Opening the AC-Unification Race

This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems

A System F accounting for scalars

The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F-like type system for the linear-algebraic lambda-calculus. We show that this "scalar" type system enjoys both the subject-reduction property and the strong-normalisation property, our main technical results. The latter yields a significant simplification of the linear-algebraic lambda-calculus itself, by removing the need for some restrictions in its reduction rules. But the more important, original feature of this scalar type system is that it keeps track of 'the amount of a type' that is present in each term. As an example of its use, we shown that it can serve as a guarantee that the normal form of a term is barycentric, i.e that its scalars are summing to one

A Prolog Technology Theorem Prover: A New Exposition and Implementation in Prolog

the full first-order predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, depth-first iterative-deepening search instead of unbounded depth-first search to make the search strategy complete, and the model elimination reduction rule that is added to Prolog inferences to make the inference system complete. This paper describes a new Prolog-based implementation of PTTP. It uses three compile-time transformations to translate formulas into Prolog clauses that directly execute, with the support of a few run-time predicates, the model elimination procedure with depth-first iterative-deepening search and unification with the occurs check. Its high performance exceeds that of Prolog-based PTTP interpreters, and it is more concise and readable than the earlier Lisp-based compiler, which makes it superior for expository purposes. Examples of inputs and outputs of the compile-time transformations provide an easy and precise way to explain how PTTP works. This Prolog-based version makes it easier to incorporate PTTP theorem-proving ideas into Prolog programs. Some suggestions are made on extensions to Prolog that could be used to improve PTTP's performance

Automated Deduction by Theory Resolution

Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theorem-proving program, thereby making it unnecessary to resolve directly upon axioms of the theory. This can greatly reduce the length of proofs and the size of the search space. Theory resolution effects a beneficial division of labor, improving the performance of the theorem prover and increasing the applicability of the specialized reasoning procedures. Total theory resolution utilizes a decision procedure that is capable of determining unsatisfiability of any set of clauses using predicates in the theory. Partial theory resolution employs a weaker decision procedure that can determine potential unsatisfiability of sets of literals. Applications include the building in of both mathematical and special decision procedures, e.g., for the taxonomic information furnished by a knowledge representation system. Theory resolution is a generalization of numerous previously known resolution refinements. Its power is demonstrated by comparing solutions of "Schubert's Steamroller" challenge problem with and without building in axioms through theory resolution. 1

A Prolog Technology Theorem Prover: Implementation by an Extended Prolog Compiler

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..