Journal of Symbolic Logic

Abstract Beth definability in institutions

Răzvan Diaconescu and Marius Petria

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This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. We generalise the concept of definability to arbitrary logics, formalised as institutions, and we develop three general definability results. One generalises the classical Beth theorem by relying on the interpolation properties of the institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for many new actual definability results, including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for ‘borrowing’ definability properties from another institution via an ‘adequate’ encoding between institutions. The power of our general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-)varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalised as institutions from computing science and logic.

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J. Symbolic Logic Volume 71, Issue 3 (2006), 1002-1028.

First available: 4 August 2006

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Petria, Marius; Diaconescu, Răzvan. Abstract Beth definability in institutions. Journal of Symbolic Logic 71 (2006), no. 3, 1002--1028. doi:10.2178/jsl/1154698588.

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  • Hajnal Andréka and István Németi, Łoś lemma holds in every category, Studia Scientiarum Mathematicarum Hungarica, vol. 13 (1978), pp. 361--376.
  • --------, A general axiomatizability theorem formulated in terms of cone-injective subcategories, Universal algebra (B. Csakany, E. Fried, and E. T. Schmidt, editors), North-Holland, 1981, Colloquia Mathematics Societas János Bolyai, 29, pp. 13--35.
  • --------, Generalization of the concept of variety and quasivariety to partial algebras through category theory, Dissertationes Mathematicae, vol. CCIV (1983).
  • Jan Bergstra, Jan Heering, and Paul Klint, Module algebra, Journal of the Association for Computing Machinery, vol. 37 (1990), no. 2, pp. 335--372.
  • Michel Bidoit and Rolf Hennicker, On the integration of the observability and reachability concepts, Proceedings of the 5th International Conference on Foundations of Software Science and Computation Structures $($FOSSACS '2002$)$, Lecture Notes in Computer Science, vol. 2303, 2002, pp. 21--36.
  • Michel Bidoit and Andrzej Tarlecki, Behavioural satisfaction and equivalence in concrete model categories, Proceedings of the 21st Colloquium on Trees in Algebra and Programming, Lecture Notes in Computer Science, vol. 1059, Springer Verlag, 1996, pp. 241--256.
  • Tomasz Borzyszkowski, Higher-order logic and theorem proving for structured specifications, Workshop on Algebraic Development Techniques 1999 (Christine Choppy, Didier Bert, and Peter Mosses, editors), Lecture Notes in Computer Science, vol. 1827, 2000, pp. 401--418.
  • --------, Generalized interpolation in CASL, Information Processing Letters, vol. 76 (2001), pp. 19--24.
  • Peter Burmeister A model theoretic oriented approach to partial algebras, Akademie-Verlag, Berlin, 1986.,
  • Maura Cerioli and José Meseguer, May I borrow your logic? $($transporting logical structures along maps$)$, Theoretical Computer Science, vol. 173 (1997), pp. 311--347.
  • C. C. Chang and H. J. Keisler Model theory, North Holland, Amsterdam, 1990.,
  • Corina Cîrstea, Institutionalising many-sorted coalgebraic modal logic, Coalgebraic Methods in Computer Science 2002, Electronic Notes in Theoretical Computer Science, 2002.
  • Răzvan Diaconescu, Extra theory morphisms for institutions: logical semantics for multi-paradigm languages, Applied Categorical Structures, vol. 6 (1998), no. 4, pp. 427--453, a preliminary version appeared as JAIST Technical Report IS-RR-97-0032F in 1997.
  • --------, Institution-independent ultraproducts, Fundamenta Informaticæ, vol. 55 (2003), no. 3--4, pp. 321--348.
  • --------, Elementary diagrams in institutions, Journal of Logic and Computation, vol. 14 (2004), no. 5, pp. 651--674.
  • --------, An institution-independent proof of Craig Interpolation Theorem, Studia Logica, vol. 77 (2004), no. 1, pp. 59--79.
  • --------, Interpolation in Grothendieck institutions, Theoretical Computer Science, vol. 311 (2004), pp. 439--461.
  • --------, Jewels of institution-independent model theory, Algebra, meaning and computation $($Essays dedicated to Joseph A. Goguen on the occasion of his 65th birthday$)$, Lecture Notes in Computer Science, vol. 4060, Springer, 2006.
  • --------, Proof systems for institutional logic, Journal of Logic and Computation, vol. 16 (2006), pp. 339--357.
  • -------- Institution-independent model theory,to appear, book draft. Ask author for current draft at,
  • Răzvan Diaconescu, Joseph Goguen, and Petros Stefaneas, Logical support for modularisation, Logical environments (Gerard Huet and Gordon Plotkin, editors), Cambridge, 1993, proceedings of a workshop held in Edinburgh, Scotland, May 1991, pp. 83--130.
  • Theodosis Dimitrakos and Tom Maibaum, On a generalized modularization theorem, Information Processing Letters, vol. 74 (2000), pp. 65--71.
  • J. L. Fiadeiro and J. F. Costa, Mirror, mirror in my hand: A duality between specifications and models of process behaviour, Mathematical Structures in Computer Science, vol. 6 (1996), no. 4, pp. 353--373.
  • Daniel Găină and Andrei Popescu, An institution-independent generalization of Tarski's Elementary Chain Theorem, Journal of Logic and Computation, to appear.
  • --------, An institution-independent proof of Robinson consistency theorem, Studia Logica, to appear.
  • Joseph Goguen and Rod Burstall, Institutions: Abstract model theory for specification and programming, Journal of the Association for Computing Machinery, vol. 39 (1992), no. 1, pp. 95--146.
  • Joseph Goguen and Răzvan Diaconescu, Towards an algebraic semantics for the object paradigm, Recent trends in data type specification (Harmut Ehrig and Fernando Orejas, editors), Lecture Notes in Computer Science, vol. 785, Springer, 1994, pp. 1--34.
  • Joseph Goguen and Grigore Roşu, Institution morphisms, Formal Aspects of Computing, vol. 13 (2002), pp. 274--307.
  • George Grätzer Universal algebra, Springer, 1979.,
  • Wilfrid Hodges Model theory, Cambridge University Press, 1993.,
  • Joachim Lambek and Phil Scott Introduction to higher order categorical logic, Cambridge Studies in Advanced Mathematics, vol. 7, Cambridge, 1986.,
  • Yngve Lamo The institution of multialgebras---a general framework for algebraic software development, PhD thesis, University of Bergen, 2003.,
  • Saunders Mac Lane Categories for the working mathematician, second ed., Springer, 1998.,
  • G. Matthiessen, Regular and strongly finitary structures over strongly algebroidal categories, Canadian Journal of Mathematics, vol. 30 (1978), pp. 250--261.
  • José Meseguer, General logics, Logic Colloquium, 1987 (H.-D. Ebbinghaus et al., editors), North-Holland, 1989, pp. 275--329.
  • --------, Conditional rewriting logic as a unified model of concurrency, Theoretical Computer Science, vol. 96 (1992), no. 1, pp. 73--155.
  • --------, Membership algebra as a logical framework for equational specification, Proceedings of the WADT'97 (F. Parisi-Pressice, editor), Lecture Notes in Computer Science, no. 1376, Springer, 1998, pp. 18--61.
  • Till Mossakowski, Specification in an arbitrary institution with symbols, Recent trends in Algebraic Development Techniques, 14th International Workshop, WADT'99, Bonas, France (C. Choppy, D. Bert, and P. Mosses, editors), Lecture Notes in Computer Science, vol. 1827, Springer-Verlag, 2000, pp. 252--270.
  • --------, Relating CASL with other specification languages: the institution level, Theoretical Computer Science, vol. 286 (2002), pp. 367--475.
  • Till Mossakowski, Joseph Goguen, Răzvan Diaconescu, and Andrzej Tarlecki, What is a logic?, Logica universalis (Jean-Yves Beziau, editor), Birkh, pp. 113--133.
  • István Németi and Ildikó Sain, Cone-implicational subcategories and some Birkhoff-type theorems, Universal algebra (B. Csakany, E. Fried, and E. T. Schmidt, editors), North-Holland, 1981, Colloquia Mathematics Societas János Bolyai, 29, pp. 535--578.
  • Andrei Popescu, Traian \c Serbănuţă, and Grigore Roşu, A semantic approach to interpolation, submitted.
  • Pieter-Hendrik Rodenburg, A simple algebraic proof of the equational interpolation theorem, Algebra Universalis, vol. 28 (1991), pp. 48--51.
  • Donald Sannella and Andrzej Tarlecki, Specifications in an arbitrary institution, Information and Control, vol. 76 (1988), pp. 165--210, earlier version in Proceedings, International Symposium on the Semantics of Data Types, Lecture Notes in Computer Science, vol. 173, Springer, 1985.
  • Lutz Schröder, Till Mossakowski, and Christoph Lüth, Type class polymorphism in an institutional framework, Recent trends in Algebraic Development Techniques, 17th International Workshop $($WADT 2004$)$ (José Fiadeiro, editor), Lecture Notes in Computer Science, vol. 3423, Springer, Berlin, 2004, pp. 234--248.
  • Joseph Shoenfield Mathematical logic, Addison-Wesley, 1967.,
  • Andrzej Tarlecki, Bits and pieces of the theory of institutions, Proceedings, Summer Workshop on Category Theory and Computer Programming (David Pitt, Samson Abramsky, Axel Poigné, and David Rydeheard, editors), Lecture Notes in Computer Science, vol. 240, Springer, 1986, pp. 334--360.
  • --------, On the existence of free models in abstract algebraic institutions, Theoretical Computer Science, vol. 37 (1986), pp. 269--304, preliminary version, University of Edinburgh, Computer Science Department, Report CSR-165-84, 1984.
  • --------, Quasi-varieties in abstract algebraic institutions, Journal of Computer and System Sciences, vol. 33 (1986), no. 3, pp. 333--360, original version, University of Edinburgh, Report CSR-173-84.
  • --------, Moving between logical systems, Recent trends in data type specification (Magne Haveraaen, Olaf Owe, and Ole-Johan Dahl, editors), Lecture Notes in Computer Science, Springer, 1996, pp. 478--502.
  • --------, Towards heterogeneous specifications, Proceedings, International Conference on Frontiers of Combining Systems $($FroCoS'98$)$ (D. Gabbay and M. van Rijke, editors), Research Studies Press, 2000, pp. 337--360.
  • Paulo Veloso, On pushout consistency, modularity and interpolation for logical specifications, Information Processing Letters, vol. 60 (1996), no. 2, pp. 59--66.