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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...
Decision Procedures and Model Building in Equational Clause Logic
 JOURNAL OF THE IGPL
, 1998
"... It is shown that a combination of semantic resolution and ordered paramodulation provides a decision procedure for a large class PVD g of clause sets with equality. It is also demonstrated how the inference system can be transformed into an algorithm that extracts finite descriptions of Herbrand ..."
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Cited by 15 (2 self)
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It is shown that a combination of semantic resolution and ordered paramodulation provides a decision procedure for a large class PVD g of clause sets with equality. It is also demonstrated how the inference system can be transformed into an algorithm that extracts finite descriptions of Herbrand models from sets of clauses. This algorithm always terminates on clause sets in PVD g and yields an appropriate model representation. Moreover, an algorithm for evaluating arbitrary clauses over the represented models is defined. Finally, it is proved that PVD g enjoys the finite model property and shown how finite models can be algorithmically extracted from model representations of this type.
Maslov's Class K Revisited
 In Proc. CADE16
, 1999
"... . This paper gives a new treatment of Maslov's class K in the framework of resolution. More specifically, we show that K and the class DK consisting of disjunction of formulae in K can be decided by a resolution refinement based on liftable orderings. We also discuss relationships to other solvable ..."
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Cited by 15 (11 self)
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. This paper gives a new treatment of Maslov's class K in the framework of resolution. More specifically, we show that K and the class DK consisting of disjunction of formulae in K can be decided by a resolution refinement based on liftable orderings. We also discuss relationships to other solvable and unsolvable classes. 1 Introduction Maslov's class K [13] is one of the most important solvable fragments of firstorder logic. It contains a variety of classical solvable fragments including the Monadic class, the initially extended Skolem class, the Godel class, and the twovariable fragment of firstorder logic FO 2 [4]. It also encompasses a range of nonclassical logics, like a number of extended modal logics, many description logics used in the field of knowledge representation [11, 4, chap. 7], and some reducts of representable relational algebras. For this reason practical decision procedures for the class K are of general interest. According to Maslov [13] the inverse method pro...
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...
Basic Paramodulation and Decidable Theories (Extended Abstract)
 in `Proceedings 11th IEEE Symposium on Logic in Computer Science, LICS'96', IEEE Computer
, 1996
"... ) Robert Nieuwenhuis Technical University of Catalonia Pau Gargallo 5, 08028 Barcelona, Spain Email: roberto@lsi.upc.es. Abstract We prove 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 simpl ..."
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Cited by 6 (0 self)
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) Robert Nieuwenhuis Technical University of Catalonia Pau Gargallo 5, 08028 Barcelona, Spain Email: roberto@lsi.upc.es. Abstract We prove 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). We define standard theories, which include and significantly extend shallow theories. Standard presentations can be finitely closed under superposition and result (ii) applies. 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 paramodulation is finite for Catalog sets, hence (i) applies. Since for shallow sets this closure is even polynom...
Automated Model Building as Future Research Topic
"... . Zabel, and N. Peltier [CZ91, CZ92, CP95b] and J. Slaney [Sla92, Sla93]. An earlier approach by S. Winker [Win82], although practically relevant and successful, did not define a general algorithmic method. Tammet's approach, like ours, is based on resolution decision procedures. His method of finit ..."
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Cited by 1 (0 self)
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. Zabel, and N. Peltier [CZ91, CZ92, CP95b] and J. Slaney [Sla92, Sla93]. An earlier approach by S. Winker [Win82], although practically relevant and successful, did not define a general algorithmic method. Tammet's approach, like ours, is based on resolution decision procedures. His method of finite Model Building applies to the monadic and Ackermann class and is based on the termination of an ordering refinement. In the resulting model description the interpretation of the function symbols is given completely, but the interpretation of predicate symbols is only partial. Moreover, Tammet uses narrowing and works with equations on the object language level. In [FL95] Model Building is based on termination sets for hyperresolution (which yield other decision classes). Finite Model Building is performed as postprocessing step and is based on the transformation of Herbrand models; it does not use equality reasoning but filtration. Caferra and Zabel define an equational extension
Paramodulation Decision Procedures
, 2003
"... The aim of this thesis is to present Paramodulation Operators deciding the (i) Ackermann class with Equality [FS93] and (ii) the monadic class with equality [BGW93]. Ad (i) a refinement of the Ordered Paramodulation calculus [HR91] is used to define the appropriate inference operator; ad (ii) a vari ..."
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The aim of this thesis is to present Paramodulation Operators deciding the (i) Ackermann class with Equality [FS93] and (ii) the monadic class with equality [BGW93]. Ad (i) a refinement of the Ordered Paramodulation calculus [HR91] is used to define the appropriate inference operator; ad (ii) a variant of the Superposition calculus [BG94] is employed for this purpose. It seemed natural to give a uniform presentation of these underlying equational calculi. Hence, the first part of this thesis is designed to be a uniform treatment of these calculi. In the definition of the paramodulation operator that decides the monadic class with equality ordering constraints are used. Therefor we choose to present all the inference rules that Ordered Paramodulation and Superposition define with respect to ordering constraints. These constraints are used to express the restrictions normally given with the description of an inference rule as part of the object language.
Automated ModelBuilding for FirstOrderLogic with Equality
"... Introduction Although automated model builing is a new branch in the field of automated deduction, the central problem is quite old and attracted attention by prominent scientists in the last three centuries. The aim of the new logic mathesis universalis formulated by G. W. Leibniz (1646 1716) i ..."
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Introduction Although automated model builing is a new branch in the field of automated deduction, the central problem is quite old and attracted attention by prominent scientists in the last three centuries. The aim of the new logic mathesis universalis formulated by G. W. Leibniz (1646 1716) in the late 17 th century was a threefold one: (i) Construction of a universal alphabetcalled characteristica universalis, whose characters should allow the representation of all other conceptions. (ii) Evolving of a calculus ratiocinator, a calculus provided to deal with the aforementioned universal characters in a purely mechanical way. And eventually (iii) an algorithmars iudicandito decide for arbitrary sentences, given through the tokens of the mathesis universalis, their validity. Leibniz' main vision was the use of the calculus ratiocinator to eliminate misleadi
Testing for Renamability to Classes of Clause Sets (Extended Abstract)
 In Bonacina and Furbach [BF97
"... This paper investigates the problem of testing clause sets for membership in classes known from literature. In particular, we are interested in classes defined via renaming: Is it possible to rename the predicates in a way such that positive and negative literals satisfy certain conditions? We sh ..."
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This paper investigates the problem of testing clause sets for membership in classes known from literature. In particular, we are interested in classes defined via renaming: Is it possible to rename the predicates in a way such that positive and negative literals satisfy certain conditions? We show that for classes like Horn or OCC1N [5] the existence of such renamings can be decided in polynomial time, whereas the same problem is NPcomplete for class PVD [5]. The decision procedures are based on hyperresolution; if a renaming exists, it can be extracted from the final saturated clause set. 1 Introduction Within the last decade the classical decision problem for predicate logic was revisited in the frame of clause logic. Instead of deciding the validity of first order formulas in prenex normal formusually without function symbolsthe aim was now to decide the satisfiability of clause sets with a rich functional structure. Many classes of clause sets with and without ...