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Interpolation for predefined types

Published online by Cambridge University Press:  31 August 2011

RĂZVAN DIACONESCU
RĂZVAN DIACONESCU*
Affiliation:
Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania
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Abstract

We give a logic-independent semantics for predefined (data) types within the categorical abstract model theoretic framework of the theory of institutions. We develop a generic interpolation result for this semantics, which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study of interpolation is motivated by a number of important applications to computing science, especially in the area of structured specifications.

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Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Issue 1 , February 2012 , pp. 1 - 24
Copyright
Copyright © Cambridge University Press 2011

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References

Amir, E. and McIlraith, S. (2000) Improving the efficiency of reasoning through structure-based reformulation. In: Choueiry, B. Y. and Walsh, T. (eds.) Proceedings of the Symposium on Abstraction, Reformulation and Approximation (SARA'2000). Springer-Verlag Lecture Notes in Artificial Intelligence 1864 247–259.Google Scholar
Bergstra, J., Heering, J. and Klint, P. (1990) Module algebra. Journal of the Association for Computing Machinery 37 (2)335–372.CrossRefGoogle Scholar
Bicarregui, J., Dimitrakos, T., Gabbay, D. and Maibaum, T. (2001) Interpolation in practical formal development. Logic Journal of the IGPL 9 (2)231–243.CrossRefGoogle Scholar
Borzyszkowski, T. (2001) Generalized interpolation in CASL. Information Processing Letters 76 19–24.CrossRefGoogle Scholar
Borzyszkowski, T. (2000) Logical systems for structured specifications. Theoretical Computer Science 286 (2)197–245.CrossRefGoogle Scholar
Borzyszkowski, T. (2005) Generalized interpolation in first-order logic. Fundamenta Informaticae 66 (3)199–219.Google Scholar
Burstall, R. and Goguen, J. (1980) The semantics of Clear, a specification language. In: Bjorner, D. (ed.) 1979 Copenhagen Winter School on Abstract Software Specification. Springer-Verlag Lecture Notes in Computer Science 86 292–332.CrossRefGoogle Scholar
Chang, C.-C. and JeromeKeisler, H. Keisler, H. (1990) Model Theory, North Holland.Google Scholar
Diaconescu, R. (1994) Category-based semantics for equational and constraint logic programming, D.Phil. thesis, University of Oxford.Google Scholar
Diaconescu, R. (2000) Category-based constraint logic. Mathematical Structures in Computer Science 10 (3)373–407.CrossRefGoogle Scholar
Diaconescu, R. (2004a) Elementary diagrams in institutions. Journal of Logic and Computation 14 (5)651–674.CrossRefGoogle Scholar
Diaconescu, R. (2004b) Herbrand theorems in arbitrary institutions. Information Processing Letters 90 29–37.CrossRefGoogle Scholar
Diaconescu, R. (2004c) An institution-independent proof of Craig Interpolation Theorem. Studia Logica 77 (1)59–79.CrossRefGoogle Scholar
Diaconescu, R. (2004d) Interpolation in Grothendieck institutions. Theoretical Computer Science 311 439–461.CrossRefGoogle Scholar
Diaconescu, R. (2008) Institution-independent Model Theory, Birkhäuser.Google Scholar
Diaconescu, R., Goguen, J. and Stefaneas, P. (1993) Logical support for modularisation. In:Huet, G. Huet, G. and Plotkin, G. (eds.) Logical Environments, Cambridge University Press 83–130. (Proceedings of a Workshop held in Edinburgh, Scotland, May 1991.)Google Scholar
Dimitrakos, T. (1998) Formal Support for Specification Design and Implementation, Ph.D. thesis, Imperial College.Google Scholar
Dimitrakos, T. and Maibaum, T. (2000) On a generalized modularization theorem. Information Processing Letters 74 65–71.CrossRefGoogle Scholar
Goguen, J. and Burstall, R. (1992) Institutions: Abstract model theory for specification and programming. Journal of the Association for Computing Machinery 39 (1)95–146.CrossRefGoogle Scholar
Goguen, J. and Meseguer, J. (1987) Models and equality for logical programming. In: Ehrig, H., Levi, G., Kowalski, R. and Montanari, U. (eds.) Proceedings, TAPSOFT 1987. Springer-Verlag Lecture Notes in Computer Science 250 1–22.CrossRefGoogle Scholar
Goguen, J., Thatcher, J., Wagner, E. and Wright, J. (1977) Initial algebra semantics and continuous algebras. Journal of the Association for Computing Machinery 24 (1)68–95. (An early version is Goguen, J. and Thatcher, J. (1974) Initial Algebra Semantics. IBM T. J. Watson Research Center, Report RC 4865.)CrossRefGoogle Scholar
Găină, D. and Popescu, A. (2004) An institution-independent proof of Robinson consistency theorem. Studia Logica 85 (1)41–73.CrossRefGoogle Scholar
Jhala, R., Majumdar, R. and Xu, R.-G. (2007) State of the union: Type inference via Craig interpolation. In: Tools and Algorithms for the Construction and Analysis of Systems. Springer-Verlag Lecture Notes in Computer Science 4424 553–567.CrossRefGoogle Scholar
Kutz, O. and Mossakowski, T. (2007) Modules in transition. Conservativity, Composition and Colimits. In: Proceedings, Second International Workshop on Modular Ontologies.Google Scholar
Mac Lane, S. (1998) Categories for the Working Mathematician, second edition, Springer-Verlag.Google Scholar
Maehara, S. (1961) On the interpolation theorem of Craig. Sugaku 235–237. (In Japanese.)Google Scholar
McIlraith, S. and Amir, E. (2001) Theorem proving with structured theories. In: Proceedings of the 17th International Conference on Artificial Intelligence (IJCAI-01) 624–631.Google Scholar
McMillan, K. (2005) Applications of Craig interpolants in model checking. In: Proceedings TACAS'2005. Springer-Verlag Lecture Notes in Computer Science 3440 1–12.CrossRefGoogle Scholar
Meseguer, J. (1989) General logics. In: Ebbinghaus, H.-D. et al. (eds.) Proceedings, Logic Colloquium, 1987, North-Holland 275–329.Google Scholar
Nelson, G. and Oppen, D. (1979) Simplication by cooperating decision procedures. ACM Transactions on Programing Languages and Systems 1 (2)245–257.CrossRefGoogle Scholar
Oppen, D. (1980) Complexity, convexity and combinations of theories. Theoretical Computer Science 12 291–302.CrossRefGoogle Scholar
Petria, M. and Diaconescu, R. (2006) Abstract Beth definability in institutions. Journal of Symbolic Logic 71 (3)1002–1028.CrossRefGoogle Scholar
Rodenburg, P.-H. (1991) A simple algebraic proof of the equational interpolation theorem. Algebra Universalis 28 48–51.CrossRefGoogle Scholar
Sannella, D. and Tarlecki, A. (1988) Specifications in an arbitrary institution. Information and Control 76 165–210.Google Scholar
Shoenfield, J. (1967) Mathematical Logic, Addison-Wesley.Google Scholar
Tarlecki, A. (1986a) Bits and pieces of the theory of institutions. In: Pitt, D., Abramsky, S., Poigné, A. and Rydeheard, D. (eds.) Proceedings, Summer Workshop on Category Theory and Computer Programming. Springer-Verlag Lecture Notes in Computer Science 240 334–360.CrossRefGoogle Scholar
Tarlecki, A. (1986b) On the existence of free models in abstract algebraic institutions. Theoretical Computer Science 37 269–304.CrossRefGoogle Scholar
Tarlecki, A., Burstall, R. and Goguen, J. (1991) Some fundamental algebraic tools for the semantics of computation, part 3: Indexed categories. Theoretical Computer Science 91 239–264.CrossRefGoogle Scholar
Veloso, P. (1996) On pushout consistency, modularity and interpolation for logical specifications. Information Processing Letters 60 (2)59–66.CrossRefGoogle Scholar