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31
Theorem Proving with Ordering and Equality Constrained Clauses
 Journal of Symbolic Computation
, 1995
"... constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples ..."
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Cited by 74 (19 self)
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constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples of clauses to sets of triples containing a clause, a constraint and an equality constraint: IR : S n \Gamma! P(hS; C; ECi) An inference system is a set of inference rules. Definition 3.2. A constraint inheritance strategy is a function mapping a clause, two constraints and an equality constraint to a clause and a constraint: H : S \Theta C \Theta C \Theta EC \Gamma! S \Theta C Inference systems and constraint inheritance strategies are combined to produce inferences in the usual sense: given constrained clauses C 1 [[T 1 ]]; : : : ; Cn [[T n ]], we obtain a conclusion C [[T ]] as follows. Applying an inference rule to C 1 ; : : : ; Cn we obtain a triple hD; OT;ET i. Then the constraint...
Set Constraints are the Monadic Class
, 1992
"... We investigate the relationship between set constraints and the monadic class of firstorder formulas and show that set constraints are essentially equivalent to the monadic class. From this equivalence we can infer that the satisfiability problem for set constraints is complete for NEXPTIME. Mor ..."
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Cited by 71 (0 self)
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We investigate the relationship between set constraints and the monadic class of firstorder formulas and show that set constraints are essentially equivalent to the monadic class. From this equivalence we can infer that the satisfiability problem for set constraints is complete for NEXPTIME. More precisely, we prove that this problem has a lower bound of NTIME(c n= log n ). The relationship between set constraints and the monadic class also gives us decidability and complexity results for certain practically useful extensions of set constraints, in particular "negative projections" and subterm equality tests.
A Superposition Decision Procedure for the Guarded Fragment with Equality
 In Proc. LICS'99
, 1999
"... We give a new decision procedure for the guarded fragment with equality. The procedure is based on resolution with superposition. We argue that this method will be more useful in practice than methods based on the enumeration of certain finite structures. It is surprising to see that one does not ne ..."
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Cited by 56 (2 self)
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We give a new decision procedure for the guarded fragment with equality. The procedure is based on resolution with superposition. We argue that this method will be more useful in practice than methods based on the enumeration of certain finite structures. It is surprising to see that one does not need any sophisticated simplification and redundancy elimination method to make superposition terminate on the class of clauses that is obtained from the clausification of guarded formulas. Yet the decision procedure obtained is optimal with regard to time complexity. We also show that the method can be extended to the loosely guarded fragment with equality. 1 Introduction The loosely guarded fragment was introduced in (Andreka, van Benthem &Nemeti 1996) as 'the modal fragment of classical logic'. It is obtained essentially by restricting quantification to the following forms: #y[R(x, y) # A(x, y)] and #y[R(x, y) # A(x, y)]. These forms naturally arise when modal formulae are transl...
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 32 (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 Method for Building Models Automatically. Experiments with an extension of OTTER
 In Proceedings of CADE12
, 1994
"... . A previous work on Herbrand model construction is extended in two ways. The first extension increases the capabilities of the method, by extending one of its key rules. The second, more important one, defines a new method for simultaneous search of refutations and models for set of equational clau ..."
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Cited by 28 (14 self)
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. A previous work on Herbrand model construction is extended in two ways. The first extension increases the capabilities of the method, by extending one of its key rules. The second, more important one, defines a new method for simultaneous search of refutations and models for set of equational clauses. The essential properties of the new method are given. The main theoretical result of the paper is the characterization of conditions assuring that models can be built. Both methods (for equational and non equational clauses) have been implemented as an extension of OTTER. Several running examples are given, in particular a new automatic solution of the ternary algebra problem first solved by Winker. The examples emphasize the unified approach to model building allowed by the ideas underlying our method and the usefulness of using constrained clauses. Several problems open by the present work are the main lines of future work. 1 Introduction It is trivial to say that the use of models o...
A resolutionbased decision procedure for SHOIQ
 Proc. of the 3rd Int. Joint Conf. on Automated Reasoning (IJCAR 2006), volume 4130 of LNAI
, 2006
"... Abstract. We present a resolutionbased decision procedure for the description logic SHOIQ—the logic underlying the Semantic Web ontology language OWLDL. Our procedure is goaloriented, and it naturally extends a similar procedure for SHIQ, which has proven itself in practice. Applying existing tec ..."
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Cited by 24 (6 self)
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Abstract. We present a resolutionbased decision procedure for the description logic SHOIQ—the logic underlying the Semantic Web ontology language OWLDL. Our procedure is goaloriented, and it naturally extends a similar procedure for SHIQ, which has proven itself in practice. Applying existing techniques for deriving saturationbased decision procedures to SHOIQ is not straightforward due to nominals, number restrictions, and inverse roles—a combination known to cause termination problems. We overcome this difficulty by using the basic superposition calculus, extended with custom simplification rules. 1
Decidability and Complexity Analysis by Basic Paramodulation
, 1998
"... It is shown that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superpositio ..."
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Cited by 12 (7 self)
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It is shown that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). These two results are applied to the following languages. For shallow presentations (equations with variables at depth at most one) we show that the closure under paramodulation can be computed in polynomial time. Applying result (i), it follows that shallow unifiability is in NP, which is optimal since unifiability in ground theories is already NPhard. The shallow word problem is even shown to be polynomial. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramo...
Computational modal logic
 Handbook of Modal Logic
, 2006
"... 2 Syntax, semantics, and reasoning problems of modal logics........................... 3 3 Translationbased methods........................................... 6 3.1 Local satisfiability in multi modal Kn................................... 6 3.2 Global satisfiability, nonlogical axioms, transitive ..."
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Cited by 10 (3 self)
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2 Syntax, semantics, and reasoning problems of modal logics........................... 3 3 Translationbased methods........................................... 6 3.1 Local satisfiability in multi modal Kn................................... 6 3.2 Global satisfiability, nonlogical axioms, transitive modalities, and K4n................. 20