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A Sahlqvist theorem for distributive modal logic
 Annals of Pure and Applied Logic 131, Issues
, 2002
"... Dedicated to Bjarni Jónsson In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For ..."
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Cited by 23 (9 self)
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Dedicated to Bjarni Jónsson In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.
Automated Theorem Proving by Resolution for FinitelyValued Logics Based on Distributive Lattices with Operators
 An International Journal of MultipleValued Logic
, 1999
"... In this paper we present a method for automated theorem proving in manyvalued logics whose algebra of truth values is a nite distributive lattice with operators. This class of manyvalued logics includes many logics that occur in a natural way in applications. The method uses the Priestley dual of t ..."
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Cited by 11 (2 self)
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In this paper we present a method for automated theorem proving in manyvalued logics whose algebra of truth values is a nite distributive lattice with operators. This class of manyvalued logics includes many logics that occur in a natural way in applications. The method uses the Priestley dual of the algebra of truth values instead of the algebra itself; this dual is used as a finite set of possible worlds. We first present a procedure that constructs, for every formula in the language of such a logic, a set of signed clauses such that is a theorem if and only if is unsatisfiable. Compared to related approaches, the method presented here leads in many cases to a reduction of the number of clauses that are generated, especially when the set of truth values is not linearly ordered. We then discuss several possibilities for checking the unsatisfiability of , among which a version of signed hyperresolution, and give several examples.
On the Universal Theory of Varieties of Distributive Lattices with Operators: Some Decidability and Complexity Results
 Proceedings of CADE16, LNAI 1632
, 1999
"... . In this paper we establish a link between satisability of universal sentences with respect to varieties of distributive lattices with operators and satisability with respect to certain classes of relational structures. We use these results for giving a method for translation to clause form of ..."
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Cited by 4 (4 self)
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. In this paper we establish a link between satisability of universal sentences with respect to varieties of distributive lattices with operators and satisability with respect to certain classes of relational structures. We use these results for giving a method for translation to clause form of universal sentences in such varieties, and then use results from automated theorem proving to obtain decidability and complexity results for the universal theory of some such varieties. 1 Introduction In this paper we give a method for automated theorem proving in the universal theory of certain varieties of distributive lattices with wellbehaved operators. For this purpose, we use extensions of Priestley's representation theorem for distributive lattices. The advantage of our method is that we avoid the explicit use of the full algebraic structure of such lattices, instead using sets endowed with a reexive and transitive relation and with additional functions and relations that corr...
Representation Theorems and Theorem Proving in NonClassical Logics
 In Proceedings of the 29th IEEE International Symposium on MultipleValued Logic. IEEE Computer Sociaty
, 1999
"... In this paper we present a method for automated theorem proving in nonclassical logics having as algebraic models bounded distributive lattices with certain types of operators. The idea is to use a Priestleystyle representation for distributive lattices with operators in order to define a class of ..."
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Cited by 1 (1 self)
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In this paper we present a method for automated theorem proving in nonclassical logics having as algebraic models bounded distributive lattices with certain types of operators. The idea is to use a Priestleystyle representation for distributive lattices with operators in order to define a class of Kripkestyle models with respect to which the logic is sound and complete. If this class of Kripkestyle models is elementary, it can then be used for a translation to clause form; satisfiability of the resulting clauses can be checked by resolution. We illustrate the ideas by several examples. 1 Introduction Efficient reasoning on incomplete, vague and imprecise knowledge requires the development of efficient manyvalued theorem provers. Since many nonclassical logics that occur in a natural way in practical applications can be proved to be sound and complete with respect to certain classes of distributive lattices with operators, in this paper we will focus on this kind of logics. One o...
TruthValues as Labels: A General Recipe for Labelled Deduction
"... We introduce a general recipe for presenting nonclassical logics in a modular and uniform way as labelled natural deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truthvalues. ..."
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Cited by 1 (1 self)
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We introduce a general recipe for presenting nonclassical logics in a modular and uniform way as labelled natural deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truthvalues.
On uniform word problems involving bridging operators on distributive lattices
 Proceedings of TABLEAUX 2002. LNAI 2381
, 2002
"... Abstract. In this paper we analyze some fragments of the universal theory of distributive lattices with many sorted bridging operators. Our interest in such algebras is motivated by the fact that, in description logics, numerical features are often expressed by using maps that associate numerical va ..."
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Abstract. In this paper we analyze some fragments of the universal theory of distributive lattices with many sorted bridging operators. Our interest in such algebras is motivated by the fact that, in description logics, numerical features are often expressed by using maps that associate numerical values to sets (more generally, to lattice elements). We first establish a link between satisfiability of universal sentences with respect to algebraic models and satisfiability with respect to certain classes of relational structures. We use these results for giving a method for translation to clause form of universal sentences, and provide some decidability results based on the use of resolution or hyperresolution. Links between hyperresolution and tableau methods are also discussed, and a tableau procedure for checking satisfiability of formulae of type t1 ≤ t2 is obtained by using a hyperresolution calculus. 1
Distributive Substructural Logics as Coalgebraic Logics over Posets
"... We show how to understand frame semantics of distributive substructural logics coalgebraically, thus opening a possibility to study them as coalgebraic logics. As an application of this approach we prove a general version of GoldblattThomason theorem that characterizes definability of classes of fr ..."
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We show how to understand frame semantics of distributive substructural logics coalgebraically, thus opening a possibility to study them as coalgebraic logics. As an application of this approach we prove a general version of GoldblattThomason theorem that characterizes definability of classes of frames for logics extending the distributive Full Lambek logic, as e.g. relevance logics, manyvalued logics or intuitionistic logic. The paper is rather conceptual and does not claim to contain significant new results. We consider a category of frames as posets equipped with monotone relations, and show that they can be understood as coalgebras for an endofunctor of the category of posets. In fact, we adopt a more general definition of frames that allows to cover a wider class of distributive modal logics. GoldblattThomason theorem for classes of resulting coalgebras for instance shows that frames for axiomatic extensions of distributive Full Lambek logic are modally definable classes of certain coalgebras, the respective modal algebras being precisely the corresponding subvarieties of distributive residuated lattices.