Jörg H. Siekmann

A short survey on the state of the art in matching and unification problems

1. Motivations There is a wide variety of areas where matching and unification problems arise: (1.1) Databases The user of a (relational) database [22] may logically AND the properties she wants to retrieve or else she may be interested in the NATURAL JOIN [17] of two stored relations. In neither case, she would appreciate if she constantly had to take into account that AND is an associative and commutative operation, or that NATURAL JOIN obeys an associative axiom, which may distribute over some other operation [68].

Paramodulated connection graphs

The connection graph proof procedure of R. Kowalski is extended to the case of equality. The extension is achieved through the introduction of special links connecting those terms that can be paramodulated upon. Completeness and consistency of the resulting proof procedure are shown.

Matching under commutativity

A complete unification algorithm for terms involving a commutative function is presented. The main results are: the unification problem is decidable and the set of unifiers is always finite. The algorithm, as presented, is not minimal, but improves over the naive solution. This paper is a short version of [21.], which contains the proofs omitted here and some additional technical material.

Studien- und Forschungsführer Künstliche Intelligenz

Inhaltsubersicht: Kunstliche Intelligenz.- Forschung und Lehre in der Bundesrepublik.- Forschung und Lehre in Osterreich.- Anmerkungen und Verweise.- Anhang: Fragebogen.

String unification is essentially infinitary

A unifier of two terms s and t is a substitution sigma such that ssigma=tsigma and for first-order terms there exists a most general unifier sigma in the sense that any other unifier delta can be composed from sigma with some substitution lambda, i.e. delta=sigmacirclambda. This notion can be generalised to E-unification , where E is an equational theory, =_{E} is equality under E andsigmaa is an E-unifier if ssigma =_{E}tsigma. Depending on the equational theory E, the set of most general unifiers is always a singleton (as above), or it may have more than one, either finitely or infinitely many unifiers and for some theories it may not even exist, in which case we call the theory of type nullary. String unification (or Lobs problem, Markovs problem, unification of word equations or Makanins problem as it is often called in the literature) is the E-unification problem, where E = {f(x,f(y,z))=f(f(x,y),z)}, i.e. unification under associativity or string unification once we drop the fs and the brackets. It is well known that this problem is infinitary and decidable. Essential unifiers, as introduced by Hoche and Szabo, generalise the notion of a most general unifier and have a dramatically pleasant effect on the set of most general unifiers: the set of essential unifiers is often much smaller than the set of most general unifiers. Essential unification may even reduce an infinitary theory to an essentially finitary theory. The most dramatic reduction known so far is obtained for idempotent semigroups or bands as they are called in computer science: bands are of type nullary, i.e. there exist two unifiable terms s and t, but the set of most general unifiers is not enumerable. This is in stark contrast to essential unification: the set of essential unifiers for bands always exists and is finite. We show in this paper that the early hope for a similar reduction of unification under associativity is not justified: string unification is essentially infinitary. But we give an enumeration algorithm for essential unifiers. And beyond, this algorithm terminates when the considered problem is finitary.

Selection Heuristics, Deletion Strategies and N-Level Terminator Configurations for the Connection Graph Proof Procedure

This report presents a quick overview of the Markgraf Karl Refutation Procedure, an automated theorem prover (TP) currently under development at the University of Karlsruhe, and then concentrates in detail on those parts of the system, which presently determine the choice of the deduction steps to be performed by the system.

Das Karlsruher Beweissystem

Wir beschreiben den gegenwartigen Stand eines Automatischen Beweissystems (ABS), das seit 1977 an der Karlsruher Universitat entwickelt wird [DS77, DS79]. Bis zur endgultigen Fertigstellung des Systems sind weitere drei Jahre geplant.

Forschung und Lehre in der Bundesrepublik

1975 wurde der erste Workshop fur Kunstliche Intelligenz in Deutschland abgehalten und dann regelmasig wiederholt [18]. 1981 wurde dieser Workshop zum ersten Mal als offizielle Fachtagung der Gesellschaft fur Informatik in Bad Honnef durchgefuhrt und wird weiterhin jahrlich abgehalten [19].

A Noetherian Rewrite System for Idempotent Semigroups

Let B be a semigroup with the additional relation n n