Showing papers by "Franz Baader published in 1991"

A scheme for integrating concrete domains into concept languages

Abstract: A drawback which concept languages based on KL-ONE have is that all the terminological knowledge has to be defined on an abstract logical level. In many applications, one would like to be able to refer to concrete domains and predicates on these domains when defining concepts. Examples for such concrete domains are the integers, the real numbers, or also non-arithmetic domains, and predicates could be equality, inequality, or more complex predicates. In the present paper we shall propose a scheme for integrating such concrete domains into concept languages rather than describing a particular extension by some specific concrete domain. We shall define a terminological and an assertional language, and consider the important inference problems such as subsumption, instantiation, and consistency. The formal semantics as well as the reasoning algorithms are given on the scheme level. In contrast to existing KL-ONE based systems, these algorithms will be not only sound but also complete. They generate subtasks which have to be solved by a special purpose reasoner of the concrete domain.

Augmenting concept languages by transitive closure of roles: an alternative to terminological cycles

Abstract: In Baader (1990,1990a), we have considered different types of semantics for terminologicial cycles in the concept language TLQ which allows only conjunction of concepts and value-restrictions It turned out that greatest fixed-point semantics (gfp-semantics) seems to be most appropriate for cycles in this language In the present paper we shall show that the concept defining facilities of FLO with cyclic definitions and gfp-semantics can also be obtained in a different way One may replace cycles by role definitions involving union, composition, and transitive closure of roles This proposes a way of retaining, in an extended language, the pleasant features of gfp-semantics for FLQ with cyclic definitions without running into the troubles caused by cycles in larger languages Starting with the language ALC of Schmidt-Schaus&Smolka (1988)--which allows negation, conjunction and disjunction of concepts as well as value-restrictions and exists-in-restrictions--we shall disallow cyclic concept definitions, but instead shall add the possibility of role definitions involving union, composition, and transitive closure of roles In contrast to other terminological KR-systems which incorporate the transitive closure operator for roles, we shall be able to give a sound and complete algorithm for concept subsumption

Qualifying number restrictions in concept languages

Abstract: We investigate the subsumption problem in logic-based knowledge representation languages of the KL-ONE family. The language presented in this paper provides the constructs for conjunction, disjunction, and negation of concepts, as well as qualifying number restrictions. The latter ones generalize the well-known role quantifications (such as value restrictions) and ordinary number restrictions, which are present in almost all KL-ONE based systems. Until now, only little attempts were made to integrate qualifying number restrictions into concept languages. It turns out that all known subsumption algorithms which try to handle these constructs are incomplete, and thus detecting only few subsumption relations between concepts. We present a subsumption algorithm for our language which is sound and complete. Subsequently we discuss why the subsumption problem in this language is rather hard from a computational point of view. This leads to an idea of how to recognize concepts which cause tractable problems.

A Terminological Knowledge Representation System with Complete Inference Algorithms

Abstract: The knowledge representation system kl-one first appeared in 1977. Since then many systems based on the idea of kl-one have been built. The formal model-theoretic semantics which has been introduced for kl-one languages [BL84] provides means for investigating soundness and completeness of inference algorithms. It turned out that almost all implemented kl-one systems such as back, kl-two, loom, nikl, sb-one use sound but incomplete algorithms.

KRIS: Knowledge Representation and Inference System

Abstract: The knowledge representation system KL-ONE first appeared in 1977. Subsequently many systems based on the idea of KL-ONE have been built. The formal model-theoretic semantics which has been introduced for KL-ONE languages [9] provides means for investigating soundness and completeness of inference algorithms. It turned out that almost all implemented KL-ONE systems such as BACK, KL-TWO, LOOM, NIKL, SB-ONE use sound but incomplete algorithms.

Profitlich: Terminological knowledge representation: A proposal for a terminological logic

Abstract: A roller particularly adapted for driving sheet material through a folding machine, which sheet material is folded by the folding machine. The roller has an elongated cylindrical body, with an axle connected to the body for rotatably supporting the body. The cylindrical body has a pair of recessed fastening flats in the body, each being adjacent to opposite ends of the body. An elongated strip friction belt has one end removably fastened to one of the fastening flats. The belt is helically wound on the cylindrical body, with adjacent edges of the belt abutting. The belt has the other end releasably fastened to the other fastening belt. The ends of the friction belt are easily released to allow the belt to be removed and replaced by a replacement belt.

Unification, weak unification, upper bound, lower bound, and generalization problems

Abstract: We introduce E-unification, weak E-unification, E-upper bound, E-lower bound, and E-generalization problems, and the corresponding notions of unification, weak unification, upper bound, lower bound, and generalization type of an equational theory. When defining instantiation preorders on solutions of these problems, one can compared substitutions w.r.t. their behaviour on all variables or on finite sets of variables. We shall study the effect which these different instantiation preorders have on the existence of most general or most specific solutions of E-unification, weak E-unification, and E-generalization problems. In addition, we shall elucidate the subtle difference between most general unifiers and coequalizers, and we shall consider generalization in the class of commutative theories.

General A- and AX-Unification via Optimized

Abstract: In a recent paper [BS91] we introduced a new unification algorithm for the combination of disjoint equational theories. Among other consequences we mentioned (1) that the algorithm provides us with a decision procedure for the solvability of general A- and AI-unification problems and (2) that Kapur and Narendran's result about the NP-decidability of the solvability of general AC- and ACI-unification problems (see [KN91]) may be obtained from our results. In [BS91] we did not give detailled proofs for these two consequences. In the present paper we will treat these problems in more detail. Moreover, we will use the two examples of general A- and AI-unification for a case study of possible optimizations of the basic combination procedure.

Adding Homomorphisms to Commutative/Monoidal Theories or How Algebra Can Help in Equational Unification

Abstract: In this paper we consider the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. This class has been introduced by the authors independently of each other as commutative theories (Baader) and monoidal theories (Nutt). The class encompasses important examples like the theories of abelian monoids, idempotent abelian monoids, and abelian groups.

General A- and AX-unification via optimized combination procedures

Abstract: In a recent paper [BS91] we introduced a new unification algorithm for the combination of disjoint equational theories. Among other consequences we mentioned (1) that the algorithm provides us with a decision procedure for the solvability of general A- and AI-unification problems and (2) that Kapur and Narendran's result about the NP-decidability of the solvability of general AC- and ACI-unification problems (see [KN91]) may be obtained from our results. In [BS91] we did not give detailled proofs for these two consequences. In the present paper we will treat these problems in more detail. Moreover, we will use the two examples of general A- and AI-unification for a case study of possible optimizations of the basic combination procedure.