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...

In this paper techniques are developed and compared which increase the inferential power of tableau systems for classical first-order logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a generalization of model elimination. Since connection tableau calculi are among the weakest proof systems with respect to proof compactness, and the (backward) cut rule is not suitable for the firstorder case, we study alternative methods for shortening proofs. The techniques we investigate are the folding up and the folding down operation. Folding up represents an efficient way of supporting the basic calculus, which is top-down oriented, with lemmata derived in a bottom-up manner. It is shown that both techniques can also be viewed as controlled integrations of the cut rule. In order to remedy the additional redundancy imported into tableau proof procedures by the new inference rules, we develop and apply an extension of the regularity condition on tableaux and the mechanism of anti-lemmata which realizes a subsumption concept on tableaux. Using the framework of the theorem prover SETHEO, we have implemented three new proof procedures which overcome the deductive weakness of cut-free tableau systems. Experimental results demonstrate the superiority of the systems with folding up over the cut-free variant and the one with folding down.

By determining what added assumptions would suffice to make the logical form of a sentence in natural language provable, abductive inference can be used in the interpretation of sentences to determine what information should be added to the listener's knowledge, i.e., what he should learn from the sentence. This is a comparatively new application of mechanized abduction. A new form of abduction---least specific abduction---is proposed as being more appropriate to the task of interpreting natural language than the forms that have been used in the traditional diagnostic and design-synthesis applications of abduction. The assignment of numerical costs to axioms and assumable literals permits specification of preferences on different abductive explanations. A new Prolog-like inference system that computes abductive explanations and their costs is given. To facilitate the computation of minimum-cost explanations, the inference system, unlike others such as Prolog, is designed to avoid the repeated use of the same instance of an axiom or assumption.

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...

Since linear resolution with clause ordering is incomplete for consequence-finding, it has been used mainly for proof-finding. In this paper, we re-evaluate consequence-finding. Firstly, consequence-finding is generalized to the problem in which only interesting clauses having a certain property (called characteristic clauses) should be found. Then, we show how adding a skip rule to ordered linear resolution makes it complete for consequence-finding in this general sense. Compared with set-of-support resolution, the proposed method generates fewer clauses to find such a subset of consequences. In the propositional case, this is an elegant tool for computing the prime implicants/implicates. The importance of the results lies in their applicability to a wide class of AI problems including procedures for nonmonotonic and abductive reasoning and truth maintenance systems. 1

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A parallel satis ability testing algorithm called Parallel Modoc is presented. Par-allel Modoc is based on Modoc, which is based on propositional Model Elimination with an added capability to prune away certain branches that cannot lead to a successful subrefutation. The pruning information is encoded in a partial truth assignment called an autarky. Parallel Modoc executes multiple instances of Modoc as separate processes and allows processes to cooperate by sharing lemmas and autarkies as they are found. When a Modoc process nds a new autarky or a new lemma, it makes the informa-tion available to other Modoc processes via a \blackboard". Combining autarkies generally is not straightforward because two autarkies found by two separate pro-cesses may have con icting assignments. The paper presents an algorithm to combine two arbitrary autarkies to form a larger autarky. Experimental results show that for many of the formulas, Parallel Modoc achieves speedup greater than the number of processors. Formulas that could not be solved in an hour by Modoc were often solved by Parallel Modoc in the order of minutes, and in some cases, in seconds.

This paper describes the architecture, implementation and performance, of a heterogeneous parallel deduction system (HPDS). The HPDS uses multiple deduction components, each of which attempts to find a refutation of the same input set, but using different deduction formats. The components cooperate by distributing clauses they generate to other components. The HPDS has been implemented in Prolog-D-Linda. Prolog-D-Linda provides appropriate data transfer and synchronisation facilities for implementing parallel deduction systems. The performance of the HPDS has been investigated. Parallel Deduction Systems A parallel deduction system is one in which multiple deduction components run in parallel on separate processors. This is distinct from those deduction systems which run multiple deduction components alternately, such as the unit preference system [Wos, Carlson & Robinson#G.A.,#1964], and those which are only conceptually parallel systems. Parallel deduction systems can be categorised ...

Goal-sensitive resolution methods, such as Model Elimination, have been observed to have a higher degree of search redundancy than model-search methods, Therefore, resolution methods have not been seen in high performance propositional satis ability testers. A method to reduce search redundancy in goal-sensitive resolution methods is introduced. The idea at the heart of the method is to attempt to construct a refutation and a model simultaneously and incrementally, based on sub-search outcomes. The method exploits the concept of \autarky", which can be informally described as a \self-su cient " model for some clauses, but which does not a ect the remaining clauses of the formula. Incorporating this method into Model Elimination leads to an algorithm called Modoc. Modoc is shown, both analytically and experimentally, to be faster than Model Elimination by an exponential factor. Modoc, unlike Model Elimination, is able to nd a model if it fails to nd a refutation, essentially by combining autarkies. Unlike the pruning strategies of most re nements of resolution, autarky-related pruning does not prune any successful refutation; it only prunes attempts that ultimately will be unsuccessful; consequently, it will not force the underlying Modoc search to nd an unnecessarily long refutation. To prove correctness and other properties, a game characterization of refutation search isintroduced, which demonstrates

Classical STRIPS-style planning problems are formulated as theorems to be proven from a new point of view. The result for a refutation-based theorem prover may be a propositional formula that is to be proven unsatisfiable. This formula is identical to the formula that may be derived directly by various "sat compilers", but the theorem-proving view provides valuable additional information not in the formula: namely, the theorem to be proven. Traditional satisfiability methods, most of which are based on model search, are unable to exploit this additional information. However, a new algorithm, called "Modoc", is able to exploit this information and has achieved performance comparable or superior to the fastest known satisfiability methods, including stochastic search methods, on planning problems that have been reported by other researchers, as well as formulas derived from other applications. Unlike most theorem provers, Modoc performs well on both satisfiable and unsatisfiable formulas...