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34
Hypertableau Reasoning for Description Logics
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 2007
"... 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 tableaubased reasoning ca ..."
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Cited by 132 (26 self)
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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 tableaubased reasoning calculi used in stateoftheart 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 stateoftheart reasoners on several wellknown ontologies.
RIQ and SROIQ are harder than SHOIQ
 In Proc. KR’08
, 2008
"... Abstract. We identify the complexity of (finite model) reasoning in the DL SROIQ to be N2ExpTimecomplete. We also prove that (finite model) reasoning in the DL SR—a fragment of SROIQ without nominals, number restrictions, and inverse roles—is 2ExpTimehard. 1 From SHIQ to SROIQ In this paper we stu ..."
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Cited by 60 (6 self)
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Abstract. We identify the complexity of (finite model) reasoning in the DL SROIQ to be N2ExpTimecomplete. We also prove that (finite model) reasoning in the DL SR—a fragment of SROIQ without nominals, number restrictions, and inverse roles—is 2ExpTimehard. 1 From SHIQ to SROIQ In this paper we study the complexity of reasoning in the DL SROIQ—the logic chosen as a candidate for OWL 2. 1 SROIQ has been introduced in [1] as an extension of SRIQ, which itself was introduced previously in [2] as an extension of RIQ [3]. These papers present tableaubased procedures for the respective DLs and prove their soundness, completeness and termination. In contrast to sublanguages of SHOIQ whose computational complexities are currently well understood [4], almost nothing was known, up until now, about the complexity of SROIQ, SRIQ and RIQ except for the hardness results inherited from their sublanbuages: SROIQ is NExpTimehard as an extension of SHOIQ, SRIQ and RIQ are ExpTimehard as extensions of SHIQ. The
A Better Uncle For OWL  Nominal Schemas for Integrating Rules and Ontologies
, 2011
"... We propose a descriptionlogic style extension of OWL 2 with nominal schemas which can be used like “variable nominal classes”within axioms. This feature allows ontology languages to express arbitrary DLsafe rules (as expressible in SWRL or RIF) in their native syntax. We show that adding nominal s ..."
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Cited by 38 (17 self)
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We propose a descriptionlogic style extension of OWL 2 with nominal schemas which can be used like “variable nominal classes”within axioms. This feature allows ontology languages to express arbitrary DLsafe rules (as expressible in SWRL or RIF) in their native syntax. We show that adding nominal schemas to OWL 2 does not increase the worstcase reasoning complexity, and we identify a novel tractable language SROELV 3(⊓, ×) that is versatile enough to capture the lightweight languages OWL EL and OWL RL.
Cheap Boolean Role Constructors for Description Logics
"... Abstract. We investigate the possibility of incorporating Boolean role constructors on simple roles into some of today’s most popular description logics, focussing on cases where those extensions do not increase complexity of reasoning. We show that the expressive DLsSHOIQ andSROIQ, serving as the l ..."
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Cited by 28 (10 self)
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Abstract. We investigate the possibility of incorporating Boolean role constructors on simple roles into some of today’s most popular description logics, focussing on cases where those extensions do not increase complexity of reasoning. We show that the expressive DLsSHOIQ andSROIQ, serving as the logical underpinning of OWL and the forthcoming OWL 2, can accommodate arbitrary Boolean expressions. The prominent OWLfragmentSHIQ can be safely extended by safe role expressions, and the tractable fragmentsEL ++ and DLP retain tractability if extended by conjunction on roles, where in the case of DLP the restriction on role simplicity can even be discarded. 1
Representing Ontologies Using Description Logics, Description Graphs, and Rules
 Artificial Intelligence
"... Description logics (DLs) are a family of stateoftheart knowledge representation languages, and their expressive power has been carefully crafted to provide useful knowledge modeling primitives while allowing for practically effective decision procedures for the basic reasoning problems. Recent ex ..."
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Cited by 28 (6 self)
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Description logics (DLs) are a family of stateoftheart knowledge representation languages, and their expressive power has been carefully crafted to provide useful knowledge modeling primitives while allowing for practically effective decision procedures for the basic reasoning problems. Recent experience with DLs, however, has shown that their expressivity is often insufficient to accurately describe structured objects—objects whose parts are interconnected in arbitrary, rather than treelike ways. DL knowledge bases describing structured objects are therefore usually underconstrained, which precludes the entailment of certain consequences and causes performance problems during reasoning. To address this problem, we propose an extension of DL languages with description graphs—a knowledge modeling construct that can accurately describe objects with parts connected in arbitrary ways. Furthermore, to enable modeling the conditional aspects of structured objects, we also extend DLs with rules. We present an indepth study of the computational properties of such a formalism. In particular, we first identify the sources of undecidability of the general, unrestricted formalism. Based on that analysis, we then investigate several restrictions of the general formalism that make reasoning decidable. We present practical evidence that such a logic can be used to model nontrivial structured objects. Finally, we present a practical decision procedure for our formalism, as well as tight complexity bounds. Key words: knowledge representation, description logics, structured objects, ontologies ⋆ This is an extended version of two papers published at WWW 2008 [29] and KR 2008 [28], respectively. ∗ Corresponding author.
A Principle for Incorporating Axioms into the FirstOrder Translation of Modal Formulae
 Automated Deduction—CADE19, volume 2741 of Lecture Notes in Artificial Intelligence
, 2003
"... In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable logics. The goal of the axiomatic translation ..."
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Cited by 20 (7 self)
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In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable logics. The goal of the axiomatic translation principle is to find simplified theories, which capture the inference problems in the original theory, but in a way that is more amenable to automation and easier to deal with by existing theorem provers. The principle of the axiomatic translation is conceptually very simple and can be largely automated. Soundness is automatic under reasonable assumptions, and termination of ordered resolution is easily achieved, but the nontrivial part of the approach is proving completeness.
Z.: Acyclicity conditions and their application to query answering in description logics
 In: KR (2012
"... Abstract Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a key problem in knowledge representation and databases. This problem can be solved using the chase (aka materialisation) algorithm; however, CQ answering is undecidable for general existential rules ..."
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Cited by 10 (3 self)
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Abstract Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a key problem in knowledge representation and databases. This problem can be solved using the chase (aka materialisation) algorithm; however, CQ answering is undecidable for general existential rules, so the chase is not guaranteed to terminate. Several acyclicity conditions provide sufficient conditions for chase termination. In this paper, we present two novel such conditionsmodelfaithful acyclicity (MFA) and modelsummarising acyclicity (MSA)that generalise many of the acyclicity conditions known so far in the literature. Materialisation provides the basis for several widelyused OWL 2 DL reasoners. In order to avoid termination problems, many of these systems handle only the OWL 2 RL profile of OWL 2 DL; furthermore, some systems go beyond OWL 2 RL, but they provide no termination guarantees. In this paper we investigate whether various acyclicity conditions can provide a principled and practical solution to these problems. On the theoretical side, we show that query answering for acyclic ontologies is of lower complexity than for general ontologies. On the practical side, we show that many of the commonly used OWL 2 DL ontologies are MSA, and that the facts obtained via materialisation are not too large. Thus, our results suggest that principled extensions to materialisationbased OWL 2 DL reasoners may be practically feasible.
Constructing finite least Kripke models for positive logic programs in serial regular grammar logics
 Logic Journal of the IGPL
"... A serial contextfree grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form ✷tϕ → ✷s1... ✷skϕ. Such an inclusion axiom corresponds to the grammar rule t → s1... sk. Thus the inclusion axioms of L capture a contextfree grammar G( ..."
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Cited by 10 (6 self)
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A serial contextfree grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form ✷tϕ → ✷s1... ✷skϕ. Such an inclusion axiom corresponds to the grammar rule t → s1... sk. Thus the inclusion axioms of L capture a contextfree grammar G(L). If for every modal index t, the set of words derivable from t using G(L) is a regular language, then L is a serial regular grammar logic. In this paper, we present an algorithm that, given a positive multimodal logic program P and a set of finite automata specifying a serial regular grammar logic L, constructs a finite least Lmodel of P. (A model M is less than or equal to model M ′ if for every positive formula ϕ, if M  = ϕ then M ′  = ϕ.) A least Lmodel M of P has the property that for every positive formula ϕ, P  = ϕ iff M  = ϕ. The algorithm runs in exponential time and returns a model with size 2 O(n3). We give examples of P and L, for both of the case when L is fixed or P is fixed, such that every finite least Lmodel of P must have size 2 Ω(n). We also prove that if G is a contextfree grammar and L is the serial grammar logic corresponding to G then there exists a finite least Lmodel of ✷sp iff the set of words derivable from s using G is a regular language. 1
Analytic cutfree tableaux for regular modal logics of agent beliefs
 Proceedings of CLIMA VIII, vol. 5056 of LNAI
, 2008
"... Abstract. We present a sound and complete tableau calculus for a class BReg of extended regular modal logics which contains useful epistemic logics for reasoning about agent beliefs. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. Applying sound ..."
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Cited by 8 (7 self)
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Abstract. We present a sound and complete tableau calculus for a class BReg of extended regular modal logics which contains useful epistemic logics for reasoning about agent beliefs. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. Applying sound global caching to the calculus, we obtain the first optimal (EXPTime) tableau decision procedure for BReg. We demonstrate the usefulness of BReg logics and our tableau calculus using the wise men puzzle and its modified version, which requires axiom (5) for single agents. 1
An extension of complex role inclusion axioms in the description logic SROIQ
 In Proceedings of IJCAR
, 2010
"... Abstract. We propose an extension of the syntactic restriction for complex role inclusion axioms in the description logic SROIQ. Like the original restriction in SROIQ, our restrictions can be checked in polynomial time and they guarantee regularity for the sets of role chains implying roles, and th ..."
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Abstract. We propose an extension of the syntactic restriction for complex role inclusion axioms in the description logic SROIQ. Like the original restriction in SROIQ, our restrictions can be checked in polynomial time and they guarantee regularity for the sets of role chains implying roles, and thereby decidability for the main reasoning problems. But unlike the original restrictions, our syntactic restrictions can represent any regular compositional properties on roles. In particular, many practically relevant complex role inclusion axioms, such as those describing various parthood relations, can be expressed in our extension, but could not be expressed in the original SROIQ. 1