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Sequent of Relations Calculi: A Framework for Analytic Deduction in ManyValued Logics
 BEYOND TWO: THEORY AND APPLICATIONS OF MULTIPLEVALUED LOGICS
, 2003
"... We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) manyvalued logics  called projective logics  characterized by a special format of their semantics. All finitevalued logics as well as infinitevalued Godel logic ..."
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We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) manyvalued logics  called projective logics  characterized by a special format of their semantics. All finitevalued logics as well as infinitevalued Godel logic are projective. As a casestudy, sequent of relations calculi for Godel logics are derived. A comparison with some other analytic calculi is provided.
Towards Specifying with Inclusions
, 1997
"... In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalizatio ..."
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Cited by 5 (2 self)
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In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalization of the conventional algebraic specification by means of equations. The main aim of this generalization is to facilitate the incremental refinement of specifications. Furthermore, inclusional specifications admit a natural visual syntax which can also be used to visualize the reasoning process. We show that reasoning with inclusions is well captured by birewriting, a rewriting technique introduced by Levy and Agust'i [15]. However, there are still key problems to be solved in order to have executable inclusional specifications, necessary for rapid prototyping purposes. The article mainly points to the potentialities and difficulties of specifying with inclusions.
NonSymmetric Rewriting
, 1996
"... Rewriting is traditionally presented as a method to compute normal forms in varieties. Conceptually, however, its essence are commutation properties. We develop rewriting as a general theory of commutation for two possibly nonsymmetric transitive relations modulo a congruence and prove a general ..."
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Rewriting is traditionally presented as a method to compute normal forms in varieties. Conceptually, however, its essence are commutation properties. We develop rewriting as a general theory of commutation for two possibly nonsymmetric transitive relations modulo a congruence and prove a generalization of the standard ChurchRosser theorem. The theorems of equational rewriting, including the existence of normal forms, derive as corollaries to this result. Completion also is purely commutational and we show how to extend it to plain transitive relations. Nevertheless the loss of symmetry introduces some unpleasant consequences: unique normal forms do not exist, rewrite proofs cannot be found by don'tcare nondeterministic rewriting and also simplification during completion requires backtracking. On the nonground level, variable critical pairs have to be considered. Keywords Transitive Relations, Rewriting, Commutation, Completion. 1 Introduction Term rewriting is one of t...
P.: A resolution procedure for description logics with nominal schemas
 Proc. 2nd Joint Int. Conf. on Semantic Technology (JIST’12). LNCS
, 2013
"... Abstract. We present a polynomial resolutionbased decision procedure for the recently introduced description logic ELHOVn(u) , which features nominal schemas as new language construct. Our algorithm is based on ordered resolution and positive superposition, together with a lifting lemma. In contra ..."
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Abstract. We present a polynomial resolutionbased decision procedure for the recently introduced description logic ELHOVn(u) , which features nominal schemas as new language construct. Our algorithm is based on ordered resolution and positive superposition, together with a lifting lemma. In contrast to previous work on resolution for description logics, we have to overcome the fact that ELHOVn(u) does not allow for a normalization resulting in clauses of globally limited size. 1
Theorem Proving with Transitive Relations from a Practical Point of View
 Research Report IIIA 95/12, Institut d'Investigaci'o en Intel\Deltalig`encia Artificial (CSIC
, 1995
"... Rewrite techniques have been typically applied to reason with the equality relation and have turned out to be among the more successful approaches to equational theorem proving. In fact, it is not only in reasoning with the equality relation where these techniques naturally apply, but in reasoning w ..."
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Cited by 3 (3 self)
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Rewrite techniques have been typically applied to reason with the equality relation and have turned out to be among the more successful approaches to equational theorem proving. In fact, it is not only in reasoning with the equality relation where these techniques naturally apply, but in reasoning with arbitrary, probably nonsymmetric, transitive relations, being the equality relation just a special case of monotone transitive relation, which is also symmetric. The work done so far in applying rewrite techniques to arbitrary transitive relations showed several important differences with the equational case. Although most equational results can be extended to nonsymmetric relations, new problems appear which must be solved in a quite different way. In this paper we review the use of rewrite techniques for reasoning with arbitrary possibly nonsymmetric transitive relations and we analyze the reasons why an efficient treatment of this generalization to nonsymmetric transitive relations...
Cancellative Abelian Monoids in Refutational Theorem Proving
 PHD THESIS, INSTITUT FÜR INFORMATIK, UNIVERSITÄT DES SAARLANDES
, 1997
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Free Variable Tableaux for a Many Sorted Logic with Preorders
, 1996
"... The proof of properties of formal systems including inequalities is currently evolving into an increasingly appealing workline in different areas of computer science. We propose a sound and complete semantic tableau method for handling manysorted preorders. As logical framework a manysorted first ..."
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The proof of properties of formal systems including inequalities is currently evolving into an increasingly appealing workline in different areas of computer science. We propose a sound and complete semantic tableau method for handling manysorted preorders. As logical framework a manysorted first order logic is supplied, where functions and predicates behave monotonically or antimonotonically on their arguments. We formulate additional expansion tableau rules as a more efficient alternative to adding the axioms characterizing a preordered structure. Efficiency of the method is improved using a free variable tableau version. Completeness of the system is proved in detail; examples and applications are introduced. 1 Introduction Almost every formal theory of interest uses the equality relation in its definition, for this reason one of the main goals of automated deduction concerns the efficient proof of properties in the context of some formal framework including equality. On the othe...
Inclusional Theories in Declarative Programming
 In Joint Conference on Declarative Programming APPIAGULPPRODE
, 1996
"... When studying specific deduction techniques and strategies for operational semantics of logic programming languages special emphasis was put on the equality relation, due to its interest in a variety of different domains. But recently special emphasis has been put on partial order relations, and spe ..."
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When studying specific deduction techniques and strategies for operational semantics of logic programming languages special emphasis was put on the equality relation, due to its interest in a variety of different domains. But recently special emphasis has been put on partial order relations, and specifically on inclusions, as a basis for several different specification frameworks. It is therefore attractive to work towards a logic programming language which deals efficiently with inclusions, and which may be useful as a rapid prototyping tool. Term rewriting appears to be a suitable technique for theorem proving with inclusional theories, since it naturally applies to arbitrary (possibly nonsymmetric) transitive relations, but turns out to be impractical in general. Therefore several restrictions need to be put on inclusional theories in order to improve the inference mechanism and to define efficient deduction strategies, which could be used in an operational semantics of an inclusio...
Birewrite Systems † JORDI LEVY ‡ § AND JAUME AGUST Í¶�
, 1995
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆,R⊇〉, that is, a pair of rewrite relations − ..."
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆,R⊇〉, that is, a pair of rewrite relations −−−→ R ⊆ and −−−→ R ⊇ , and seek a common term c such that a −−−→ R c and b −−−→ R c. Each component of the birewrite system −−−→
Birewriting Rewriting Logic
, 1996
"... Rewriting logic appears to have good properties as logical framework, and can be useful for the development of programming languages which attempt to integrate various paradigms of declarative programming. In this paper I propose to tend towards the operational semantics for such languages by basing ..."
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Rewriting logic appears to have good properties as logical framework, and can be useful for the development of programming languages which attempt to integrate various paradigms of declarative programming. In this paper I propose to tend towards the operational semantics for such languages by basing it on birewrite systems and ordered chaining calculi which apply rewrite techniques to firstorder theories with arbitrary possibly nonsymmetric transitive relations, because this was an important breakthrough for the automation of deduction in these kind of theories. I show that a proof calculus based on the birewriting technique may serve as framework of different proof calculi, by analizing those of equational logic and Horn logic, and presenting them as specific cases of birewrite systems. Deduction is then essentially birewriting a theory of rewriting logic. Since recently the interest in specifications based on theories with transitive relations has arisen, the result of this res...