Wolfgang Bibel

SETHEO: a high-performance theorem prover

A sound and complete theorem prover for first-order logic is presented, which is based on the connection method. The inference machine is implemented using PROLOG technology, an approach taken also with other systems, most prominently with Stickels PTTP. But SETHEO differs from those in essential characteristics, among which are the following ones. It incorporates a powerful preprocessing module for a reduction of the input formula. The main proof procedure is realized as a variant of Warrens abstract machine. For search pruning we perform subsumption and regular proofs. Factorization, lemma generation, and the application of proof schemata are offered as options. The entire system is implemented in C and is running on several machines. The most remarkable feature of SETHEO is its performance of up to 70 Klips on a SUN SPARC station 1 with 12 Mips. The paper comprises the theoretical background, the system architecture as well as details of the implementation.

On Matrices with Connections

Theorem proving is considered as the problem of verifying that each path through a matrix consisting of a set of clauses can be made complementary. By introducing connections to such a matrix the followmg three results are derived from that conceptual basis. First, a sunple and short proof for the consistency and completeness of the connection graph procedure as given Second, a macrosimplificatlon rule for the preparatory step of any ATP-method is defined which, like the deletion or subsumptlon rules, properly reduces a given matrix whenever it apphes It can be regarded as a generalization to arbitrary clauses of the well-known fact that sets of two-lateral clauses can be decided quickly Finally, m view of the relation between resolution-based and natural-deduction-based methods, a constructive transformation is specified which explicitly relates each resolution step to a pair of complementary literals in an axiom of a natural deduction, and wce versa. Although the treatment is restricted to the ground case, it is obvious that all results can be easily lifted to the general case in the usual way

Matings in matrices

This paper gives an overview over the connection method in Automated Theorem Proving, developed earlier by the author. Its prominent features are illustrated with a number of examples. These features provide it with provable advantages over any standard proof method. Some of them also have been obtained by Andrews in an independent approach using matings. The relationship between these two methods is clarified.

Syntax-directed, semantics-supported program synthesis

A number of strategies for the synthesis of algorithms from a given input-output specification of a problem are presented which are centered around a few basic principles. It has been verified for more than ten different algorithms that their uniform application in all cases results in a successful deductive synthesis. They include a spanning-tree algorithm, a graph-circuits algorithm, a finding-the-ith-smallest-element algorithm, and a linear (string) pattern-matching algorithm. Two of them are presented here.

Constraint satisfaction from a deductive viewpoint (Research Note)

Abstract This paper reports the result of testing the authors proof techniques on the class of constraint satisfaction problems (CSPs). This experiment has been successful in the sense that a completely general proof technique turns out to behave well also for this special class of problems which itself has received considerable attention in the community. So, at the same time, the paper happens to present a new (deductive) mechanism for solving constraint satisfaction problems that is of interest in its own right. This mechanism may be characterized as a bottom-up, lazy-evaluation technique which reduces any such problem to the problem of evaluating a database expression typically involving a number of joins. A way of computing such an expression is proposed.

Fundamentals of Artificial Intelligence

This book covers the following topics: knowledge representation; features of knowledge; knowledge processing; deduction and computation; an introduction to automated deduction; fundamental mechanisms in machine learning and inductive inference; methods of automated reasoning; knowledge programming; term rewriting as a basis for the design of a functional and parallel programming language; a case study: the language FP2; and concurrent PROLOG: a progress report.

A comparative study of several proof procedures

Abstract In this paper three algorithms for testing the complementarity of a matrix (representing a propositional formula) are developed in stages. Any of these algorithms is distinguished from its predecessor by a specific feature (linearity, jump, non-normal form) which endows it with a provable advantage w.r.t. its performance. For well-known proof procedures it is shown that they can be simulated by at least one of these algorithms which provides a hierarchy among several ATP-methods.

Tautology testing with a generalized matrix reduction method

Abstract A formalization of the tautology problem in terms of matrices is given. From that a generalized matrix reduction method is derived. Its application to a couple of selected examples indicates a relatively efficient behaviour in testing the validity of a given formula in propositional logic—not only for machines but also for humans. A further result from that formalization is a reduction of the tautology problem to a part of Presburger arithmetic which involves formulas of the ∀∃∃⋯∃-type where all quantifiers have finite range.

Plan Generation by Linear Proofs: On Semantics

In realistic circumstances reasoning almost always encounters changes in time. In [BIB 86] a logical calculus—the so-called linear proofs—was proposed, which integrates such changes in the form of transition rules into a classical logic framework. This paper presents three ways towards the semantics for this calculus. First, we prove its soundness through a translation into a suitable situational calculus. Second, we interpret it in a particular modal semantics. Third, we propose how to extend the semantics of first-order logic for the expression of planning problems.

A Multi-level Approach to Program Synthesis

We present an approach to a coherent program synthesis system which integrates a variety of interactively controlled and automated techniques from theorem proving and algorithm design at different levels of abstraction. Besides providing an overall view we summarize the individual research results achieved in the course of this development.