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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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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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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.
BOOLEAN TOPOLOGICAL DISTRIBUTIVE LATTICES AND CANONICAL EXTENSIONS
"... Abstract. This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a 2element underlying set; additional structure may be algebraic (l ..."
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Abstract. This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a 2element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices. 1.
Latticebased relation algebras and their representability II
 In Theory and Applications of Relational Structures as Knowledge Instruments, de
, 2003
"... www.cosc.brocku.ca Lattice–based relation algebras and their representability ⋆ ..."
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www.cosc.brocku.ca Lattice–based relation algebras and their representability ⋆
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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. 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...
Rewitzky, Coalgebraic representation of distributive lattices with operators, Topology and its Applications 154
, 2007
"... We present a framework for extending Stone’s representation theorem for distributive lattices to representation theorems for distributive lattices with operators. We proceed by introducing the definition of algebraic theory of operators over distributive lattices. Each such theory induces a functor ..."
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We present a framework for extending Stone’s representation theorem for distributive lattices to representation theorems for distributive lattices with operators. We proceed by introducing the definition of algebraic theory of operators over distributive lattices. Each such theory induces a functor on the category of distributive lattices such that its algebras are exactly the distributive lattices with operators in the original theory. We characterize the topological counterpart of these algebras in terms of suitable coalgebras on spectral spaces. We work out some of these coalgebraic representations, including a new representation theorem for distributive lattices with monotone operators. 1
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
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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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.
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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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...