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35
Modular construction of modal logics
 Concurrency Theory, CONCUR 04, volume 3170 of Lect. Notes Comput. Sci
, 2004
"... Abstract. We present a modular approach to defining logics for a wide variety of statebased systems. We use coalgebras to model the behaviour of systems, and modal logics to specify behavioural properties of systems. We show that the syntax, semantics and proof systems associated to such logics can ..."
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Cited by 22 (7 self)
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Abstract. We present a modular approach to defining logics for a wide variety of statebased systems. We use coalgebras to model the behaviour of systems, and modal logics to specify behavioural properties of systems. We show that the syntax, semantics and proof systems associated to such logics can all be derived in a modular way. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems. 1
Modulated Fibring and the Collapsing Problem
, 2001
"... Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse ..."
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Cited by 20 (12 self)
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Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem. 1
Fibring Labelled Deduction Systems
 Journal of Logic and Computation
, 2002
"... We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial ..."
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Cited by 13 (9 self)
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We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial characterization of fibring of models. Based on this, we establish the conditions under which our systems are sound and complete with respect to the general semantics for the corresponding logics, and establish requirements on logics and systems so that completeness is preserved by both forms of fibring.
Fibring Modal FirstOrder Logics: Completeness Preservation
 Logic Journal of the IGPL
, 2002
"... Fibring is de ned as a mechanism for combining logics with a rstorder base, at both the semantic and deductive levels. A completeness theorem is established for a wide class of such logics, using a variation of the Henkin method that takes advantage of the presence of equality and inequality i ..."
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Cited by 12 (5 self)
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Fibring is de ned as a mechanism for combining logics with a rstorder base, at both the semantic and deductive levels. A completeness theorem is established for a wide class of such logics, using a variation of the Henkin method that takes advantage of the presence of equality and inequality in the logic. As a corollary, completeness is shown to be preserved when bring logics in that class. A modal rstorder logic is obtained as a bring where neither the Barcan formula nor its converse hold.
Combining Valuations with Society Semantics
, 2003
"... Society Semantics, introduced in [5] by W. Carnielli and M. LimaMarques, is a method for obtaining new logics from the combination of agents (valuations) of a given logic. The goal of this paper is to present several generalizations of this method, as well as to show some applications to manyvalued ..."
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Cited by 11 (7 self)
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Society Semantics, introduced in [5] by W. Carnielli and M. LimaMarques, is a method for obtaining new logics from the combination of agents (valuations) of a given logic. The goal of this paper is to present several generalizations of this method, as well as to show some applications to manyvalued logics. After a reformulation of Society Semantics in a wider setting, we develop in detail two examples of application of the new formalism, characterizing a hierarchy of paraconsistent logics called P n (for n 2 N) and a hierarchy of paracomplete logics I n (for n 2 N). We also propose three increasing generalizations, obtaining Society Semantics for several manyvalued logics, including a hierarchy of logics called I n P k which are both paraconsistent and paracomplete. Keywords: society semantics, paraconsistent logics, paracomplete logics, manyvalued logics, combinations of logics, agents.
Transfers between Logics and their Applications
 STUDIA LOGICA
, 2002
"... In this paper, logics are conceived as twosorted firstorder structures, and we argue that this broad definition encompasses a wide class of logics with theoretical interest as well as interest from the point of view of applications. The language, concepts and methods of model theory can thus be ..."
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Cited by 11 (4 self)
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In this paper, logics are conceived as twosorted firstorder structures, and we argue that this broad definition encompasses a wide class of logics with theoretical interest as well as interest from the point of view of applications. The language, concepts and methods of model theory can thus be used to describe the relationship between logics through morphisms of structures called transfers. This leads to a formal framework for studying several properties of abstract logics and their attributes such as consequence operator, syntactical structure, and internal transformations. In particular,
Fibring Logics with Topos Semantics
, 2002
"... The concept of fibring is extended to higherorder logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the metatheorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich ..."
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Cited by 11 (6 self)
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The concept of fibring is extended to higherorder logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the metatheorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich logics. This result is extended to weaker logics in the cases where fibring preserves conservativeness of HOLenrichments. Soundness is shown to be preserved by fibring without any further assumptions.
Combining Logics: Parchments Revisited
 In Recent Trends in Algebraic Development Techniques, volume 2267 of LNCS
, 2001
"... generalizes the common situation when truthvalues are ordered, we require a whole Tarskian closure operation as in [2]. In the sequel, AlgSig denotes the category of algebraic many sorted signatures with a distinguished sort (for formulae) and morphisms preserving it. Given such a signature hS; ..."
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Cited by 7 (5 self)
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generalizes the common situation when truthvalues are ordered, we require a whole Tarskian closure operation as in [2]. In the sequel, AlgSig denotes the category of algebraic many sorted signatures with a distinguished sort (for formulae) and morphisms preserving it. Given such a signature hS; Oi, we denote by Alg(hS; Oi) the category of hS; Oi algebras, and by cAlg(hS; Oi) the class of all pairs hA; i with A 2 jAlg(hS; Oi)j and a closure operation on A . Denition 1. A layered parchment is a tuple P = hSig; L; Mi where: { Sig is a category (of abstract<F13
NonTruthFunctional Fibred Semantics
, 2001
"... wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are n ..."
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Cited by 7 (4 self)
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wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are non{truth{functional (ntf) rooms . For simplicity, we shall only work at this level of abstraction. As shown in [3], everything can be smoothly lifted to the fully edged indexed case. In the sequel, AlgSig' denotes the category of algebraic many sorted signatures with a distinguished sort ' (for formulae) and morphisms preserving it. Given one such signature , we denote by Alg() the category of {algebras and {algebra homomorphisms, and by cAlg() the class of all pairs hA; i with A a<
Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
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Cited by 7 (5 self)
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Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result