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Fibring of logics as a categorial construction
 Journal of Logic and Computation
, 1999
"... Much attention has been given recently to the mechanism of fibring of logics, allowing free mixing of the connectives and using proof rules from both logics. Fibring seems to be a rather useful and general form of combination of logics that deserves detailed study. It is now well understood at the p ..."
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Cited by 51 (31 self)
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Much attention has been given recently to the mechanism of fibring of logics, allowing free mixing of the connectives and using proof rules from both logics. Fibring seems to be a rather useful and general form of combination of logics that deserves detailed study. It is now well understood at the prooftheoretic level. However, the semantics of fibring is still insufficiently understood. Herein we provide a categorial definition of both prooftheoretic and modeltheoretic fibring for logics without terms. To this end, we introduce the categories of Hilbert calculi, interpretation systems and logic system presentations. By choosing appropriate notions of morphism it is possible to obtain pure fibring as a coproduct. Fibring with shared symbols is then easily obtained by cocartesian lifting from the category of signatures. Soundness is shown to be preserved by these constructions. We illustrate the constructions within propositional modal logic.
Extra Theory Morphisms for Institutions: logical semantics for multiparadigm languages
, 1996
"... We extend the ordinary concept of theory morphism in institutions to extra theory morphisms. Extra theory morphism map theories belonging to different institutions across institution morphisms. We investigate the basic mathematical properties of extra theory morphisms supporting the semantics of log ..."
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Cited by 26 (7 self)
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We extend the ordinary concept of theory morphism in institutions to extra theory morphisms. Extra theory morphism map theories belonging to different institutions across institution morphisms. We investigate the basic mathematical properties of extra theory morphisms supporting the semantics of logical multiparadigm languages, especially structuring specifications (module systems) a la OBJClear. They include model reducts, free constructions (liberality), colimits, model amalgamation (exactness), and inclusion systems. We outline a general logical semantics for languages whose semantics satisfy certain "logical" principles by extending the institutional semantics developed within the ClearOBJ tradition. Finally, in the Appendix, we briefly illustrate it with the concrete example of CafeOBJ. Keywords Algebraic specification, Institutions, Theory morphism. AMS Classifications 68Q65, 18C10, 03G30, 08A70 2 1 Introduction Computing Motivation This work belongs to the research are...
Theoroidal maps as algebraic simulations
 WADT 2004, LNCS 3423
, 2005
"... Abstract. Computational systems are often represented by means of Kripke structures, and related using simulations. We propose rewriting logic as a flexible and executable framework in which to formally specify these mathematical models, and introduce a particular and elegant way of representing sim ..."
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Cited by 12 (8 self)
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Abstract. Computational systems are often represented by means of Kripke structures, and related using simulations. We propose rewriting logic as a flexible and executable framework in which to formally specify these mathematical models, and introduce a particular and elegant way of representing simulations in it: theoroidal maps. A categorical viewpoint is very natural in the study of these structures and we show how to organize Kripke structures in categories that afterwards are lifted to the rewriting logic’s level. We illustrate the use of theoroidal maps with two applications: predicate abstraction and the study of fairness constraints. 1
Synchronization of Logics
 Studia Logica
, 1996
"... Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, onestep derivation systems and theory spaces, as well as their functorial ..."
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Cited by 12 (9 self)
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Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, onestep derivation systems and theory spaces, as well as their functorial relationships. We define the synchronization on formulae of two consequence systems and provide a categorial characterization of the construction. For illustration we consider the synchronization of linear temporal logic and equational logic. We define the synchronization on models of two satisfaction systems and provide a categorial characterization of the construction. We illustrate the technique in two cases: linear temporal logic versus equational logic; and linear temporal logic versus branching temporal logic. Finally, we lift the synchronization on formulae to the category of logics over consequence systems. Key words: combination of logics, synchronization on formulae, sync...
Foundations of Heterogeneous Specification
"... We provide a semantic basis for heterogeneous specifications that not only involve different logics, but also different kinds of translations between these. We show that Grothendieck institutions based on spans of (co)morphisms can serve as a unifying framework providing a simple but powerful semant ..."
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Cited by 10 (3 self)
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We provide a semantic basis for heterogeneous specifications that not only involve different logics, but also different kinds of translations between these. We show that Grothendieck institutions based on spans of (co)morphisms can serve as a unifying framework providing a simple but powerful semantics for heterogeneous specification.
Synchronization of Logics with Mixed Rules: Completeness Preservation
 In Algebraic Methodology and Software Technology  AMAST'97
, 1997
"... . Several mechanisms for combining logics have appeared in the literature. Synchronization is one of the simplest: the language of the combined logic is the disjoint union of the given languages, but the class of models of the resulting logic is a subset of the cartesian product of the given classes ..."
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Cited by 9 (5 self)
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. Several mechanisms for combining logics have appeared in the literature. Synchronization is one of the simplest: the language of the combined logic is the disjoint union of the given languages, but the class of models of the resulting logic is a subset of the cartesian product of the given classes of models (the interaction between the two logics is imposed by constraining the class of pairs of models). Herein, we give both a modeltheoretic and a prooftheoretic account of synchronization as a categorial construction (using coproducts and cocartesian liftings) . We also prove that soundness is preserved by possibly constrained synchronization and state sufficient conditions for preservation of model existence and strong completeness. We provide an application to the combination of dynamic logic and linear temporal logic. Keywords: combination of logics, synchronization of logics, model existence, completeness, dynamic logic, temporal logic. 1 Introduction There has been a recent g...
A categorical approach to simulations, in
 of Lecture Notes in Computer Science
, 2005
"... Abstract. Simulations are a very natural way of relating concurrent systems, which are mathematically modeled by Kripke structures. The range of available notions of simulations makes it very natural to adopt a categorical viewpoint in which Kripke structures become the objects of several categories ..."
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Cited by 8 (3 self)
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Abstract. Simulations are a very natural way of relating concurrent systems, which are mathematically modeled by Kripke structures. The range of available notions of simulations makes it very natural to adopt a categorical viewpoint in which Kripke structures become the objects of several categories while the morphisms are obtained from the corresponding notion of simulation. Here we define in detail several of those categories, collect them together in various institutions, and study their most interesting properties. 1
Representations, Hierarchies, and Graphs of Institutions
, 1996
"... For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em ..."
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Cited by 5 (4 self)
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For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of reusing proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various wellknown institutions of total, ordersorted and partial algebras and firstorder structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, ordersorted and partial algebras and firstorder structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of socalled parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd KriegBr\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.
Shedding New Light in the World of Logical Systems
 Category Theory and Computer Science, 7th International Conference, CTCS'97
, 1997
"... The notion of an Institution [5] is here taken as the precise formulation for the notion of a logical system. By using elementary tools from the core of category theory, we are able to reveal the underlying mathematical structures lying "behind" the logical formulation of the satisfaction condit ..."
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Cited by 4 (1 self)
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The notion of an Institution [5] is here taken as the precise formulation for the notion of a logical system. By using elementary tools from the core of category theory, we are able to reveal the underlying mathematical structures lying "behind" the logical formulation of the satisfaction condition, and hence to acquire a both suitable and deeper understanding of the institution concept. This allows us to systematically approach the problem of describing and analyzing relations between logical systems. Theorem 2.10 redesigns the notion of an institution to a purely categorical level, so that the satisfaction condition becomes a functorial (and natural) transformation from specifications to (subcategories of) models and vice versa. This systematic procedure is also applied to discuss and give a natural description for the notions of institution morphism and institution map. The last technical discussion is a careful and detailed analysis of two examples, which tries to outl...
Kan Extensions of Institutions
"... GB92] to formally capture the notion of logical system. Interpreting institutions as functors, and morphisms and representations of institutions as natural transformations, we give elegant proofs for the completeness of the categories of institutions with morphisms and representations, respectively, ..."
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Cited by 1 (0 self)
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GB92] to formally capture the notion of logical system. Interpreting institutions as functors, and morphisms and representations of institutions as natural transformations, we give elegant proofs for the completeness of the categories of institutions with morphisms and representations, respectively, show that the duality between morphisms and representations of institutions comes from an adjointness between categories of functors, and prove the cocompleteness of the categories of institutions over small signatures with morphisms and representations, respectively. Category: F.3, F.4