system.

Fast Tactic-Based Theorem Proving

Lecture Notes in Computer Science, 2000

Theorem provers for higher-order logics often use tactics to implement automated proof search. Tactics use a general-purpose metalanguage to implement both general-purpose reasoning and computationally intensive domain-specific proof procedures. The generality of tactic provers has a performance penalty; the speed of proof search lags far behind special-purpose provers. We present a new modular proving architecture that significantly increases the speed of the core logic engine. Our speedup is due to efficient data structures and modularity, which allows parts of the prover to be customized on a domain-specific basis. Our architecture is used in the MetaPRL logical framework, with speedups of more than two orders of magnitude over traditional tactic-based proof search.

LEO II: An Effective Higher-Order Theorem Prover

2008

Background Automatic theorem provers (ATPs) based on the resolution principle, such as SPASS [22] and Vampire [19], have reached a high degree of sophistication. They can often find long proofs even for problems having thousands of axioms. A fundamental limitation, however, is that they reason in first-order logic. Higher-order logic extends first-order logic with λ-notation for functions and with function and predicate variables. It supports reasoning in set theory, using the obvious representation of sets by predicates.

Proof strategies in linear logic

Journal of Automated Reasoning, 1994

Linear logic, introduced by J.-Y.Girard, is a re nement of classical logic providing means for controlling the allocation of \resources". It has aroused considerable interest both from proof theorists and computer scientists.