Pinpointing in Terminating Forest Tableaux

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. The pinpointing algorithms described in the DL literature 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 paper 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

Axiom Pinpointing in General Tableaux

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. The pinpointing algorithms described in the DL literature 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 paper 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 „tableaux algorithms,' which captures many of the known tableau-based algorithms employed in DLs, but also other kinds of reasoning procedures

PSpace Automata with Blocking for Description Logics

In Description Logics (DLs), both tableau-based and automatabased algorithms are frequently used to show decidability and complexity results for basic inference problems such as satisfiability of concepts. Whereas tableau-based algorithms usually yield worst-case optimal algorithms in the case of PSpace-complete logics, it is often very hard to design optimal tableau-based algorithms for ExpTime-complete DLs. In contrast, the automata-based approach is usually well-suited to prove ExpTime upper-bounds, but its direct application will usually also yield an ExpTime-algorithm for a PSpace-complete logic since the (tree) automaton constructed for a given concept is usually exponentially large. In the present paper, we formulate conditions under which an on-the-fly construction of such an exponentially large automaton can be used to obtain a PSpace-algorithm. We illustrate the usefulness of this approach by proving a new PSpace upper-bound for satisfiability of concepts w.r.t. acyclic terminologies in the DL SI, which extends the basic DL ALC with transitive and inverse roles

Blocking and Pinpointing in Forest Tableaux

Axiom pinpointing has been introduced in description logics (DLs) to help the used understand the reasons why consequences hold by computing minimal subsets of the knowledge base that have the consequence in consideration. Several pinpointing algorithms have been described as extensions of the standard tableau-based reasoning algorithms for deciding 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, using specific traits of them. In the past, we have developed a general approach for extending tableau-based algorithms into pinpointing algorithms. In this paper we explore some issues of termination of general tableaux and their pinpointing extensions. We also define a subclass of tableaux that allows the use of so-called blocking conditions, which stop the execution of the algorithm once a pattern is found, and adapt the pinpointing extensions accordingly, guaranteeing its correctness and termination

An Abstract Tableau Calculus for the Description Logic SHOI Using UnrestrictedBlocking and Rewriting

Abstract This paper presents an abstract tableau calculus for the description logic SHOI. SHOI is the extension of ALC with singleton concepts, role inverse, transitive roles and role inclusion axioms. The presented tableau calculus is inspired by a recently introduced tableau synthesis framework. Termination is achieved by a variation of the unrestricted blocking mechanism that immediately rewrites terms with respect to the conjectured equalities. This approach leads to reduced search space for decision procedures based on the calculus. We also discuss restrictions of the application of the blocking rule by means of additional side conditions and/or additional premises.

Hypertableau Reasoning for Description Logics

We present a novel reasoning calculus for the description logic SHOIQ^+---a knowledge representation formalism with applications in areas such as the Semantic Web. Unnecessary nondeterminism and the construction of large models are two primary sources of inefficiency in the tableau-based reasoning calculi used in state-of-the-art reasoners. In order to reduce nondeterminism, we base our calculus on hypertableau and hyperresolution calculi, which we extend with a blocking condition to ensure termination. In order to reduce the size of the constructed models, we introduce anywhere pairwise blocking. We also present an improved nominal introduction rule that ensures termination in the presence of nominals, inverse roles, and number restrictions---a combination of DL constructs that has proven notoriously difficult to handle. Our implementation shows significant performance improvements over state-of-the-art reasoners on several well-known ontologies

Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

Practical Reasoning for Very Expressive Description Logics

Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles and functional restrictions with respect to general concept inclusion axioms and role hierarchies; early experiments indicate that this algorithm is well-suited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSPACE. We investigate the limits of decidability for this family of DLs, showing that relaxing the constraints placed on the kinds of roles used in number restrictions leads to the undecidability of all inference problems. Finally, we describe a number of optimisation techniques that are crucial in obtaining implementations of the decision procedures, which, despite the worst-case complexity of the problem, exhibit good performance with real-life problems

Reasoning with Individuals for the Description Logic SHIQ

While there has been a great deal of work on the development of reasoning algorithms for expressive description logics, in most cases only Tbox reasoning is considered. In this paper we present an algorithm for combined Tbox and Abox reasoning in the SHIQ description logic. This algorithm is of particular interest as it can be used to decide the problem of (database) conjunctive query containment w.r.t. a schema. Moreover, the realisation of an efficient implementation should be relatively straightforward as it can be based on an existing highly optimised implementation of the Tbox algorithm in the FaCT system.Comment: To appear at CADE-1