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30
PSPACE bounds for rank 1 modal logics
 IN LICS’06
, 2006
"... For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a sh ..."
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Cited by 34 (16 self)
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For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACEbounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.
Combining Tableaux and Algebraic Methods for Reasoning with Qualified Number Restrictions
 In this volume
, 2001
"... This paper investigates an optimization technique for reasoning with qualified number restrictions in the description logic ..."
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Cited by 22 (12 self)
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This paper investigates an optimization technique for reasoning with qualified number restrictions in the description logic
A Hybrid Tableau Algorithm for ALCQ
"... Abstract. We propose an approach for extending a tableaubased satisfiability algorithm by an arithmetic component. The result is a hybrid satisfiability algorithm for the Description Logic (DL) ALCQ which extends ALC with qualified number restrictions. The hybrid approach ensures a more informed ca ..."
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Cited by 15 (10 self)
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Abstract. We propose an approach for extending a tableaubased satisfiability algorithm by an arithmetic component. The result is a hybrid satisfiability algorithm for the Description Logic (DL) ALCQ which extends ALC with qualified number restrictions. The hybrid approach ensures a more informed calculus which, on the one hand, adequately handles the interaction between numerical and logical restrictions of descriptions, and on the other hand, when applied is a very promising framework for average case optimizations. 1
Algebraic tableau reasoning for the description logic SHOQ
 Logic Journal of the IGPL, Special Issue on Hybrid Logics
"... Semantic web applications based on the web ontology language (OWL) often require the use of numbers in class descriptions for expressing cardinality restrictions on properties or even classes. Some of these cardinalities are specified explicitly but quite a few are entailed and need to be discovered ..."
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Cited by 10 (6 self)
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Semantic web applications based on the web ontology language (OWL) often require the use of numbers in class descriptions for expressing cardinality restrictions on properties or even classes. Some of these cardinalities are specified explicitly but quite a few are entailed and need to be discovered by reasoning procedures. Due to the description logic (DL) foundation of OWL those reasoning services are offered by DL reasoners which employ reasoning procedures that are arithmetically uninformed and substitute arithmetic reasoning by “don’t know ” nondeterminism in order to cover all possible cases. This lack of information about arithmetic problems dramatically degrades the performance of DL reasoners in many cases, especially with ontologies relying on the use of nominals (O) and qualified cardinality restrictions (Q). In this article we present a new algebraic tableau reasoning procedure for the DL SHOQ that combines tableau procedures and algebraic methods, namely linear integer programming, to ensure arithmetically better informed reasoning procedures. SHOQ extends the standard DL ALC (which is equivalent to the multimodal logic Km) with transitive roles, role hierarchies, qualified cardinality restrictions, and nominals, and forms an expressive subset of the web ontology language OWL 2. Although the proposed algebraic tableau (in analogy to standard tableau) is still double exponential in the worst case, it deals with cardinalities in a very informed way due to its arithmetic component and can be considered as a novel foundation for informed reasoning procedures addressing cardinality restrictions.
Quantifierelimination for the firstorder theory of boolean algebras with linear cardinality constraints
 In Proc. Advances in Databases and Information Systems (ADBIS’04), volume 3255 of LNCS
, 2004
"... Abstract. We present for the firstorder theory of atomic Boolean algebras of sets with linear cardinality constraints a quantifier elimination algorithm. In the case of atomic Boolean algebras of sets, this is a new generalization of Boole’s wellknown variable elimination method for conjunctions o ..."
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Cited by 7 (0 self)
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Abstract. We present for the firstorder theory of atomic Boolean algebras of sets with linear cardinality constraints a quantifier elimination algorithm. In the case of atomic Boolean algebras of sets, this is a new generalization of Boole’s wellknown variable elimination method for conjunctions of Boolean equality constraints. We also explain the connection of this new logical result with the evaluation of relational calculus queries on constraint databases that contain Boolean linear cardinality constraints. 1
Practical Reasoning with Qualified Number Restrictions: A Hybrid Abox Calculus for the Description Logic SHQ
"... This article presents a hybrid Abox tableau calculus for SHQ which extends the basic description logic ALC with role hierarchies, transitive roles, and qualified number restrictions. The prominent feature of our hybrid calculus is that it reduces reasoning about qualified number restrictions to inte ..."
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Cited by 6 (4 self)
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This article presents a hybrid Abox tableau calculus for SHQ which extends the basic description logic ALC with role hierarchies, transitive roles, and qualified number restrictions. The prominent feature of our hybrid calculus is that it reduces reasoning about qualified number restrictions to integer linear programming. The calculus decides SHQ Abox consistency w.r.t. a Tbox containing general axioms. The presented approach ensures a more informed calculus which adequately handles the interaction between numerical and logical restrictions in SHQ concept and individual descriptions. A prototype reasoner for deciding ALCHQ concept satisfiability has been implemented. An empirical evaluation of our hybrid reasoner and its integrated optimization techniques for a set of synthesized benchmarks featuring qualified number restrictions clearly demonstrates the effectiveness of our hybrid calculus.
Modal logics with counting
 in WoLLIC, ser. LNCS 6188, 2010
"... Abstract. We present a modal language that includes explicit operators to count the number of elements that a model might include in the extension of a formula, and we discuss how this logic has been previously investigated under different guises. We show that the language is related to graded moda ..."
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Cited by 5 (0 self)
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Abstract. We present a modal language that includes explicit operators to count the number of elements that a model might include in the extension of a formula, and we discuss how this logic has been previously investigated under different guises. We show that the language is related to graded modalities and to hybrid logics. We illustrate a possible application of the language to the treatment of plural objects and queries in natural language. We investigate the expressive power of this logic via bisimulations, discuss the complexity of its satisfiability problem, define a new reasoning task that retrieves the cardinality bound of the extension of a given input formula, and provide an algorithm to solve it. 1 Counting, Modally Suppose there are at least two apples (say, on the table, but we don’t care at the moment where the apples are). Firstorder logic (FOL) with equality has no problem expressing this fact1: ∃x.∃y.(x 6 = y ∧Apple(x) ∧Apple(y)).
Collections, Cardinalities, and Relations
"... Abstract. Logics that involve collections (sets, multisets), and cardinality constraints are useful for reasoning about unbounded data structures and concurrent processes. To make such logics more useful in verification this paper extends them with the ability to compute direct and inverse relation ..."
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Cited by 5 (3 self)
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Abstract. Logics that involve collections (sets, multisets), and cardinality constraints are useful for reasoning about unbounded data structures and concurrent processes. To make such logics more useful in verification this paper extends them with the ability to compute direct and inverse relation and function images. We establish decidability and complexity bounds for the extended logics. 1
Algebraic Tableau Algorithm for ALCOQ
"... Description Logics (DLs) are a family of knowledge representation formalisms used to represent and reason about an application’s domain elements. They are applicable in the semantic web as they provide the basis for the Web Ontology Language (OWL). Decision procedures for expressive DLs enabling bot ..."
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Cited by 4 (3 self)
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Description Logics (DLs) are a family of knowledge representation formalisms used to represent and reason about an application’s domain elements. They are applicable in the semantic web as they provide the basis for the Web Ontology Language (OWL). Decision procedures for expressive DLs enabling both nominals and QCRs were
ExpTime tableaux for the description logic SHIQ based on global state caching and integer linear feasibility checking. arXiv:1205.5838
, 2012
"... Abstract. We give the first ExpTime (complexityoptimal) tableau decision procedure for checking satisfiability of a knowledge base in the description logic SHIQ when numbers are coded in unary. Our procedure is based on global state caching and integer linear feasibility checking. ..."
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Cited by 3 (3 self)
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Abstract. We give the first ExpTime (complexityoptimal) tableau decision procedure for checking satisfiability of a knowledge base in the description logic SHIQ when numbers are coded in unary. Our procedure is based on global state caching and integer linear feasibility checking.