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Finite equational bases in process algebra: Results and open questions
 Processes, Terms and Cycles: Steps on the Road to Infinity, LNCS 3838
, 2005
"... Abstract. Van Glabbeek (1990) presented the linear time/branching time spectrum of behavioral equivalences for finitely branching, concrete, sequential processes. He studied these semantics in the setting of the basic process algebra BCCSP, and tried to give finite complete axiomatizations for them. ..."
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Cited by 29 (19 self)
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Abstract. Van Glabbeek (1990) presented the linear time/branching time spectrum of behavioral equivalences for finitely branching, concrete, sequential processes. He studied these semantics in the setting of the basic process algebra BCCSP, and tried to give finite complete axiomatizations for them. Obtaining such axiomatizations in concurrency theory often turns out to be difficult, even in the setting of simple languages like BCCSP. This has raised a host of open questions that have been the subject of intensive research in recent years. Most of these questions have been settled over BCCSP, either positively by giving a finite complete axiomatization, or negatively by proving that such an axiomatization does not exist. Still some open questions remain. This paper reports on these results, and on the stateoftheart in axiomatizations for richer process algebras with constructs like sequential and parallel composition. 1
CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to the conjecture of BangJensen and Hell)
"... ... of graph homomorphisms) a CSP dichotomy for digraphs with no sources or sinks. The conjecture states that the constraint satisfaction problem for such a digraph is tractable if each component of its core is a circle and is NPcomplete otherwise. In this paper we prove this conjecture, and, as a ..."
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Cited by 25 (8 self)
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... of graph homomorphisms) a CSP dichotomy for digraphs with no sources or sinks. The conjecture states that the constraint satisfaction problem for such a digraph is tractable if each component of its core is a circle and is NPcomplete otherwise. In this paper we prove this conjecture, and, as a consequence, a conjecture of BangJensen, Hell and MacGillivray from 1995 classifying hereditarily hard digraphs. Further, we show that the CSP dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by Bulatov, Jeavons and Krokhin in 2005.
Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of NonClassical Logics I
 Studia Logica
, 1998
"... The main goal of this paper is to explain the link between the algebraic and the Kripkestyle models for certain classes of propositional logics. We start by presenting a Priestleytype duality for distributive lattices endowed with a general class of wellbehaved operators. We then show that fin ..."
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Cited by 12 (6 self)
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The main goal of this paper is to explain the link between the algebraic and the Kripkestyle models for certain classes of propositional logics. We start by presenting a Priestleytype duality for distributive lattices endowed with a general class of wellbehaved operators. We then show that finitelygenerated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and nontopological Kripkestyle models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the abovementioned classes. Introduction In the study of nonclassical propositional logics (and especially of modal logics) there are two main ways of defining interpretations or models. One possibility is to use algebras  usually lattices with operators  as models. Propositional variables are interpreted over elements of these algebraic models, an...
Erdös Graphs Resolve Fine's Canonicity Problem
 The Bulletin of Symbolic Logic
, 2003
"... We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting a b ..."
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Cited by 11 (8 self)
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We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any firstorder definable class of Kripke frames. The constructions use the result of Erd os that there are finite graphs with arbitrarily large chromatic number and girth.
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.
Automated theorem proving by resolution in nonclassical logics
 Annals of Mathematics and Artificial Intelligence
, 2007
"... This paper is an overview of a variety of results, all centered around a common theme, namely embedding of nonclassical logics into first order logic and resolution theorem proving. We present several classes of nonclassical logics, many of which are of great practical relevance in knowledge repre ..."
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Cited by 8 (4 self)
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This paper is an overview of a variety of results, all centered around a common theme, namely embedding of nonclassical logics into first order logic and resolution theorem proving. We present several classes of nonclassical logics, many of which are of great practical relevance in knowledge representation, which can be translated into tractable and relatively simple fragments of classical logic. In this context, we show that refinements of resolution can often be used successfully for automated theorem proving, and in many interesting cases yield optimal decision procedures. 1
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 5 (3 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...
Behavioral algebraization of logics
, 2008
"... Abstract. We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of manysorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL towards providing a meaningful a ..."
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Cited by 4 (4 self)
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Abstract. We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of manysorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL towards providing a meaningful algebraic counterpart also to logics with a manysorted language, and possibly including nontruthfunctional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new algebraic perspective to logics which are algebraizable using the standard tools of AAL. Furthermore, we pave the way towards a robust behavioral theory of AAL, namely by providing a behavioral version of the Leibniz operator which allows us to generalize the traditional Leibniz hierarchy, as well as several wellknown characterization results. A number of meaningful examples will be used to illustrate the novelties and advantages of the approach.
Combinatorial Trees in Priestley Spaces
 Comment. Math. Univ. Carolinae
, 2004
"... We show that prohibiting a combinatorial tree in the Priestley duals determines an axomatizable class of distributive lattices. ..."
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Cited by 3 (3 self)
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We show that prohibiting a combinatorial tree in the Priestley duals determines an axomatizable class of distributive lattices.