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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 (25 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.
A FirstOrder Logic DavisPutnamLogemannLoveland Procedure
"... The DavisPutnamLogemannLoveland procedure (DPLL) was introduced in the early ..."
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Cited by 38 (6 self)
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The DavisPutnamLogemannLoveland procedure (DPLL) was introduced in the early
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 34 (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.
Positive Unit Hyperresolution Tableaux and Their Application to Minimal Model Generation
 Journal of Automated Reasoning
, 2000
"... . Minimal Herbrand models of sets of firstorder clauses are useful in several areas of computer science, e.g. automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are inappropriate bec ..."
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Cited by 15 (0 self)
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. Minimal Herbrand models of sets of firstorder clauses are useful in several areas of computer science, e.g. automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are inappropriate because they generate nonminimal Herbrand models and can be inefficient. This article describes an approach for generating the minimal Herbrand models of sets of firstorder clauses. The approach builds upon positive unit hyperresolution (PUHR) tableaux, that are in general smaller than conventional tableaux. PUHR tableaux formalize the approach initially introduced with the theorem prover SATCHMO. Two minimal model generation procedures are described. The first one expands PUHR tableaux depthfirst relying on a complement splitting expansion rule and on a form of backtracking involving constraints. A Prolog implementation, named MMSATCHMO, of this procedure is given and its performance on ben...
Individual Reuse in Description Logic Reasoning
 In Proc. of the 4th Int. Joint Conf. on Automated Reasoning (IJCAR 2008
, 2008
"... Abstract. Tableau calculi are the stateoftheart for reasoning in description logics (DL). Despite recent improvements, tableaubased reasoners still cannot process certain knowledge bases (KBs), mainly because they end up building very large models. To address this, we propose a tableau calculus ..."
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Cited by 14 (7 self)
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Abstract. Tableau calculi are the stateoftheart for reasoning in description logics (DL). Despite recent improvements, tableaubased reasoners still cannot process certain knowledge bases (KBs), mainly because they end up building very large models. To address this, we propose a tableau calculus with individual reuse: to satisfy an existential assertion, our calculus nondeterministically tries to reuse individuals from the model generated thus far. We present two expansion strategies: one is applicable to the DL ELOH and gives us a worstcase optimal algorithm, and the other is applicable to the DL SHOIQ. Using this technique, our reasoner can process several KBs that no other reasoner can. 1
A Survey of Decidable FirstOrder Fragments and Description Logics
 Journal of Relational Methods in Computer Science
, 2004
"... The guarded fragment and its extensions and subfragments are often considered as a framework for investigating the properties of description logics. There are also other, some less wellknown, decidable fragments of firstorder logic which all have in common that they generalise the standard tran ..."
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Cited by 14 (2 self)
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The guarded fragment and its extensions and subfragments are often considered as a framework for investigating the properties of description logics. There are also other, some less wellknown, decidable fragments of firstorder logic which all have in common that they generalise the standard translation of to firstorder logic. We provide a short survey of some of these fragments and motivate why they are interesting with respect to description logics, mentioning also connections to other nonclassical logics.
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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Cited by 12 (11 self)
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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.
Model Generation without Normal Forms and Applications in NaturalLanguage Semantics
, 1998
"... . I present a new tableauxbased model generation method for firstorder formulas without function symbols. Unlike comparable approaches, the Relational Models (RM) tableaux calculus does not require clausal input theories. I propose some applications of the RM calculus in naturallanguage semantics ..."
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Cited by 1 (1 self)
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. I present a new tableauxbased model generation method for firstorder formulas without function symbols. Unlike comparable approaches, the Relational Models (RM) tableaux calculus does not require clausal input theories. I propose some applications of the RM calculus in naturallanguage semantics and discuss its usefulness as an inference procedure in naturallanguage processing. 1 Introduction Refutational methods in automated deduction prove the unsatisfiability of logical theories. For many applications, the interpretations of a theory that show its satisfiability are at least as interesting as proofs. Model generation refers to the automatic construction of such interpretations from firstorder theories. In the recent years, there has been a growing interest in the automated deduction community in developing model generation methods for various application areas such as finite mathematics [25, 22], deductive databases [7], diagnosis [13, 1], and planning [19]. As a result, mode...