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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.
Computing finite models by reduction to functionfree clause logic
 Journal of Applied Logic
, 2007
"... Recent years have seen considerable interest in procedures for computing finite models of firstorder logic specifications. One of the major paradigms, MACEstyle model building, is based on reducing model search to a sequence of propositional satisfiability problems and applying (efficient) SAT sol ..."
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Cited by 32 (9 self)
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Recent years have seen considerable interest in procedures for computing finite models of firstorder logic specifications. One of the major paradigms, MACEstyle model building, is based on reducing model search to a sequence of propositional satisfiability problems and applying (efficient) SAT solvers to them. A problem with this method is that it does not scale well because the propositional formulas to be considered may become very large. We propose instead to reduce model search to a sequence of satisfiability problems consisting of functionfree firstorder clause sets, and to apply (efficient) theorem provers capable of deciding such problems. The main appeal of this method is that firstorder clause sets grow more slowly than their propositional counterparts, thus allowing for more space efficient reasoning. In this paper we describe our proposed reduction in detail and discuss how it is integrated into the Darwin prover, our implementation of the Model Evolution calculus. The results are general, however, as our approach can be used in principle with any system that decides the satisfiability of functionfree firstorder clause sets. To demonstrate its practical feasibility, we tested our approach on all satisfiable problems from the TPTP library. Our methods can solve a significant subset of these problems, which overlaps but is not included in the subset of problems solvable by stateoftheart finite model builders such as Paradox and Mace4.
Hyper Tableaux with Equality
 PROCEEDINGS OF THE 21ST INTERNATIONAL CONFERENCE ON AUTOMATED DEDUCTION, NUMBER 4603 IN LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2007
"... In most theorem proving applications, a proper treatment of equational theories or equality is mandatory. In this paper we show how to integrate a modern treatment of equality in the hyper tableau calculus. It is based on splitting of positive clauses and an adapted version of the superposition in ..."
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Cited by 24 (5 self)
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In most theorem proving applications, a proper treatment of equational theories or equality is mandatory. In this paper we show how to integrate a modern treatment of equality in the hyper tableau calculus. It is based on splitting of positive clauses and an adapted version of the superposition inference rule, where equations used for paramodulation are drawn (only) from a set of positive unit clauses, the candidate model. The calculus also features a generic, semantically justified simplification rule which covers many redundancy elimination techniques known from superpositionstyle theorem proving. Our main theoretical result is the soundness and completeness of the calculus. The calculus is implemented, and we also report on practical experiments.
System Description: EKRHyper
"... Abstract. The EKRHyper system is a model generator and theorem prover for firstorder logic with equality. It implements the new Ehyper tableau calculus, which integrates a superpositionbased handling of equality into the hyper tableau calculus. EKRHyper extends our previous KRHyper system, whic ..."
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Cited by 21 (4 self)
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Abstract. The EKRHyper system is a model generator and theorem prover for firstorder logic with equality. It implements the new Ehyper tableau calculus, which integrates a superpositionbased handling of equality into the hyper tableau calculus. EKRHyper extends our previous KRHyper system, which has been used in a number of applications in the field of knowledge representation. In contrast to most first order theorem provers, it supports features important for such applications, for example queries with predicate extensions as answers, handling of large sets of uniformly structured input facts, arithmetic evaluation and stratified negation as failure. It is our goal to extend the range of application possibilities of KRHyper by adding equality reasoning. 1
Optimized Description Logic Reasoning via Core Blocking
"... Abstract. State of the art reasoners for expressive description logics, such as those that underpin the OWL ontology language, are typically based on highly optimized implementations of (hyper)tableau algorithms. Despite numerous optimizations, certain ontologies encountered in practice still pose s ..."
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Cited by 13 (2 self)
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Abstract. State of the art reasoners for expressive description logics, such as those that underpin the OWL ontology language, are typically based on highly optimized implementations of (hyper)tableau algorithms. Despite numerous optimizations, certain ontologies encountered in practice still pose significant challenges to such reasoners, mainly because of the size of the model abstractions that they construct. To address this problem, we propose a new blocking technique that tries to identify and halt redundant construction at a much earlier stage than standard blocking techniques. An evaluation of a prototypical implementation in the HermiT reasoner shows that our technique can dramatically reduce the size of constructed model abstractions and reduce reasoning time. 1
Automated Synthesis of Tableau Calculi
"... Abstract This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules which can then be used to reason within the logic. The method guarantees that the generated rules ..."
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Abstract This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules which can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a welldefined firstorder semantics then adding a general blocking mechanism produces a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for propositional intuitionistic logic. 1
D.: Using tableau to decide description logics with full role negation and identity (2011), manuscript, http://www.mettelprover.org/ papers/ALBOid.pdf
"... This paper presents a tableau approach for deciding expressive description logics with full role negation and role identity. We consider the description logic ALBO id, which is ALC extended with the Boolean role operators, inverse of roles, the identity role, and includes full support for individual ..."
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Cited by 10 (9 self)
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This paper presents a tableau approach for deciding expressive description logics with full role negation and role identity. We consider the description logic ALBO id, which is ALC extended with the Boolean role operators, inverse of roles, the identity role, and includes full support for individuals and singleton concepts. ALBO id is expressively equivalent to the twovariable fragment of firstorder logic with equality and subsumes Boolean modal logic. In this paper we define a sound, complete and terminating tableau calculus for ALBO id that provides the basis for decision procedures for this logic and all its sublogics. An important novelty of our approach is the use of a generic unrestricted blocking mechanism. Unrestricted blocking is based on equality reasoning and a conceptually simple rule, which performs case distinctions over the identity of individuals. The blocking mechanism ties the proof of termination of tableau derivations to the finite model property of ALBO id.
InstGen – A Modular Approach to InstantiationBased Automated Reasoning
"... Abstract. InstGen is an instantiationbased reasoning method for firstorder logic introduced in [18]. One of the distinctive features of InstGen is a modular combination of firstorder reasoning with efficient ground reasoning. Thus, InstGen provides a framework for utilising efficient offthes ..."
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Cited by 6 (2 self)
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Abstract. InstGen is an instantiationbased reasoning method for firstorder logic introduced in [18]. One of the distinctive features of InstGen is a modular combination of firstorder reasoning with efficient ground reasoning. Thus, InstGen provides a framework for utilising efficient offtheshelf propositional SAT and SMT solvers as part of general firstorder reasoning. In this paper we present a unified view on the developments of the InstGen method: (i) completeness proofs; (ii) abstract and concrete criteria for redundancy elimination, including dismatching constraints and global subsumption; (iii) implementation details and evaluation. 1
Noncyclic sorts for firstorder satisfiability
"... Abstract. In this paper we investigate the finite satisfiability problem for firstorder logic. We show that the finite satisfiability problem can be represented as a sequence of satisfiability problems in a fragment of manysorted logic, which we call the noncyclic fragment. The noncyclic fragment ..."
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Abstract. In this paper we investigate the finite satisfiability problem for firstorder logic. We show that the finite satisfiability problem can be represented as a sequence of satisfiability problems in a fragment of manysorted logic, which we call the noncyclic fragment. The noncyclic fragment can be seen as a generalisation of the effectively propositional fragment (EPR) in the manysorted setting. We show that the noncyclic fragment is decidable by instantiationbased methods and present a linear time algorithm for checking whether a given clause set is in this fragment. One of the distinctive features of our finite satisfiability translation is that it avoids unnecessary flattening of terms, which can be crucial for efficiency. We implemented our finite model finding translation in iProver and evaluated it over the TPTP library. Using our translation it was possible solve a large class of problems which could not be solved by other systems. 1