A Prolog technology theorem prover (PTTP) is an extension of Prolog that is complete for the full first-order predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, the model-elimination reduction rule that is added to Prolog inferences to make the inference system complete, and depth-first iterative-deepening search instead of unbounded depth-first search to make the search strategy complete. A Prolog technology theorem prover has been implemented by an extended Prolog-to-LISP compiler that supports these additional features. It is capable of proving theorems in the full first-order predicate calculus at a rate of thousands of inferences per second. 1 This is a revised and expanded version of a paper presented at the 8th International Conference on Automated Deduction, Oxford, England, July 1986, and is to appear in Journal of Automated Reasoning. This research was supported by the Defense Advanced Research Projects Agency under Co...

Theorem provers based on model elimination have exhibited extremely high inference rates but have lacked a redundancy control mechanism such as subsumption. In this paper we report on work done to modify a model elimination theorem prover using two techniques, caching and lemmaizing, that have reduced by more than an order of magnitude the time required to find proofs of several problems and that have enabled the prover to prove theorems previously unobtainable by top-down model elimination theorem provers.

In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and first-order logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is two-fold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward message-passing and using backward message-passing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our message-passing ...

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We propose a method for combining the clause linking theorem proving method with theorem proving methods based on orderings. This may be useful for incorporating term-rewriting based approaches into clause linking. In this way, some of the propositional inefficiencies of ordering-based approaches may be overcome, while at the same time incorporating the advantages of ordering methods into clause linking. The combination also provides a natural way to combine resolution on non-ground clauses, with the clause linking method, which is essentially a ground method. We describe the method, prove completeness, and show that the enumeration part of clause linking with semantics can be reduced to polynomial time in certain cases. We analyze the complexity of the proposed method, and also give some plausibility arguments concerning its expected performance. 1 Introduction There are at least two basic approaches to the study of automated deduction. One approach concentrates on solving...

A general theory of deduction systems is presented. The theory is illustrated with deduction systems based on the resolution calculus, in particular with clause graphs. This theory distinguishes four constituents of a deduction system: ffl the logic, which establishes a notion of semantic entailment; ffl the calculus, whose rules of inference provide the syntactic counterpart of entailment; ffl the logical state transition system, which determines the representation of formulae or sets of formulae together with their interrelationships, and also may allow additional operations reducing the search space; ffl the control, which comprises the criteria used to choose the most promising from among all applicable inference steps. Much of the standard material on resolution is presented in this framework. For the last two levels many alternatives are discussed. Appropriately adjusted notions of soundness, completeness, confluence, and Noetherianness are introduced in order to characterize...

We analyze the search efficiency of a number of common refutational theorem proving strategies for first-order logic. Search efficiency is concerned with the total number of proofs and partial proofs generated, rather than with the sizes of the proofs. We show 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, and we discuss it in some detail. We show some advantages of A-ordering over other strategies, which may help to explain its efficiency in practice. We also point out some of its combinatorial inefficiencies, especially in relation to goal-sensitivity and irrelevant clauses. In addition, SLD-reso...

When the Model Elimination (ME) procedure was rst proposed, a notion of lemma was put forth as a promising augmentation to the basic complete proof procedure. Here the lemmas that are used are also discovered by the procedure in the same proof run. Several implementations of ME now exist but only a 1970's implementation explicitly examined this lemma mechanism, with indi erent results. We report on the successful use of lemmas using the METEOR implementation of ME. Not only does the lemma device permit METEOR to obtain proofs not otherwise obtainable by METEOR, or any other ME prover not using lemmas, but some well-known challenge problems are solved. We discuss several of these more di cult problems, including two challenge problems for uniform general-purpose provers, where METEOR was rst in obtaining the proof. The problems are not selected simply to show o the lemma device, but rather to understand it better. Thus, we choose problems with widely di erent characteristics, including one where very few lemmas are created automatically, the opposite of normal behavior. This selection points out the potential of, and the problems with, lemma use. The biggest problem normally is the selection of appropriate lemmas to retain from the large number generated. 1

This paper highlights a phenomenon that causes deductively learned knowledge to be harmful when used for problem solving. The problem occurs when deductive problem solvers encounter a failure branch of the search tree. The backtracking mechanism of such problem solvers will force the program to traverse the whole subtree thus visiting many nodes twice - once by using the deductively learned rule and once by using the rules that generated the learned rule in the first place. We suggest an approach called utilization filtering to solve that problem. Learners that use this approach submit to the problem solver a filter function together with the knowledge that was acquired. The function decides for each problem whether to use the learned knowledge and what part of it to use. We have tested the idea in the context of a lemma learning system, where the filter uses the probability of a subgoal failing to decide whether to turn lemma usage off. Experiments show an improvement of performance b...

In this paper we describe the theorem prover METEOR which is a high-performance Model Elimination prover running in sequential, parallel and distributed computing environments. METEOR has a very high inference rate, but as is the case with better chess-playing programs speed alone is not sufficient when exploring large search spaces; intelligent search is necessary. We describe modifications to traditional iterative deepening search mechanisms whose implementation in METEOR result in performance improvements of several orders of magnitude and that have permitted the discovery of proofs unobtainable by top-down Model Elimination provers. 1 Introduction Model Elimination (ME) [Lov68, Lov69, Lov78] is the basis for the underlying inference mechanism of several high-performance theorem provers. The design of these provers is adapted from the architecture of the WAM (Warren Abstract Machine) [War83] --- the de facto standard for efficient Prolog implementations. Such provers includ...