614 research outputs found
Unification in the Description Logic EL
The Description Logic EL has recently drawn considerable attention since, on
the one hand, important inference problems such as the subsumption problem are
polynomial. On the other hand, EL is used to define large biomedical
ontologies. Unification in Description Logics has been proposed as a novel
inference service that can, for example, be used to detect redundancies in
ontologies. The main result of this paper is that unification in EL is
decidable. More precisely, EL-unification is NP-complete, and thus has the same
complexity as EL-matching. We also show that, w.r.t. the unification type, EL
is less well-behaved: it is of type zero, which in particular implies that
there are unification problems that have no finite complete set of unifiers.Comment: 31page
Unification theory
The purpose of this paper is not to give an overview of the state of art in unification theory. It is intended to be a short introduction into the area of equational unification which should give the reader a feeling for what unification theory might be about. The basic notions such as complete and minimal complete sets of unifiers, and unification types of equational theories are introduced and illustrated by examples. Then we shall describe the original motivations for considering unification (in the empty theory) in resolution theorem proving and term rewriting. Starting with Robinson\u27s first unification algorithm it will be sketched how more efficient unification algorithms can be derived.
We shall then explain the reasons which lead to the introduction of unification in non-empty theories into the above mentioned areas theorem proving and term rewriting. For theory unification it makes a difference whether single equations or systems of equations are considered. In addition, one has to be careful with regard to the signature over which the terms of the unification problems can be built. This leads to the distinction between elementary unification, unification with constants, and general unification (where arbitrary free function symbols may occur). Going from elementary unification to general unification is an instance of the so-called combination problem for equational theories which can be formulated as follows: Let E, F be equational theories over disjoint signatures. How can unification algorithms for E, F be combined to a unification algorithm for the theory E cup F
A formal definition for the expressive power of knowledge representation languages
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 used in an intuitive sense. The papers contain only informal descriptions of what is meant by expressiveness. There are several reasons which 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 which can be used in mathematical proofs.
Though we shall only consider KL-ONE-based KR-language in our motivation and in the examples, the definition of expressive power which will be given in this paper can be used for all KR-languages with 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. Sometimes, the tradeoff lies between convenience and computational tractability. The paper contains several examples which demonstrate how the definition of expressive power can be used in positive proofs -- that is, proofs where it is shown that one language can be expressed by another language -- as well as for negative proofs -- which show that a given language cannot be expressed by the other language
Terminological cycles in KL-ONE-based knowledge representation languages
Cyclic definitions are often prohibited in terminological knowledge representation languages, because, from a theoretical point of view, their semantics is not clear and, from a practical point of view, existing inference algorithms may go astray in the presence of cycles. In this paper we consider terminological cycles in a very small KL-ONE-based language. For this language, the effect of the three types of semantics introduced by Nebel (1987, 1989, 1989a) can be completely described with the help of finite automata. These descriptions provide a rather intuitive understanding of terminologies with cyclic definitions and give insight into the essential features of the respective semantics. In addition, one obtains algorithms and complexity results for subsumption determination. The results of this paper may help to decide what kind of semantics is most appropriate for cyclic definitions, not only for this small language, but also for extended languages. As it stands, the greatest fixed-point semantics comes off best. The characterization of this semantics is easy and has an obvious intuitive interpretation. Furthermore, important constructs--such as value-restriction with respect to the transitive or reflexive-transitive closure of a role--can easily be expressed
Terminological cycles in a description logic with existential restrictions
Cyclic definitions in description logics have until now been investigated only for description logics allowing for value restrictions. Even for the most basic language FLā, which allows for conjunction and value restrictions only, deciding subsumption in the presence of terminological cycles is a PSPACE-complete problem. This report investigates subsumption in the presence of terminological cycles for the language EL, which allows for conjunction and existential restrictions. In contrast to the results for FLā, subsumption in EL remains polynomial, independent of wether we use least fixpoint semantics, greatest fixpoint semantics, or descriptive semantics. These results are shown via a characterization of subsumption through the existence of certain simulation relations between nodes of the description graph associated with a given cyclic terminology.This is an updated version of the original report, in which some errors in Section 3.1 of the original report have been corrected
The instance problem and the most specific concept in the description logic EL w.r.t. terminological cycles with descriptive semantics
In two previous reports we have investigated both standard and non-standard inferences in the presence of terminological cycles for the description logic EL, which allows for conjunctions, existential restrictions, and the top concept. Regarding standard inference problems, it was shown there that the subsumption problem remains polynomial for all three types of semantics usually considered for cyclic definitions in description logics, and that the instance problem remains polynomial for greatest fixpoint semantics. Regarding non-standard inference problems, it was shown that, w.r.t. greatest fixpoint semantics, the least common subsumer and the most specific concept always exist and can be computed in ploynomial time, and that, w.r.t. descriptive semantics, the least common subsumer need not exist. The present report is concerned with two problems left open by this previous work, namely the instance problem and the problem of computing most specific concepts w.r.t. descriptive semantics, which is the usual first-order semantics for description logic. We will show that the instance problem is polynomial also in this context. Similar to the case of the least common subsumer, the most specific concept w.r.t. descriptive semantics need not exist, but we are able to characterize the cases in which it exists and give a decidable sufficient condition for the existence of the most specific concept. Under this condition, it can be computed in polynomial time
Concept Descriptions with Set Constraints and Cardinality Constraints
We introduce a new description logic that extends the well-known logic ALCQ by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of ALCQ. To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than ALCQ, we are able to show that the complexity of reasoning in it is the same as in ALCQ, both without and with TBoxes.The first version of this report was put online on April 6, 2017. The current version, containing more information on related
work, was put online on July 13, 2017.
This is an extended version of a paper published in the proceedings of FroCoS 2017
Unification, Weak Unification, Upper Bound, Lower Bound, and Generalization Problems
We deļ¬ne E-uniļ¬cation, weak E-uniļ¬cation, E-upper bound, E-lower bound and E-generalization problems and the corresponding notions of uniļ¬cation, weak unification, upper bound, lower bound and generalization type of an equational theory. Most general uniļ¬ers, most general weak uniļ¬ers, suprema, infima and most speciļ¬c generalizers correspond to "weak versions" of well-known categorical concepts.
The problems are ļ¬rst studied for the empty theory using the restricted instantiation ordering ( i.e., substitutions are compared w.r.t. their behaviour on ļ¬nite sets of variables ) and the unrestricted instantiation ordering ( i.e., substitutions are compared w.r.t. their behaviour on all variables ). This shows that the unrestricted instantiation ordering should only be used for uniļ¬cation. For the other problems the restricted ordering yields much better results. We shall also show that there exists an equational theory where unification problems always have most general uniļ¬ers w.r.t. the restricted instantiation ordering but not w.r.t. the unrestricted instantiation ordering. This accounts for the fact that equational uniļ¬cation is mostly done with restricted instantiation.
Most general uniļ¬ers ( i.e., weak coequalizers ) modulo commutative theories cannot always be chosen as coequalizers. But we can give algebraic conditions under which this is possible. For the class of commutative theories there always exist least speciļ¬c generalizers. That means that all commutative theories have generalization type "unitary"
Unification in Commutative Theories, Hilbert's Basis Theorem, and Grƶbner Bases
Uniļ¬cation in a commutative theory E may be reduced to solving linear equations in the corresponding semiring S(E) (Nutt (1988)). The uniļ¬cation type of E can thus be characterized by algebraic properties of S(E). The theory of abelian groups with n commuting homomorphisms corresponds to the semiring Z[X1,...,Xn]. Thus Hilbertās Basis Theorem can be used to show that this theory is unitary. But this argument does not yield a uniļ¬cation algorithm. Linear equations in Z[X1,..,Xn] can be solved with the help of Grƶbner Base methods, which thus provide the desired algorithm. The theory of abelian monoids with a homomorphism is of type zero (Baader (1988)). This can also be proved by using the fact that the corresponding semiring, namely N[X], is not noetherian. An other example of a semiring (even ring), which is not noetherian, is the ring Z, where X1, ..., Xn ( n > 1 ) are non-commuting indeterminates. This semiring corresponds to the theory of abelian groups with n non-commuting homomorphisms. Surprisingly, by construction of a Grƶbner Base algorithm for right ideals in Z, it can be shown that this theory is unitary unifying
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