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44
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.
New Decidability Results for Fragments of FirstOrder Logic and Application to Cryptographic Protocols
, 2003
"... We consider a new extension of the Skolem class for firstorder logic and prove its decidability by resolution techniques. We then extend this class including the builtin equational theory of exclusive or. Again, we prove the decidability of the class by resolution techniques. ..."
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Cited by 54 (18 self)
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We consider a new extension of the Skolem class for firstorder logic and prove its decidability by resolution techniques. We then extend this class including the builtin equational theory of exclusive or. Again, we prove the decidability of the class by resolution techniques.
New Directions in InstantiationBased Theorem Proving
"... We consider instantiationbased theorem proving whereby instances of clauses are generated by certain inferences, and where inconsistency is detected by propositional tests. We give a model construction proof of completeness by which restrictive inference systems as well as admissible simplification ..."
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Cited by 38 (3 self)
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We consider instantiationbased theorem proving whereby instances of clauses are generated by certain inferences, and where inconsistency is detected by propositional tests. We give a model construction proof of completeness by which restrictive inference systems as well as admissible simplification techniques can be justified. Another contribution of the paper are novel inference systems that allow one to also employ decision procedures for firstorder fragments more complex than propositional logic. The decision procedure provides for an approximative consistency test, and the instance generation inference system is a means of successively refining the approximation.
A Decomposition Rule for Decision Procedures by Resolutionbased Calculi
 In: Proc. 11th Int. Conf. on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR
, 2004
"... Abstract. Resolutionbased calculi are among the most widely used calculi for theorem proving in firstorder logic. Numerous refinements of resolution are nowadays available, such as e.g. basic superposition, a calculus highly optimized for theorem proving with equality. However, even such an advanc ..."
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Cited by 36 (11 self)
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Abstract. Resolutionbased calculi are among the most widely used calculi for theorem proving in firstorder logic. Numerous refinements of resolution are nowadays available, such as e.g. basic superposition, a calculus highly optimized for theorem proving with equality. However, even such an advanced calculus does not restrict inferences enough to obtain decision procedures for complex logics, such as SHIQ. In this paper, we present a new decomposition inference rule, which can be combined with any resolutionbased calculus compatible with the standard notion of redundancy. We combine decomposition with basic superposition to obtain three new decision procedures: (i) for the description logic SHIQ, (ii) for the description logic ALCHIQb, and (iii) for answering conjunctive queries over SHIQ knowledge bases. The first two procedures are worstcase optimal and, based on the vast experience in building efficient theorem provers, we expect them to be suitable for practical usage. 1
Monodic temporal resolution
 ACM Transactions on Computational Logic
, 2003
"... Until recently, FirstOrder Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a f ..."
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Cited by 31 (15 self)
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Until recently, FirstOrder Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a finitely axiomatisable fragment, termed the monodic fragment, has led to improved understanding of FOTL. Yet, in order to utilise these theoretical advances, it is important to have appropriate proof techniques for this monodic fragment. In this paper, we modify and extend the clausal temporal resolution technique, originally developed for propositional temporal logics, to enable its use in such monodic fragments. We develop a specific normal form for monodic formulae in FOTL, and provide a complete resolution calculus for formulae in this form. Not only is this clausal resolution technique useful as a practical proof technique for certain monodic classes, but the use of this approach provides us with increased understanding of the monodic fragment. In particular, we here show how several features of monodic FOTL can be established as corollaries of the completeness result for the clausal temporal resolution method. These include definitions of new decidable monodic classes, simplification of existing monodic classes by reductions, and completeness of clausal temporal resolution in the case of
Deciding Effectively Propositional Logic using DPLL and substitution sets
"... We introduce a DPLL calculus that is a decision procedure for the BernaysSchönfinkel class, also known as EPR. Our calculus allows combining techniques for efficient propositional search with datastructures, such as Binary Decision Diagrams, that can efficiently and succinctly encode finite sets o ..."
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Cited by 27 (3 self)
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We introduce a DPLL calculus that is a decision procedure for the BernaysSchönfinkel class, also known as EPR. Our calculus allows combining techniques for efficient propositional search with datastructures, such as Binary Decision Diagrams, that can efficiently and succinctly encode finite sets of substitutions and operations on these. In the calculus, clauses comprise of a sequence of literals together with a finite set of substitutions; truth assignments are also represented using substitution sets. The calculus works directly at the level of sets, and admits performing parallel constraint propagation and decisions, resulting in potentially exponential speedups over existing approaches.
Clause/term resolution and learning in the evaluation of quantified boolean formulas
 Journal of Artificial Intelligence Research (JAIR
"... Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as a set of clauses. Deduction starts by inferring new clauses ..."
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Cited by 23 (6 self)
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Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as a set of clauses. Deduction starts by inferring new clauses by resolution, and goes on until the empty clause is generated or satisfiability of the set of clauses is proven, e.g., because no new clauses can be generated. In this paper, we restrict our attention to the problem of evaluating Quantified Boolean Formulas (QBFs). In this setting, the above outlined deduction process is known to be sound and complete if given a formula in CNF and if a form of resolution, called “Qresolution”, is used. We introduce Qresolution on terms, to be used for formulas in disjunctive normal form. We show that the computation performed by most of the available procedures for QBFs –based on the DavisLogemannLoveland procedure (DLL) for propositional satisfiability – corresponds to a tree in which Qresolution on terms and clauses alternate. This poses the theoretical bases for the introduction of learning, corresponding to recording Qresolution formulas associated with the nodes of the tree. We discuss the problems related to the introduction of learning in DLL based procedures, and present solutions extending stateoftheart proposals coming from the literature on propositional satisfiability. Finally, we show that our DLL based solver extended with learning, performs significantly better on benchmarks used in the 2003 QBF solvers comparative evaluation. 1.
Hyperresolution for guarded formulae
 J. Symbolic Computat
, 2000
"... Abstract. This paper investigates the use of hyperresolution as a decision procedure and model builder for guarded formulae. In general hyperresolution is not a decision procedure for the entire guarded fragment. However we show that there are natural fragments which can be decided by hyperresolutio ..."
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Cited by 16 (9 self)
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Abstract. This paper investigates the use of hyperresolution as a decision procedure and model builder for guarded formulae. In general hyperresolution is not a decision procedure for the entire guarded fragment. However we show that there are natural fragments which can be decided by hyperresolution. In particular, we prove decidability of hyperresolution with or without splitting for the fragment GF1 − and point out several ways of extending this fragment without loosing decidability. As hyperresolution is closely related to various tableaux methods the present work is also relevant for tableaux methods. We compare our approach to hypertableaux, and mention the relationship to other clausal classes which are decidable by hyperresolution. 1