Abstract Standard Description Logics (DLs) can encode quantitative aspects of an application domain through either number restrictions , which constrain the number of individuals that are in a certain relationship with an individual, or concrete domains , which can be used to assign concrete values to individuals using so-called features. These two mechanisms have been extended towards very expressive DLs, for which reasoning nevertheless remains decidable. Number restrictions have been generalized to more powerful comparisons of sets of role successors in $$\mathcal {ALCSCC}$$ ALCSCC , while the comparison of feature values of different individuals in $$\mathcal {ALC} (\mathfrak {D})$$ ALC ( D ) has been studied in the context of $$\omega $$ ω -admissible concrete domains $$\mathfrak {D}$$ D . In this paper, we combine both formalisms and investigate the complexity of reasoning in the thus obtained DL $$\mathcal {ALCOSCC}(\mathfrak {D})$$ ALCOSCC ( D ) , which additionally includes the ability to refer to specific individuals by name. We show that, in spite of its high expressivity, the consistency problem for this DL is ExpTime -complete, assuming that the constraint satisfaction problem of $$\mathfrak {D}$$ D is also decidable in exponential time. It is thus not higher than the complexity of the basic DL $$\mathcal {ALC}$$ ALC . At the same time, we show that many natural extensions to this DL, including a tighter integration of the concrete domain and number restrictions, lead to undecidability.

In the area of belief change, contraction operations are used to modify a given belief set or belief base such that certain unwanted consequences no longer follow. In previous work we have introduced a framework for constructing contraction operations that generalizes the well-known partial meet contraction approach, called partial product contractions (PPCs). The main idea was to replace the remainders employed by partial meet contractions with optimal repairs, which were first considered in ontology engineering. We were able to characterize PPCs with variants of well-known rationality postulates, and provided a large number of concrete instances of the general framework. In the present work, we start to investigate whether the rather weak conditions imposed by our framework are sufficient to generalize further classical results from belief change to this setting. To this purpose, we consider Gärdenfors’s supplementary postulates for belief contractions. We are able to show that, under two reasonable additional conditions, PPCs induced by maximizingly and transitively relational selection functions indeed satisfy these postulates, similarly to the classical case. However, unlike the classical case, in our general framework these conditions are not strong enough to prove a characterization theorem. In fact, we provide an example that shows that there are partial product contractions satisfying the supplementary postulates that cannot be obtained from a weakly maximizingly and transitively relational selection function. We also introduce a third condition that ensures that already transitively relational selection functions yield PPCs that satisfy the supplementary postulates.

The unification type of an equational theory is defined using a preorder on substitutions, called the instantiation preorder, whose scope is either restricted to the variables occurring in the unification problem, or unrestricted such that all variables are considered. It has been known for more than three decades that the unification type of an equational theory may vary, depending on which instantiation preorder is used. More precisely, it was shown in 1991 that the theory ACUI of an associative, commutative, and idempotent binary function symbol with a unit is unitary w.r.t. the restricted instantiation preorder, but not unitary w.r.t. the unrestricted one. In 2016 this result was strengthened by showing that the unrestricted type of this theory also cannot be finitary. Here, we considerably improve on this result by proving that ACUI is infinitary w.r.t. the unrestricted instantiation preorder, thus precluding type zero. We also show that, w.r.t. this preorder, the unification type of ACU (where idempotency is removed from the axioms) and of AC (where additionally the unit is removed) is infinitary, though it is respectively unitary and finitary in the restricted case. In the other direction, we prove (using the example of unification in the description logic EL) that the unification type may actually improve from type zero to infinitary when switching from the restricted instantiation preorder to the unrestricted one. In addition, we establish some general results on the relationship between the two instantiation preorders.

Removing unwanted consequences from a knowledge base has been investigated in belief change under the name contraction and is called repair in ontology engineering. Simple repair and contraction approaches based on removing statements from the knowledge base (respectively called belief base contractions and classical repairs) have the disadvantage that they are syntax-dependent and may remove more consequences than necessary. Belief set contractions do not have these problems, but may result in belief sets that have no finite representation. Similarly, optimal repairs, which are syntax-independent and maximize the retained consequences, may not exist. Our KR 2024 paper leverage advances in characterizing and computing optimal repairs of ontologies based on the description logics EL to obtain contraction operations that combine the advantages of belief set and belief base contractions. It introduces this new approach in a very general setting, and proves a characterization theorem that relates the obtained contractions with well-known rationality postulates. Then, it describes a variety of interesting instances, not only in the standard repair/contraction setting where one wants to get rid of a consequence, but also in other settings such as variants of forgetting in propositional and description logic.

Motivated by an application where we try to make proofs for Description Logic inferences smaller by rewriting, we consider the following decision problem, which we call the small term reachability problem: given a term rewriting system $R$, a term $s$, and a natural number $n$, decide whether there is a term $t$ of size $\leq n$ reachable from $s$ using the rules of $R$. We investigate the complexity of this problem depending on how termination of $R$ can be established. We show that the problem is in general NP-complete for length-reducing term rewriting systems. Its complexity increases to N2ExpTime-complete (NExpTime-complete) if termination is proved using a (linear) polynomial order and to PSpace-complete for systems whose termination can be shown using a restricted class of Knuth-Bendix orders. Confluence reduces the complexity to P for the length-reducing case, but has no effect on the worst-case complexity in the other two cases. Finally, we consider the large term reachability problem, a variant of the problem where we are interested in reachability of a term of size $\geq n$. It turns out that this seemingly innocuous modification in some cases changes the complexity of the problem, which may also become dependent on whether the number $n$ is is represented in unary or binary encoding, whereas this makes no difference for the complexity of the small term reachability problem.

Abstract Unification has been introduced in Description Logic (DL) as a means to detect redundancies in ontologies. In particular, it was shown that testing unifiability in the DL $$\mathcal{E}\mathcal{L}$$ E L is an NP-complete problem, and this result has been extended in several directions. Surprisingly, it turned out that the complexity increases to PSpace if one disallows the use of the top concept in concept descriptions. Motivated by features of the medical ontology SNOMED CT, we extend this result to a setting where the top concept is disallowed, but there is a background ontology consisting of restricted forms of concept and role inclusion axioms. We are able to show that the presence of such axioms does not increase the complexity of unification without top, i.e., testing for unifiability remains a PSpace-complete problem.

Removing unwanted consequences from a knowledge base has been investigated in belief change under the name contraction and is called repair in ontology engineering. Simple repair and contraction approaches based on removing statements from the knowledge base (respectively called belief base contractions and classical repairs) have the disadvantage that they are syntax-dependent and may remove more consequences than necessary. Belief set contractions do not have these problems, but may result in belief sets that have no finite representation if one works with logics that are not fragments of propositional logic. Similarly, optimal repairs, which are syntax-independent and maximize the retained consequences, may not exist. In this paper, we want to leverage advances in characterizing and computing optimal repairs of ontologies based on the description logics EL to obtain contraction operations that combine the advantages of belief set and belief base contractions. The basic idea is to employ, in the partial meet contraction approach, optimal repairs instead of optimal classical repairs as remainders. We introduce this new approach in a very general setting, and prove a characterization theorem that relates the obtained contractions with well-known postulates. Then, we consider several interesting instances, not only in the standard repair/contraction setting were one wants to get rid of a consequence, but also in other settings such as variants of forgetting in propositional and description logic. We also show that classical belief set contraction is an instance of our approach.

In applications of AI systems where exact definitions of the important notions of the application domain are hard to come by, the use of traditional logic-based knowledge representation languages such as Description Logics may lead to very large and unintuitive definitions, and high complexity of reasoning. To overcome this problem, we define new concept constructors that allow us to define concepts in an approximate way. To be more precise, we present a family τEL(m) of extensions of the lightweight Description Logic EL that use threshold constructors for this purpose. To define the semantics of these constructors we employ graded membership functions m, which for each individual in an interpretation and concept yield a number in the interval [0,1] expressing the degree to which the individual belongs to the concept in the interpretation. Threshold concepts C⋈t for ⋈∈{<,≤,>,≥} then collect all the individuals that belong to C with degree ⋈t. The logic τEL(m) extends EL with threshold concepts whose semantics is defined relative to a function m. To construct appropriate graded membership functions, we show how concept measures ∼ (which are graded generalizations of subsumption or equivalence between concepts) can be used to define graded membership functions m∼. Then we introduce a large class of concept measures, called simi-d, for which the logics τEL(m∼) have good algorithmic properties. Basically, we show that reasoning in τEL(m∼) is NP/coNP-complete without TBox, PSpace-complete w.r.t. acyclic TBoxes, and ExpTime-complete w.r.t. general TBoxes. The exception is the instance problem, which is already PSpace-complete without TBox w.r.t. combined complexity. While the upper bounds hold for all elements of simi-d, we could prove some of the hardness results only for a subclass of simi-d. This article considerably improves on and generalizes results we have shown in three previous conference papers and it provides detailed proofs of all our results.

Abstract OWL is a powerful language to formalize terminologies in an ontology. Its main strength lies in its foundation on description logics, allowing systems to automatically deduce implicit information through logical reasoning. However, since ontologies are often complex, understanding the outcome of the reasoning process is not always straightforward. Unlike already existing tools for exploring ontologies, our visualization tool Evonne is tailored towards explaining logical consequences. In addition, it supports the debugging of unwanted consequences and allows for an interactive comparison of the impact of removing statements from the ontology. Our visual approach combines (1) specialized views for the explanation of logical consequences and the structure of the ontology, (2) employing multiple layout modes for iteratively exploring explanations, (3) detailed explanations of specific reasoning steps, (4) cross‐view highlighting and colour coding of the visualization components, (5) features for dealing with visual complexity and (6) comparison and exploration of possible fixes to the ontology. We evaluated Evonne in a qualitative study with 16 experts in logics, and their positive feedback confirms the value of our concepts for explaining reasoning and debugging ontologies.

Abstract The word problem for a finite set of ground identities is known to be decidable in polynomial time using congruence closure, and this is also the case if some of the function symbols are assumed to be commutative or defined by certain shallow identities, called strongly shallow. We show that decidability in P is preserved if we add the assumption that certain function symbolsfareextensionalin the sense that $$f(s_1,\ldots ,s_n) \mathrel {\approx }f(t_1,\ldots ,t_n)$$ f(s1,…,sn)≈f(t1,…,tn) implies $$s_1 \mathrel {\approx }t_1,\ldots ,s_n \mathrel {\approx }t_n$$ s1≈t1,…,sn≈tn . In addition, we investigate a variant of extensionality that is more appropriate for commutative function symbols, but which raises the complexity of the word problem to coNP.

Abstract Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL $$\mathcal {ALC}$$ ALC in the presence of GCIs, quite strong restrictions, in sum called $$\omega $$ ω -admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of $$\omega $$ ω -admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate $$\omega $$ ω -admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield $$\omega $$ ω -admissible concrete domains. This allows us to show $$\omega $$ ω -admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the $$\mathcal {EL}$$ EL family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although $$\omega $$ ω -admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into $$\mathcal {ALC}$$ ALC yields decidable DLs.

Abstract The inexpressive Description Logic (DL) ${\cal F}{{\cal L}_0}$ , which has conjunction and value restriction as its only concept constructors, had fallen into disrepute when it turned out that reasoning in ${\cal F}{{\cal L}_0}$ w.r.t. general TBoxes is ExpTime-complete, that is, as hard as in the considerably more expressive logic ${\cal A}{\cal L}{\cal C}$ . In this paper, we rehabilitate ${\cal F}{{\cal L}_0}$ by presenting a dedicated subsumption algorithm for ${\cal F}{{\cal L}_0}$ , which is much simpler than the tableau-based algorithms employed by highly optimized DL reasoners. Our experiments show that the performance of our novel algorithm, as prototypically implemented in our ${\cal F}{{\cal L}_0}$ wer reasoner, compares very well with that of the highly optimized reasoners. ${\cal F}{{\cal L}_0}$ wer can also deal with ontologies written in the extension ${\cal F}{{\cal L}_ \bot }$ of ${\cal F}{{\cal L}_0}$ with the top and the bottom concept by employing a polynomial-time reduction, shown in this paper, which eliminates top and bottom. We also investigate the complexity of reasoning in DLs related to the Horn-fragments of ${\cal F}{{\cal L}_0}$ and ${\cal F}{{\cal L}_ \bot }$ .

Abstract Logic-based approaches to AI have the advantage that their behavior can in principle be explained to a user. If, for instance, a Description Logic reasoner derives a consequence that triggers some action of the overall system, then one can explain such an entailment by presenting a proof of the consequence in an appropriate calculus. How comprehensible such a proof is depends not only on the employed calculus, but also on the properties of the particular proof, such as its overall size, its depth, the complexity of the employed sentences and proof steps, etc. For this reason, we want to determine the complexity of generating proofs that are below a certain threshold w.r.t. a given measure of proof quality. Rather than investigating this problem for a fixed proof calculus and a fixed measure, we aim for general results that hold for wide classes of calculi and measures. In previous work, we first restricted the attention to a setting where proof size is used to measure the quality of a proof. We then extended the approach to a more general setting, but important measures such as proof depth were not covered. In the present paper, we provide results for a class of measures called recursive, which yields lower complexities and also encompasses proof depth. In addition, we close some gaps left open in our previous work, thus providing a comprehensive picture of the complexity landscape.

Abstract The application of automated reasoning approaches to Description Logic (DL) ontologies may produce certain consequences that either are deemed to be wrong or should be hidden for privacy reasons. The question is then how to repair the ontology such that the unwanted consequences can no longer be deduced. An optimal repair is one where the least amount of other consequences is removed. Most of the previous approaches to ontology repair are of a syntactic nature in that they remove or weaken the axioms explicitly present in the ontology, and thus cannot achieve semantic optimality. In previous work, we have addressed the problem of computing optimal repairs of (quantified) ABoxes, where the unwanted consequences are described by concept assertions of the lightweight DL $$\mathcal {EL}$$ EL . In the present paper, we improve on the results achieved so far in two ways. First, we allow for the presence of terminological knowledge in the form of an $$\mathcal {EL}$$ EL TBox. This TBox is assumed to be static in the sense that it cannot be changed in the repair process. Second, the construction of optimal repairs described in our previous work is best case exponential. We introduce an optimized construction that is exponential only in the worst case. First experimental results indicate that this reduces the size of the computed optimal repairs considerably.

We introduce and investigate the expressive description logic (DL) ALCSCC++, in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we prove that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, the satisfiability problem becomes undecidable if inverse roles are added to the languages. In addition, even without inverse roles, conjunctive query entailment in this DL turns out to be undecidable. We prove that decidability of querying can be regained if global and local constraints are not mixed and the global constraints are appropriately restricted. The latter result is based on a locally-acyclic model construction, and it reduces query entailment to ABox consistency in the restricted setting, i.e., to ABox consistency w.r.t. restricted cardinality constraints in ALCSCC, for which we can show an ExpTime upper bound.

Logic-based approaches to AI have the advantage that their behaviour can in principle be explained by providing their users with proofs for the derived consequences. However, if such proofs get very large, then it may be hard to understand a consequence even if the individual derivation steps are easy to comprehend. This motivates our interest in finding small proofs for Description Logic (DL) entailments. Instead of concentrating on a specific DL and proof calculus for this DL, we introduce a general framework in which proofs are represented as labeled, directed hypergraphs, where each hyperedge corresponds to a single sound derivation step. On the theoretical side, we investigate the complexity of deciding whether a certain consequence has a proof of size at most n along the following orthogonal dimensions: (i) the underlying proof system is polynomial or exponential; (ii) proofs may or may not reuse already derived consequences; and (iii) the number n is represented in unary or binary. We have determined the exact worst-case complexity of this decision problem for all but one of the possible combinations of these options. On the practical side, we have developed and implemented an approach for generating proofs for expressive DLs based on a non-standard reasoning task called forgetting. We have evaluated this approach on a set of realistic ontologies and compared the obtained proofs with proofs generated by the DL reasoner ELK, finding that forgetting-based proofs are often better w.r.t. different measures of proof complexity.

Abstract The theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit0was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite setGof ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.

In contrast to qualitative linear temporal logics, which can be used to state that some property will eventually be satisfied, metric temporal logics allow us to formulate constraints on how long it may take until the property is satisfied. While most of the work on combining description logics (DLs) with temporal logics has concentrated on qualitative temporal logics, there is a growing interest in extending this work to the quantitative case. In this article, we complement existing results on the combination of DLs with metric temporal logics by introducing interval-rigid concept and role names. Elements included in an interval-rigid concept or role name are required to stay in it for some specified amount of time. We investigate several combinations of (metric) temporal logics with A ℒ C by either allowing temporal operators only on the level of axioms or also applying them to concepts. In contrast to most existing work on the topic, we consider a timeline based on the integers and also allow assertional axioms. We show that the worst-case complexity does not increase beyond the previously known bound of 2-E xp S pace and investigate in detail how this complexity can be reduced by restricting the temporal logic and the occurrences of interval-rigid names.

Matching concept descriptions against concept patterns was introduced as a new inference task in Description Logics two decades ago, motivated by applications in the Classic system. Shortly afterwards, a polynomial-time matching algorithm was developed for the DL FL0. However, this algorithm cannot deal with general TBoxes (i.e., finite sets of general concept inclusions). Here we show that matching in FL0 w.r.t. general TBoxes is in ExpTime, which is the best possible complexity for this problem since already subsumption w.r.t. general TBoxes is ExpTime-hard in FL0. We also show that, w.r.t. a restricted form of TBoxes, the complexity of matching in FL0 can be lowered to PSpace.

We investigate ontology-based query answering (OBQA) in a setting where both the ontology and the query can refer to concrete values such as numbers and strings. In contrast to previous work on this topic, the built-in predicates used to compare values are not restricted to being unary. We introduce restrictions on these predicates and on the ontology language that allow us to reduce OBQA to query answering in databases using the so-called combined rewriting approach. Though at first sight our restrictions are different from the ones used in previous work, we show that our results strictly subsume some of the existing first-order rewritability results for unary predicates.

In a previous paper, we have introduced an extension of the lightweight Description Logic EL that allows us to define concepts in an approximate way. For this purpose, we have defined a graded membership function deg, which for each individual and concept yields a number in the interval [0, 1] expressing the degree to which the individual belongs to the concept. Threshold concepts C~tfor ~∈{<,≤,>,≥} then collect all the individuals that belong to C with degree ~t. We have then investigated the complexity of reasoning in the Description Logic, which is obtained fromby adding such threshold concepts. In the present paper, we extend these results, which were obtained for reasoning without TBoxes, to the case of reasoning w.r.t. acyclic TBoxes. Surprisingly, this is not as easy as might have been expected. On the one hand, one must be quite careful to define acyclic TBoxes such that they still just introduce abbreviations for complex concepts, and thus can be unfolded. On the other hand, it turns out that, in contrast to the case of EL, adding acyclic TBoxes toincreases the complexity of reasoning by at least on level of the polynomial hierarchy.

In ontology-based data access (OBDA), database querying is enriched with an ontology that provides domain knowledge and additional vocabulary for query formulation. We identify query emptiness and predicate emptiness as two central reasoning services in this context. Query emptiness asks whether a given query has an empty answer over all databases formulated in a given vocabulary. Predicate emptiness is defined analogously, but quantifies universally over all queries that contain a given predicate. In this paper, we determine the computational complexity of query emptiness and predicate emptiness in the EL, DL-Lite, and ALC-families of description logics, investigate the connection to ontology modules, and perform a practical case study to evaluate the new reasoning services.

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems.

Language equations are equations where both the constants occurring in the equations and the solutions are formal languages. They have first been introduced in formal language theory, but are now also considered in other areas of computer science. In the present paper, we restrict the attention to language equations with one-sided concatenation, but in contrast to previous work on these equations, we allow not just union but all Boolean operations to be used when formulating them. In addition, we are not just interested in deciding solvability of such equations, but also in deciding other properties of the set of solutions, like its cardinality (finite, infinite, uncountable) and whether it contains least/greatest solutions. We show that all these decision problems are EXPTIME-complete.

Most of the research on temporalized Description Logics (DLs) has concentrated on the case where temporal operators can be applied to concepts, and sometimes additionally to TBox axioms and ABox assertions. The aim of this article is to study temporalized DLs where temporal operators on TBox axioms and ABox assertions are available, but temporal operators on concepts are not. While the main application of existing temporalized DLs is the representation of conceptual models that explicitly incorporate temporal aspects, the family of DLs studied in this article addresses applications that focus on the temporal evolution of data and of ontologies. Our results show that disallowing temporal operators on concepts can significantly decrease the complexity of reasoning. In particular, reasoning with rigid roles (whose interpretation does not change over time) is typically undecidable without such a syntactic restriction, whereas our logics are decidable in elementary time even in the presence of rigid roles. We analyze the effects on computational complexity of dropping rigid roles, dropping rigid concepts, replacing temporal TBoxes with global ones, and restricting the set of available temporal operators. In this way, we obtain a novel family of temporalized DLs whose complexity ranges from 2- ExpTime-complete via NExpTime-complete to ExpTime-complete.

This paper concentrates on a fuzzy Description Logic with product t-norm and involutive negation. It does not answer the question posed in its title for this logic, but it gives strong indications that the answer might in fact be "no." On the one hand, it shows that an algorithm that was claimed to answer the question affirmatively for this logic is actually incorrect. On the other hand, it proves undecidability of a variant of this logic.

Axiom pinpointing has been introduced in description logics (DLs) to help the user to understand the reasons why consequences hold and to remove unwanted consequences by computing minimal (maximal) subsets of the knowledge base that have (do not have) the consequence in question. Most of the pinpointing algorithms described in the DLliterature are obtained as extensions of the standard tableau-based reasoning algorithms for computing consequences from DL knowledge bases. Although these extensions are based on similar ideas, they are all introduced for a particular tableau-based algorithm for a particular DL. The purpose of this article is to develop a general approach for extending a tableau-based algorithm to a pinpointing algorithm. This approach is based on a general definition of ‘tableau algorithms,’ which captures many of the known tableau-based algorithms employed in DLs, but also other kinds of reasoning procedures.

Abstract Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called -connection approach for combining modal logics to an algebraic setting.

Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

Matching concepts against patterns (concepts with variables) is a relatively new operation that has been introduced in the context of concept description languages (description logics). The original goal was to help filter out unimportant aspects of complicated concepts appearing in large industrial knowledge bases. We propose a new approach to performing matching, based on a 'concept-centred' normal form, rather than the more standard 'structural subsumption' normal form for concepts. As a result, matching can be performed (in polynomial time) using arbitrary concept patterns of the description language ALN, thus removing restrictions from previous work. The paper also addresses the question of matching problems with additional 'side conditions', which were motivated by practical needs. Key words: Knowledge representation, description logics, matching.

Number restrictions are concept constructors that are available in almost all implemented Description Logic systems. However, they are mostly available only in a rather weak form, which considerably restricts their expressive power. On the one hand, the roles that may occur in number restrictions are usually of a very restricted type, namely atomic roles or complex roles built using either intersection or inversion. In the present paper, we increase the expressive power of Description Logics by allowing for more complex roles in number restrictions. As role constructors, we consider composition of roles (which will be present in all our logics) and intersection, union, and inversion of roles in different combinations. We will present two decidability results (for the basic logic that extends ALC by number restrictions on roles with composition, and for one extension of this logic), and three undecidability results for three other extensions of the basic logic. On the other hand, with the rather weak form of number restrictions available in implemented systems, the number of role successors of an individual can only be restricted by a fixed non-negative integer. To overcome this lack of expressiveness, we allow for variables ranging over the non-negative integers in place of the fixed numbers in number restrictions. The expressive power of this constructor is increased even further by introducing explicit quantifiers for the numerical variables. The Description Logic obtained this way turns out to have an undecidable satisfiability problem. For a restricted logic we show that concept satisfiability is decidable. Key words: Knowledge representation, description logics, number restrictions

Reduction orderings that are compatible with an equational theory E and total on (the E-equivalence classes of) ground terms play an important role in automated deduction. This paper presents a general approach for combining such orderings: it shows how E/sub 1/-compatible reduction orderings total on /spl Sigma//sub 1/-ground terms and E/sub 2/-compatible reduction orderings total on /spl Sigma//sub 2/-ground terms can be used to construct an (E/sub 1//spl cup/E/sub 2/)-compatible reduction ordering total on (/spl Sigma//sub 1//spl cup//spl Sigma//sub 2/)-ground terms, provided that the signatures are disjoint and some other (rather weak) restrictions are satisfied. This work was motivated by the observation that it is often easier to construct such orderings for "small" signatures and theories separately, rather than directly for their union.

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 eepressiveness 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.

article Free AccessUnification in commutative theories, Hilbert's basis theorem, and Gröbner bases Author: Franz Baader German Research Center for Artificial Intelligence (DFKI), Saarbru¨cken, Germany German Research Center for Artificial Intelligence (DFKI), Saarbru¨cken, GermanyView Profile Authors Info & Claims Journal of the ACMVolume 40Issue 3July 1993 pp 477–503https://doi.org/10.1145/174130.174133Published:01 July 1993Publication History 19citation746DownloadsMetricsTotal Citations19Total Downloads746Last 12 Months35Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF