Showing papers by "Franz Baader published in 1996"

Using automata theory for characterizing the semantics of terminological cycles

Abstract: In most of the implemented terminological knowledge representation systems it is not possible to state recursive concept definitions, so-called terminological cycles. One reason is that it is not clear what kind of semantics to use for such cycles. In addition, the inference algorithms used in such systems may go astray in the presence of terminological cycles. In this paper we consider terminological cycles in a very small terminological representation language. For this language, the effect of the three types of semantics introduced by B. Nebel can completely be described with the help of finite automata. These descriptions provide for a rather intuitive understanding of terminologies with recursive definitions, and they give an insight into the essential features of the respective semantics. In addition, one obtains algorithms and complexity results for the subsumption problem and for related inference tasks. The results of this paper may help to decide what kind of semantics is most appropriate for cyclic definitions, depending on the representation task.

A Formal Definition for the Expressive Power of Terminological Knowledge Representation Languages

Abstract: The notions 'expressive power' or 'expressiveness' of knowledge representation languages (KR languages) can be found in most papers on knowledge representation; but these terms are usually just employed in an intuitive sense. The papers contain only informal descriptions of what is meant by expressiveness. There are several reasons that speak in favour of a formal definition of expressiveness: for example, if we want to show that certain expressions in one language cannot be expressed in another language, we need a strict formalism that can be used in mathematical proofs. Even though we shall only consider terminological KR languages—i.e. KR languages descending from the original system KL-ONE—in our motivation and in the examples, the definition of expressive power that will be given in this paper can be used for all KR languages with Tarski-style model-theoretic semantics. This definition will shed a new light on the tradeoff between expressiveness of a representation language and its computational tractability. There are KR languages with identical expressive power, but different complexity results for reasoning, which comes from the fact that sometimes the tradeoff lies between convenience and computational tractability. The definition of expressive power will be applied to compare various terminological KR languages known from the literature with respect to their expressiveness. This will yield examples for how to utilize the definition both in positive proofs— that is, proofs where it is shown that one language can be expressed by another language—and, more interestingly, in negative proofs—which show that a given language cannot be expressed by the other language.

A Description Logic Based Schema for the Classification of Medical Data

Abstract: The European Galen project aims to promote the sharing and re-use of medical data by providing a concept model which can be used by application designers as a exible and extensible classiication schema. A description logic style terminological knowledge representation system called Grail has been developed speciically for this task. Using a description logic based schema has a number of important beneets including coherence checking, schema enrichment and query optimisation. In order to support a variety of design requirements Grail includes transitive closure of roles and general concept inclusions. Replacing the Grail classiier's existing structural subsumption algorithm with a sound, provably complete and decidable tableaux calculus based algorithm would have many attractions if the intractability problem could be mitigated by suitable optimisations. The optimisation of non-deterministic constraint expansion would be of particular importance as large numbers of these constraints can be introduced by general concept inclusions. Both intelligent back-tracking and the use of meta-knowledge to guide constraint expansion are being studied as possible methods of tackling this problem.

Number Restrictions on Complex Roles in Description Logics

Abstract: Number restrictions are concept constructors that are available in almost all implemented description logic systems. However, even though there has lately been considerable effort on integrating expressive role constructors into description logics, the roles that may occur in number restrictions are usually of a very restricted type. Until now, only languages with number restrictions on atomic roles and inversion of atomic roles, or with number restrictions on intersection of atomic roles have been investigated in detail. In the present paper, we increase the expressive power of description languages by allowing for more complex roles in number restrictions. As role constructors, we consider composition of roles (which will be present in all our languages), and intersection, union and inversion of roles in different combinations. We will present two decidability results (for the basic language that extends A~U by number restrictions on roles with composition, and for one extension of this language), and three undecidability results for three other extensions of the basic language. 1 Motivation and introduction Description logics is a field of knowledge representation in which there is a rather close interaction between theory and practice. On the one hand, there are various implemented systems based on description logics, which offer a palette of description formalisms with differing expressive power [Peltason,1991; Brachman et a/.,1991; MacGregor,1991; Mays et al.,1991; Baader et al.,1994; Bresciani et al.,1995]. On the other hand, the computational properties (like de*This author is supported by the Deutsche Forschungsgemeinschaft under Grant No. Sp 230\ 6-6. cidability, complexity) of various description formalisms have thoroughly been investigated [Nebel,1988; Schmidt-Schauss,1989; Patel-Schneider,1989; Donini et a/.,1991a; 1991b]. These investigation were often motivated by the use of certain constructors in systems or the need for these constructors in specific applications [Baader & Hanschke,1993; Franconi,1994], and the results have influenced the design of new systems. The terminological formalisms of knowledge representation systems based on description logics provide constructors that can be used to build complex concepts and roles out of atomic concepts (unary predicates) and roles (binary predicates). Until recently, the main emphasis, both in implemented systems and in theoretical research, was on constructors for building complex concepts. The need for rich role constructors in certain application domains (such as representing rich schema languages for databases [Calvanese et a/.,1994; 1995], or domains that require the appropriate modeling of part-whole relations [Padgham & Lambrix,1994; Artale et a/.,1994; Sattler,1995]) has triggered research on description languages that also provide for expressive role constructors [Baader,1990; De Giacomo & Lenzerini,1995]. These investigations were facilitated by the observation that the formalisms considered in description logics are very similar to certain modal logics [Schild,1991; De Giacomo & Lenzerini,1994]. In particular, well-known modal logics, such as propositional dynamic logics (PDL) and its extensions [Fischer & Ladner,1979; BenAri et a/.,1982; Harel,1984], provide for role constructors like composition, union, transitive closure, and inversion. Number restrictions are concept constructors that are available in almost all implemented description logic systems. They allow to restrict the number of role successors of an individual w.r.t, a given role. For example, if has-child is an atomic role and person is an atomic concept, then we can describe all persons having at most 2 children by the concept person ~ (_< 2 has-child). In contrast to the rather prominent rSle that number restrictions play in deFrom: AAAI Technical Report WS-96-05. Compilation copyright © 1996, AAAI (www.aaai.org). All rights reserved. scription logics, the corresponding constructors in modal logic--so-called "graded modalities" [Fine,1972; van der HoekD intersection of roles can prohibit that a parent marries his/her own child: (_< 0 has-childYlis-married-to); union and composition can be used to describe that all children have the same name as their parent: (= 1 has-name [_] (has-childohas-name)). Number restrictions on complex roles are not only of interest in toy examples like the family domain used above. Our original motivation for considering these constructs comes from a process engineering application, where planning and optimization of large chemical plants is supported by building process models. The engineering knowledge concerning standard building blocks of these models is to be represented in a description logic system. For example, the concept (device V1 (= 1 controlled-by))describes devices that are controlled by a single control unit. If we want to describe a device such that all devices connected to it are controlled by the same control unit, we need composition in the number restriction: (device [-1 (= 1 connected-to o controlled-by)). To assure that the device itself is also controlled by the same unit controlling the devices connected to it, we additionally need union in the number restriction: (device [-1 (---1 controlled-by U connected-toocontrolled-by)). Inversion of roles comes in if we need the role controls as well. There are also more complex properties of devices and other parts of process models that could be expressed with number restrictions on complex roles. However, to be useful in practice, it is not sufficient to have a description language that can just be used to represent the relevant properties of objects. The description logic system must also be able to reason about the descriptions. As a positive result in this direction, we show that the subsumption and the satisfiability problem for the language AggAf(o), which extends AlL with number restrictions on roles built with composition, are decidable. On the other hand, three extensions of this language turn out to be undecidable: Af-E+with number restrictions on roles built with composition and union; MEg with number restrictions on roles built with composition and intersection; and MEg with number restrictions on roles built with composition, union, and inversion. However, if union and intersection are restricted to role chains of the same length, then we obtain a decidable extension of Af_~. In the next section, we introduce syntax and semantics of the concept and role constructors that will be considered. Section 3.1 describes the algorithm that decides satisfiability of AE_gAf(o)-concepts, and Section 3.2 extends this decidability result to number restrictions on union and intersection of role chains of the same length. The subsequent section sketches the undecidability proofs, which all use a reduction of the domino problem. In Section 5, we mention related decidability and undecidability results from modal and description logics. 2 Concept and role constructors We define syntax and semantics of all the constructors considered in the present paper, and introduce the description languages that will be investigated in more detail. Definition 1 Starting with atomic roles from a set NR of role names, complex roles are built using the role constructors composition (RoS), union (R 0 intersection (R ~ S), inversion (R-I), and transitive closure ( R+ ). The set of Mr_E-concepts is built from a set Nc of concept names using the concept constructors disjunction (C U D), conjunction (C N D), negation (-~C),

Description Logics with Symbolic Number Restrictions.

Abstract: Motivated by a chemical engineering application, we introduce an extension of the concept description language ALCN by symbolic number restrictions. This first extension turns out to have an undecidable concept satisfiability problem. For a restricted language—whose expressive power is sufficient for our application— we show that concept satisfiability is decidable.

Knowledge Representation in Process Engineering

Abstract: In process engineering, as in many other application domains, the domain specific knowledge is far too complex to be described entirely using description logics. Hence this knowledge is often stored using an object-oriented system, which, because of its high expressiveness, provides only weak inference services. In particular, the process engineers at RWTH Aachen have developed a frame-like language for describing process models. In this paper, we investigate how the powerful inference services provided by a DL system can support the users of this frame-based system. In addition, we consider extensions of description languages that are necessary to represent the relevant process engineering knowledge. The application domain Process engineering is concerned with the design and operation of chemical processes that take place in large chemical plants. This engineering task includes activities like deciding on an appropriate flowsheet structure (e.g. configuration of reaction and separation systems), mathematical modeling and simulation of the process behavior (e.g. stating mathematical equations and performing numerical simulations), sizing of components (like reactors, heat exchangers etc.) well as budgeting and engineering economics. These highly complex tasks can be supported by building computer models of the chemical plants and processes, using appropriate software tools such CAD, decision support systems and numerical tools. Rather than designing each new model from scratch, one wants a system that offers standard building blocks that can easily be put together. Standard building blocks [Marquardt, 1994; Bogusch&Marquardt, 1995] are objects representing

Combination problems for commutative/monoidal theories or how algebra can help in equational unification

Abstract: We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the authors independently of each other as “commutative theories” (Baader) and “monoidal theories” (Nutt).

Number restrictions on complex roles in description logics: a preliminary report

Abstract: Number restrictions are concept constructors that are available in almost all implemented description logic systems. However, even though there has lately been considerable effort on integrating expressive role constructors into description logics, the roles that may occur in number restrictions are usually of a very restricted type. Until now, only languages with number restrictions on atomic roles and inversion of atomic roles, or with number restrictions on intersection of atomic roles have been investigated in detail. In the present paper, we increase the expressive power of description languages by allowing for more complex roles in number restrictions. As role constructors, we consider composition of roles (which will be present in all our languages), and intersection, union and inversion of roles in different combinations. We will present two decidability results (for the basic language that extends A~U by number restrictions on roles with composition, and for one extension of this language), and three undecidability results for three other extensions of the basic language. 1 Motivation and introduction Description logics is a field of knowledge representation in which there is a rather close interaction between theory and practice. On the one hand, there are various implemented systems based on description logics, which offer a palette of description formalisms with differing expressive power [Peltason,1991; Brachman et a/.,1991; MacGregor,1991; Mays et al.,1991; Baader et al.,1994; Bresciani et al.,1995]. On the other hand, the computational properties (like de*This author is supported by the Deutsche Forschungsgemeinschaft under Grant No. Sp 230\ 6-6. cidability, complexity) of various description formalisms have thoroughly been investigated [Nebel,1988; Schmidt-Schauss,1989; Patel-Schneider,1989; Donini et a/.,1991a; 1991b]. These investigation were often motivated by the use of certain constructors in systems or the need for these constructors in specific applications [Baader & Hanschke,1993; Franconi,1994], and the results have influenced the design of new systems. The terminological formalisms of knowledge representation systems based on description logics provide constructors that can be used to build complex concepts and roles out of atomic concepts (unary predicates) and roles (binary predicates). Until recently, the main emphasis, both in implemented systems and in theoretical research, was on constructors for building complex concepts. The need for rich role constructors in certain application domains (such as representing rich schema languages for databases [Calvanese et a/.,1994; 1995], or domains that require the appropriate modeling of part-whole relations [Padgham & Lambrix,1994; Artale et a/.,1994; Sattler,1995]) has triggered research on description languages that also provide for expressive role constructors [Baader,1990; De Giacomo & Lenzerini,1995]. These investigations were facilitated by the observation that the formalisms considered in description logics are very similar to certain modal logics [Schild,1991; De Giacomo & Lenzerini,1994]. In particular, well-known modal logics, such as propositional dynamic logics (PDL) and its extensions [Fischer & Ladner,1979; BenAri et a/.,1982; Harel,1984], provide for role constructors like composition, union, transitive closure, and inversion. Number restrictions are concept constructors that are available in almost all implemented description logic systems. They allow to restrict the number of role successors of an individual w.r.t, a given role. For example, if has-child is an atomic role and person is an atomic concept, then we can describe all persons having at most 2 children by the concept person ~ (_< 2 has-child). In contrast to the rather prominent rSle that number restrictions play in deFrom: AAAI Technical Report WS-96-05. Compilation copyright © 1996, AAAI (www.aaai.org). All rights reserved. scription logics, the corresponding constructors in modal logic--so-called "graded modalities" [Fine,1972; van der HoekD intersection of roles can prohibit that a parent marries his/her own child: (_< 0 has-childYlis-married-to); union and composition can be used to describe that all children have the same name as their parent: (= 1 has-name [_] (has-childohas-name)). Number restrictions on complex roles are not only of interest in toy examples like the family domain used above. Our original motivation for considering these constructs comes from a process engineering application, where planning and optimization of large chemical plants is supported by building process models. The engineering knowledge concerning standard building blocks of these models is to be represented in a description logic system. For example, the concept (device V1 (= 1 controlled-by))describes devices that are controlled by a single control unit. If we want to describe a device such that all devices connected to it are controlled by the same control unit, we need composition in the number restriction: (device [-1 (= 1 connected-to o controlled-by)). To assure that the device itself is also controlled by the same unit controlling the devices connected to it, we additionally need union in the number restriction: (device [-1 (---1 controlled-by U connected-toocontrolled-by)). Inversion of roles comes in if we need the role controls as well. There are also more complex properties of devices and other parts of process models that could be expressed with number restrictions on complex roles. However, to be useful in practice, it is not sufficient to have a description language that can just be used to represent the relevant properties of objects. The description logic system must also be able to reason about the descriptions. As a positive result in this direction, we show that the subsumption and the satisfiability problem for the language AggAf(o), which extends AlL with number restrictions on roles built with composition, are decidable. On the other hand, three extensions of this language turn out to be undecidable: Af-E+with number restrictions on roles built with composition and union; MEg with number restrictions on roles built with composition and intersection; and MEg with number restrictions on roles built with composition, union, and inversion. However, if union and intersection are restricted to role chains of the same length, then we obtain a decidable extension of Af_~. In the next section, we introduce syntax and semantics of the concept and role constructors that will be considered. Section 3.1 describes the algorithm that decides satisfiability of AE_gAf(o)-concepts, and Section 3.2 extends this decidability result to number restrictions on union and intersection of role chains of the same length. The subsequent section sketches the undecidability proofs, which all use a reduction of the domino problem. In Section 5, we mention related decidability and undecidability results from modal and description logics. 2 Concept and role constructors We define syntax and semantics of all the constructors considered in the present paper, and introduce the description languages that will be investigated in more detail. Definition 1 Starting with atomic roles from a set NR of role names, complex roles are built using the role constructors composition (RoS), union (R 0 intersection (R ~ S), inversion (R-I), and transitive closure ( R+ ). The set of Mr_E-concepts is built from a set Nc of concept names using the concept constructors disjunction (C U D), conjunction (C N D), negation (-~C),

Proceedings of the Workshop on Knowledge Representation and Configuration, WRKP'96

Abstract: We consider two extensions of traditional terminological knowledge representation languages, which are motivated by technical applications such as configuration of technical systems. The first extension integrates "concrete" domains (such as numbers) and concrete predicates on these domains into the abstract terminological language. The second extension introduces transitive closure of roles, which can, for example, be used to model transitivity of the "part-of" relation.