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23
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 62 (19 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 35 (5 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
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 19 (5 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 18 (6 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.
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 11 (4 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
On the complexity of Horn description logics
 PROCEEDINGS OF THE 2ND WORKSHOP ON OWL: EXPERIENCES AND DIRECTIONS. VOLUME 216 OF CEUR WORKSHOP PROCEEDINGS
, 2006
"... HornSHIQ has been identified as a fragment of the description logic SHIQ for which inferencing is in PTime with respect to the size of the ABox. This enables reasoning with larger ABoxes in situations where the TBox is static, and represents one approach towards tractable description logic reasoni ..."
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Cited by 9 (6 self)
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HornSHIQ has been identified as a fragment of the description logic SHIQ for which inferencing is in PTime with respect to the size of the ABox. This enables reasoning with larger ABoxes in situations where the TBox is static, and represents one approach towards tractable description logic reasoning. In this paper, we show that reasoning in HornSHIQ, in spite of its low datacomplexity, is ExpTimehard with respect to the overall size of the knowledge base. While this result is not unexpected, the proof is not a mere modification of existing reductions since it has to account for the restrictions of Hornness. We establish the result for HornFLE, showing that Hornness does not simplify TBox reasoning even for very restricted description logics. Moreover, we derive a contextfree grammar that defines HornSHIQ in a simpler and more intuitive way than existing characterisations.
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 9 (4 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.
Acyclicity conditions and their application to query answering in description logics
 In KR
, 2012
"... Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a fundamental reasoning problem although undecidable due to nontermination of the main reasoning algorithm used—the chase. Several acyclicity conditions have been formulated that ensure chase termination. In t ..."
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Cited by 8 (4 self)
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Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a fundamental reasoning problem although undecidable due to nontermination of the main reasoning algorithm used—the chase. Several acyclicity conditions have been formulated that ensure chase termination. In this paper, we show that acyclicity can also be practically relevant for description logic (DL) reasoning. Due to the high complexity of answering CQs over DL ontologies, applications often solve this problem using materialisation, in which ontology consequences are precomputed using variants of the chase. Due to the nontermination problem, the execution of the algorithm is restricted only to rules that fall within the OWL 2 RL profile, which results in incomplete reasoning. After presenting two novel acyclicity conditions (modelfaithful acyclicity (MFA) and modelsummarising acyclicity (MSA)), we investigate the practical applicability of these and other acyclicity conditions for DL query answering. Our experiments reveal that many existing ontologies are MSA and that materialisation is typically not too large. Thus, our results suggest that principled, materialisationbased reasoning for ontologies beyond the OWL 2 RL profile 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 6 (4 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
Weakening Horn knowledge bases in regular description logics to have PTIME data complexity
 Proceedings of ADDCT’2007
, 2007
"... This work is a continuation of our previous works [4,5]. We assume that the reader is familiar with description logics (DLs). A knowledge base in a description logic is a tuple (R,T,A) consisting of an RBox R of assertions about roles, a TBox T of global assumptions about concepts, and an ABox A of ..."
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Cited by 5 (4 self)
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This work is a continuation of our previous works [4,5]. We assume that the reader is familiar with description logics (DLs). A knowledge base in a description logic is a tuple (R,T,A) consisting of an RBox R of assertions about roles, a TBox T of global assumptions about concepts, and an ABox A of facts about individuals (objects) and roles. The instance checking problem in a DL is to check whether a given individual a is an instance of a concept C w.r.t. a knowledge base (R,T,A), written as (R,T,A)  = C(a). This problem in DLs including the basic description logic ALC (with R = /0) is EXPTIMEhard. From the point of view of deductive databases, A is assumed to be much larger than R and T, and it makes sense to consider the data complexity, which is measured when the query consisting of R, T, C, a is fixed while A varies as input data. It is desirable to find and study fragments of DLs with PTIME data complexity. Several authors have recently introduced a number of Horn fragments of DLs with PTIME data complexity [2,1,3]. The most expressive fragment from those is HornSHIQ introduced by Hustadt et al. [3]. It assumes, however, that the constructor ∀R.C does not occur in bodies of program clauses and goals. The data complexity of the “general Horn fragment of ALC ” is coNPhard [6]. So, to obtain PTIME data complexity one has to adopt some restrictions for the “general Horn fragments of DLs”. The goal is to find as less restrictive conditions as possible. A RBox is a finite set of assertions of the form Rs1 ◦... ◦ Rs ⊑ Rt, whereRs1,..., k Rs, Rt are role names. A regular RBox is an RBox whose set of corresponding grammar k rules t → s1...sk forms a grammar such that the set of words derivable from any symbol s using the grammar is a regular language specified by a finite automaton. We assume that the corresponding finite automata specifying R are given when R is considered. By R eg we denote ALC extended with regular RBoxes. We extend the language of ALC and R eg with the concept constructor ∀∃, which creates a concept ∀∃Rt.C from a role name Rt and a concept C.LetSem1(∀∃Rt.C)={∀Rt.C,∃Rt.⊤} and Sem2,R (∀∃Rt.C)= {∀Rt.C} ∪{∀Rs1...∀Rsi−1∃Rsi.⊤Rs1 ◦ ···◦Rs ⊑ Rt is a consequence of R and 1 ≤