Synchronization of Logics

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, one-step 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 consequences systems.

[1]  K. J. Barwise,et al.  Axioms for abstract model theory , 1974 .

[2]  Mordechai Ben-Ari,et al.  The Temporal Logic of Branching Time , 1981, POPL.

[3]  Dov M. Gabbay,et al.  Adding a temporal dimension to a logic system , 1992, J. Log. Lang. Inf..

[4]  María Manzano,et al.  Extensions of First-Order Logic , 1996 .

[5]  Joseph A. Goguen,et al.  Putting Theories Together to Make Specifications , 1977, IJCAI.

[6]  Joseph A. Goguen,et al.  Institutions: abstract model theory for specification and programming , 1992, JACM.

[7]  M. de Rijke,et al.  Why Combine Logics? , 1997, Stud Logica.

[8]  José Luiz Fiadeiro,et al.  Structuring Theories on Consequence , 1988, ADT.

[9]  S. Lane.  Categories for the Working Mathematician , 1971 .

[10]  José Luiz Fiadeiro,et al.  Unifying Theories in Different Institutions , 1995, COMPASS/ADT.

[11]  J. Etchemendy.  The Concept of Logical Consequence , 1991 .

[12]  Wojciech Penczek,et al.  A Temporal Logic for Event Structures , 1990 .

[13]  Dov M. Gabbay,et al.  Combining Temporal Logic Systems , 1996, Notre Dame J. Formal Log..

[14]  Mordechai Ben-Ari,et al.  The temporal logic of branching time , 1981, POPL '81.

[15]  D. Gabbay.  What is a logical system , 1994 .

[16]  José Meseguer,et al.  May I Borrow Your Logic? , 1993, MFCS.

[17]  Marcus Kracht,et al.  Properties of independently axiomatizable bimodal logics , 1991, Journal of Symbolic Logic.

[18]  Dov M. Gabbay,et al.  An Overview of Fibred Semantics and the Combination of Logics , 1996, FroCoS.

[19]  Liz Sonenberg,et al.  Fixed Point Theorems and Semantics: A Folk Tale , 1982, Inf. Process. Lett..

[20]  A. Tarski.  A LATTICE-THEORETICAL FIXPOINT THEOREM AND ITS APPLICATIONS , 1955 .