Fibring Logics: Past, Present and Future

This paper is a guided tour through the theory of fibring as a general mechanism for combining logics. We present the main ideas, constructions and difficulties of fibring, from both a model and a proof-theoretic perspective, and give an outline of soundness, completeness and interpolation preservation results. Along the way, we show how the current algebraic semantics of fibring relates with the original ideas of Dov Gabbay. We also analyze the collapsing problem, the challenges it raises, and discuss a number of future research

[1]  Yannick Chevalier,et al.  Combining Intruder Theories , 2005, ICALP.

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

[3]  Jaime Ramos,et al.  Cryptomorphisms at Work , 2004, WADT.

[4]  M. de Rijke,et al.  Model Checking for Combined Logics with an Application to Mobile Systems , 2004, Automated Software Engineering.

[5]  Cristina Sernadas,et al.  Parameterisation of Logics , 1998, WADT.

[6]  Cristina Sernadas,et al.  Fibring: completeness preservation , 2001, Journal of Symbolic Logic.

[7]  P. Mateus,et al.  Exogenous Quantum Logic , 2004 .

[8]  Maarten Marx,et al.  Repairing the interpolation theorem in quantified modal logic , 2003, Ann. Pure Appl. Log..

[9]  Maarten Marx,et al.  Complexity of Products of Modal Logics , 1999, J. Log. Comput..

[10]  Amílcar Sernadas,et al.  Combining Logics: Parchments Revisited , 2001, WADT.

[11]  Dov M. Gabbay,et al.  Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics , 1996, Journal of Symbolic Logic.

[12]  Luca Viganò,et al.  Fibring Labelled Deduction Systems , 2002, J. Log. Comput..

[13]  Dov M. Gabbay,et al.  Craig interpolation theorem for intuitionistic logic and extensions Part III , 1977, Journal of Symbolic Logic.

[14]  Guido Governatori,et al.  On Fibring Semantics for BDI Logics , 2002, JELIA.

[15]  Cristina Sernadas,et al.  Combining logic systems: Why, how, what for? , 2003 .

[16]  Amílcar Sernadas,et al.  Weakly complete axiomatization of exogenous quantum propositional logic , 2005, Inf. Comput..

[17]  Frank Wolter,et al.  Fusions of Modal Logics Revisited , 1996, Advances in Modal Logic.

[18]  P. Mateus,et al.  Exogenous Semantics Approach to Enriching Logics , 2005 .

[19]  Cristina Sernadas,et al.  Synchronization of Logics with Mixed Rules: Completeness Preservation , 1997, AMAST.

[20]  Josep Maria Font,et al.  A Survey of Abstract Algebraic Logic , 2003, Stud Logica.

[21]  Cristina Sernadas,et al.  Fibring Modal First-Order Logics: Completeness Preservation , 2002, Log. J. IGPL.

[22]  Cristina Sernadas,et al.  Fibring of Logics as a Universal Construction , 2005 .

[23]  Jan J. M. M. Rutten,et al.  Mathematical techniques for analyzing concurrent and probabilistic systems , 2004, CRM monograph series.

[24]  L. F. D. Cerro,et al.  Combining Classical and Intuitionistic Logic , 1996 .

[25]  F. Wolter,et al.  Temporalizing Description Logics , 1998, FroCoS.

[26]  John Fox,et al.  Probability, logic and the cognitive foundations of rational belief , 2003, Journal of Applied Logic.

[27]  M. Kracht.  Tools and Techniques in Modal Logic , 1999 .

[28]  Hans-Jörg Kreowski,et al.  Recent Trends in Data Type Specification , 1985, Informatik-Fachberichte.

[29]  J. Michael Dunn,et al.  Relevance Logic and Entailment , 1986 .

[30]  Cristina Sernadas,et al.  Categorial bring of logics with terms and binding operators , 2000 .

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

[32]  Marcus Kracht,et al.  Simulation and Transfer Results in Modal Logic – A Survey , 1997, Stud Logica.

[33]  William Craig,et al.  Linear reasoning. A new form of the Herbrand-Gentzen theorem , 1957, Journal of Symbolic Logic.

[34]  L. L. Maksimova,et al.  Interpolation in superintuitionistic predicate logics with equality , 1997 .

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

[36]  Marcelo Finger,et al.  The Unrestricted Combination of Temporal Logic Systems , 2002, Log. J. IGPL.

[37]  Helmut Veith,et al.  Interpolation in fuzzy logic , 1999, Arch. Math. Log..

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

[39]  Cristina Sernadas,et al.  Fibring Logics with Topos Semantics , 2003, J. Log. Comput..

[40]  Amílcar Sernadas,et al.  Reasoning About Quantum Systems , 2004, JELIA.

[41]  D. Gabbay,et al.  Many-Dimensional Modal Logics: Theory and Applications , 2003 .

[42]  Larisa Maksimova,et al.  Complexity of interpolation and related problems in positive calculi , 2002, Journal of Symbolic Logic.

[43]  R. Thomason.  Combinations of Tense and Modality , 2002 .

[44]  Dov M. Gabbay,et al.  Products of modal logics. Part 2: Relativised quantifiers in classical logic , 2000, Log. J. IGPL.

[45]  Maarten Marx,et al.  Hybrid logics: characterization, interpolation and complexity , 2001, Journal of Symbolic Logic.

[46]  Till Mossakowski,et al.  Combining and representing logical systems using model-theoretic parchments , 1997, WADT.

[47]  Till Mossakowski,et al.  Combining and Representing Logical Systems , 1997, Category Theory and Computer Science.

[48]  J. Czelakowski,et al.  Amalgamation and Interpolation in Abstract Algebraic Logic , 2003 .

[49]  A. E. Eiben,et al.  Combining Algebraizable Logics , 1996, Notre Dame J. Formal Log..

[50]  Cristina Sernadas,et al.  Fibring Non-Truth-Functional Logics: Completeness Preservation , 2003, J. Log. Lang. Inf..

[51]  Till Mossakowski,et al.  Using Limits of Parchment to Systematically Construct Institutions of partial Algebras , 1995, COMPASS/ADT.

[52]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[53]  J. Davenport.  Editor , 1960 .

[54]  Luca Viganò,et al.  Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity , 2004, Log. J. IGPL.

[55]  Cristina Sernadas,et al.  Modulated Fibring and The Collapsing Problem , 2002, J. Symb. Log..

[56]  Jaime Ramos,et al.  Completeness Results for Fibred Parchments: Beyond the Propositional Base , 2002, WADT.

[57]  Dov M. Gabbay,et al.  Fibring Semantic Tableaux , 1998, TABLEAUX.

[58]  Cristina Sernadas,et al.  Synchronization of Logics , 1997, Stud Logica.

[59]  Alessandra Carbone,et al.  Interpolants, Cut Elimination and Flow Graphs for the Propositional Calculus , 1997, Ann. Pure Appl. Log..

[60]  Cristina Sernadas,et al.  Fibring of Logics as a Categorial Construction , 1999, J. Log. Comput..

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

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

[63]  Cristina Sernadas,et al.  Preservation of interpolation by fibring , 2004 .