We investigate the computational complexity of axiom pinpointing, which is the task of finding minimal subsets of a Description Logic knowledge base that have a given consequence. We consider the problems of enumerating such subsets with and without order, and show hardness results that already hold for the propositional Horn fragment, or for the Description Logic EL. We show complexity results for several other related decision and enumeration problems for these fragments that extend to more expressive logics. In particular we show that hardness of these problems depends not only on expressivity of the fragment but also on the shape of the axioms used.

This paper proves that validity and satisfiability of assertions in the Fuzzy Description Logic based on infinite-valued Product Logic with universal and existential quantifiers (which are non-interdefinable) is decidable when we only consider quasi-witnessed interpretations. We prove that this restriction is neither necessary for the validity problem (i.e., the validity of assertions in the Fuzzy Description Logic based on infinite-valued Product Logic is decidable) nor for the positive satisfiability problem, because quasi-witnessed interpretations are particularly adequate for the infinite-valued Product Logic. We give an algorithm that reduces the problem of validity (and satisfiability) of assertions in our Fuzzy Description Logic (restricted to quasi-witnessed interpretations) to a semantic consequence problem, with finite number of hypothesis, on infinite-valued propositional Product Logic.

Ontologies can be used to provide an enriched vocabulary for the formulation of queries over instance data. 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 data sets formulated in a given signature. 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.

Research effort, with its emphasis on evaluation and measurable progress, things began to change. Instead SHRDLU (WIN72) is perhaps the canonical example. of systems whose architecture and vocabulary were The rapid growth of efforts to found the next generation of based on linguistic theory (in this case acoustic phonetics), systems on general-purpose knowledge representation languages new approaches based on statistical modelling and Bayesian (I'm thinking of several varieties of semantic nets, probability emerged and quickly spread. "Every time I fire a from plain to partitioned, as well as KRL, KL-ONE and linguist my system's performance improves" (Fred Jellinek, their successors, ending (not yet, of course) with CYC (See head of speech recognition at IBM, c. 1980, latterly repudiated (BRA08) for all these) stumbled to a halt once their failure by Fred but widely attested). As advanced from resolution theorem provers through a number more and more problems are re-conceived as instances of of stages to the current proliferation of a range of Description the noisy channel model, the empiricist paradigm continually Logic'reasoners'; Whereas in the 1970s and 1980s there grew, so did the need to manage the impact of change and was real energy and optimism at the interface between computational conflict: enter'truth maintenance', subsequently renamed and theoretical linguistics, the overwhelming success'reason maintenance'. While still using some of But outflanking these'normal science' advances of AI, the terminology of linguistic theory, computational linguistics the paradigm shifters were coming up fast on the outside: practioners are increasingly detached from theory itself, over the last ten years machine learning has spread from which has suffered a, perhaps connected, loss of energy and small specialist niches such as speech recognition to become sense of progress.

Description Logics (DLs) [2] are a logical reconstruction of the so-called frame-based knowledge representation languages, with the aim of providing a simple well-established Tarski-style declarative semantics to capture the meaning of the most popular features of structured representation of knowledge. Nowadays, DLs have gained even more popularity due to their application in 1 the context of the Semantic Web [4]. For example, the current standard language for specifying ontologies, the Web Ontology Language OWL is based on Description Logics. It is very natural to extend DLs to the fuzzy case in order to manage fuzzy/vague/imprecise pieces of knowledge for which a clear and precise definition is not possible. For a good and recent survey on the advances in the field of fuzzy DLs, we refer the reader to [14]. One of the challenges of the research in this community is the fact that different families of fuzzy operators (or fuzzy logics) lead to fuzzy DLs with different properties. In fuzzy logic, there are a lot of families of fuzzy operators (or fuzzy logics). Table 1 shows the connectives involved in what are considered the main four families. The most famous families correspond to the three basic continuous t-norms (i.e., Lukasiewicz, Gödel and Product [10]) together with an R-implication

We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ).

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.

In the last two decades, modal and description logics have been applied to numerous areas of computer science, including knowledge representation, formal verification, database theory, distributed computing and, more recently, semantic web and ontologies. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In particular, many approaches have been proposed for efficiently handling the satisfiability of the core normal modal logic K(m), and of its notational variant, the description logic ALC. Although simple in structure, K(m)/ALC is computationally very hard to reason on, its satisfiability being PSPACE-complete. In this paper we start exploring the idea of performing automated reasoning tasks in modal and description logics by encoding them into SAT, so that to be handled by state-of-the-art SAT tools; as with most previous approaches, we begin our investigation from the satisfiability in K(m). We propose an efficient encoding, and we test it on an extensive set of benchmarks, comparing the approach with the main state-of-the-art tools available. Although the encoding is necessarily worst-case exponential, from our experiments we notice that, in practice, this approach can handle most or all the problems which are at the reach of the other approaches, with performances which are comparable with, or even better than, those of the current state-of-the-art tools.

We review and extend earlier work on the logic CFD, a description logic that allows terminological cycles with universal restrictions over functional roles. In particular, we consider the problem of reasoning about concept subsumption and the problem of computing certain answers for a family of attribute-connected conjunctive queries, showing that both problems are in PTIME. We then consider the effect on the complexity of these problems after adding a concept constructor that expresses concept union, or after adding a concept constructor for the bottom class. Finally, we show that adding both constructors makes both problems EXPTIME-complete.

We develop a framework for forgetting concepts and roles (aka uniform interpolation) in terminologies in the lightweight description logic EL extended with role inclusions and domain and range restrictions. Three different notions of forgetting, preserving, respectively, concept inclusions, concept instances, and answers to conjunctive queries, with corresponding languages for uniform interpolants are investigated. Experiments based on SNOMED CT (Systematised Nomenclature of Medicine Clinical Terms) and NCI (National Cancer Institute Ontology) demonstrate that forgetting is often feasible in practice for large-scale terminologies.