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Acyclicity Notions for Existential Rules and Their Application to Query Answering in Ontologies
"... Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a prominent problem in knowledge representation and databases. This problem can be solved using the chase algorithm, which extends the given set of facts with fresh facts in order to satisfy the rules. If the ..."
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Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a prominent problem in knowledge representation and databases. This problem can be solved using the chase algorithm, which extends the given set of facts with fresh facts in order to satisfy the rules. If the chase terminates, then CQs can be evaluated directly in the resulting set of facts. The chase, however, does not terminate necessarily, and checking whether the chase terminates on a given set of rules and facts is undecidable. Numerous acyclicity notions were proposed as sufficient conditions for chase termination. In this paper, we present two new acyclicity notions called modelfaithful acyclicity (MFA) and modelsummarising acyclicity (MSA). Furthermore, we investigate the landscape of the known acyclicity notions and establish a complete taxonomy of all notions known to us. Finally, we show that MFA and MSA generalise most of these notions. Existential rules are closely related to the Horn fragments of the OWL 2 ontology language; furthermore, several prominent OWL 2 reasoners implement CQ answering by using the chase to materialise all relevant facts. In order to avoid termination problems,
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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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 ≤
Firstorder resolution methods for modal logics
 In Volume in Memoriam of Harald Ganzinger, LNCS
, 2006
"... Abstract. In this paper we give an overview of results for modal logic which can be shown using techniques and methods from firstorder logic and resolution. Because of the breadth of the area and the many applications we focus on the use of firstorder resolution methods for modal logics. In additi ..."
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Abstract. In this paper we give an overview of results for modal logic which can be shown using techniques and methods from firstorder logic and resolution. Because of the breadth of the area and the many applications we focus on the use of firstorder resolution methods for modal logics. In addition to traditional propositional modal logics we consider more expressive PDLlike dynamic modal logics which are closely related to description logics. Without going into too much detail, we survey different ways of translating modal logics into firstorder logic, we explore different ways of using firstorder resolution theorem provers, and we discuss a variety of results which have been obtained in the setting of firstorder resolution. 1
The Axiomatic Translation Principle for Modal Logic
, 2007
"... 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 fragments of firstorder logic. The goal of the ax ..."
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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 fragments of firstorder logic. 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 can be readily automated and is easier to deal with by existing (firstorder) theorem provers than the standard translation. The principle of the axiomatic translation is conceptually very simple and can be almost completely automated. Soundness is automatic under reasonable assumptions, general decidability results can be stated and termination of ordered resolution is easily achieved. The nontrivial part of the approach is proving completeness. We prove results of completeness, decidability, model generation, the small model property and the interpolation property for a number of common and less common modal logics. We also present results of experiments with a number of firstorder logic theorem provers which are very encouraging.
A cutfree exptime tableau decision procedure for the description logic SHI
 In Piotr Jedrzejowicz, Ngoc Thanh Nguyen, and Kiem Hoang, editors, ICCCI’2011, volume 6922 of LNCS
, 2011
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ExpTime Tableau Decision Procedures for Regular Grammar Logics with Converse ⋆
"... Abstract. Grammar logics were introduced by Fariñas del Cerro and Penttonen in [7] and have been widely studied. In this paper we consider regular grammar logics with converse (REG c logics) and present sound and complete tableau calculi for the general satisfiability problem of REG c logics and the ..."
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Abstract. Grammar logics were introduced by Fariñas del Cerro and Penttonen in [7] and have been widely studied. In this paper we consider regular grammar logics with converse (REG c logics) and present sound and complete tableau calculi for the general satisfiability problem of REG c logics and the problem of checking consistency of an ABox w.r.t. a TBox in a REG c logic. Using our calculi we develop ExpTime (optimal) tableau decision procedures for the mentioned problems, to which various optimization techniques can be applied. We also prove a new result that the data complexity of the instance checking problem in REG c logics is coNPcomplete. 1
An Extension of Regularity Conditions for Complex Role Inclusion Axioms
"... The description logic (DL) SROIQ [1] provides a logical foundation for the new version of the web ontology language OWL 2. 1 In comparison to the DL SHOIN which underpins the first version of OWL, 2 SROIQ provides several new constructors for classes and axioms. One of the new powerful features of ..."
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The description logic (DL) SROIQ [1] provides a logical foundation for the new version of the web ontology language OWL 2. 1 In comparison to the DL SHOIN which underpins the first version of OWL, 2 SROIQ provides several new constructors for classes and axioms. One of the new powerful features of
Clausal Tableaux for Multimodal Logics of Belief
"... Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multidegree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multiagent systems. Our tableau calculi are sound, complete, cutf ..."
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Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multidegree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multiagent systems. Our tableau calculi are sound, complete, cutfree and have the analytic superformula property, thereby giving decision procedures for all of these logics. We also use our calculi to obtain complexity results for five of these logics. The complexity of one was known and that of the seventh remains open.
A Tableau Calculus with AutomatonLabelled Formulae for Regular Grammar Logics
"... Abstract. We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automatonlabelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cutfree and has the analytic superf ..."
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Abstract. We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automatonlabelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. We show that the known EXPTIME upper bound for regular grammar logics can be obtained using our tableau calculus. We also give an effective Craig interpolation lemma for regular grammar logics using our calculus. 1
Nominal Schemas for Integrating Rules and Description Logics
"... Abstract. We propose an extension of SROIQ with nominal schemas which can be used like “variable nominal concepts ” within axioms. This feature allows us to express arbitrary DLsafe rules in description logic syntax. We show that adding nominal schemas to SROIQ does not increase its worstcase reas ..."
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Abstract. We propose an extension of SROIQ with nominal schemas which can be used like “variable nominal concepts ” within axioms. This feature allows us to express arbitrary DLsafe rules in description logic syntax. We show that adding nominal schemas to SROIQ does not increase its worstcase reasoning complexity, and we identify a family of tractable DLs SROELV n that allow for restricted use of nominal schemas. 1