Abstract
Higher-order logic with shallow type class polymorphism is widely used as a specification formalism. Its polymorphic entities (types, operators, axioms) can easily be equipped with a ‘naive’ semantics defined in terms of collections of instances. However, this semantics has the unpleasant property that while model reduction preserves satisfaction of sentences, model expansion generally does not. In other words, unless further measures are taken, type class polymorphism fails to constitute a proper institution, being only a so-called rps preinstitution; this is unfortunate, as it means that one cannot use institution-independent or heterogeneous structuring languages, proof calculi, and tools with it.
Here, we suggest to remedy this problem by modifying the notion of model to include information also about its potential future extensions. Our construction works at a high level of generality in the sense that it provides, for any preinstitution, an institution in which the original preinstitution can be represented. The semantics of polymorphism used in the specification language HasCasl makes use of this result. In fact, HasCasl’s polymorphism is a special case of a general notion of polymorphism in institutions introduced here, and our construction leads to the right notion of semantic consequence when applied to this generic polymorphism. The appropriateness of the construction for other frameworks that share the same problem depends on methodological questions to be decided case by case. In particular, it turns out that our method is apparently unsuitable for observational logics, while it works well with abstract state machine formalisms such as state-based Casl.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories. Wiley Interscience, Hoboken (1990)
Barwise, J.: Axioms for abstract model theory. Ann. Math. Logic 7, 221–265 (1974)
Baumeister, H.: An institution for SB-CASL. In: Talk presented at the 15th International Workshop on Algebraic Development Techniques, Genova (2001)
Baumeister, H., Zamulin, A.: State-based extension of Casl. In: Grieskamp, W., Santen, T., Stoddart, B. (eds.) IFM 2000. LNCS, vol. 1945, pp. 3–24. Springer, Heidelberg (2000)
Bidoit, M., Hennicker, R.: On the integration of observability and reachability concepts. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 21–36. Springer, Heidelberg (2002)
Bidoit, M., Mosses, P.D. (eds.): CASL User Manual. LNCS, vol. 2900. Springer, Heidelberg (2004)
Borzyszkowski, T.: Higher-order logic and theorem proving for structured specifications. In: Bert, D., Choppy, C., Mosses, P.D. (eds.) WADT 1999. LNCS, vol. 1827, pp. 401–418. Springer, Heidelberg (2000)
Bossi, A., Focardi, R., Piazza, C., Rossi, S.: Refinement operators and information flow security. In: Software Engineering and Formal Methods, pp. 44–53. IEEE Computer Society Press, Los Alamitos (2003)
Cerioli, M., Reggio, G.: Very abstract specifications: a formalism independent approach. Math. Struct. Comput. Sci. 8, 17–66 (1998)
Coquand, T.: An analysis of Girard’s paradox. In: Logic in Computer Science, pp. 227–236. IEEE, Los Alamitos (1986)
Diaconescu, R., Goguen, J., Stefaneas, P.: Logical support for modularisation. In: Workshop on Logical Frameworks, Programming Research Group. Oxford University, Oxford (1991)
Durán, F., Meseguer, J.: Structured theories and institutions. In: Category Theory and Computer Science, ENTCS, vol. 29 (1999)
Ehrig, H., Mahr, B.: Fundamentals of algebraic specification 2. Springer, Heidelberg (1990)
George, C., Haff, P., Havelund, K., Haxthausen, A.E., Milne, R., Bendix Nielson, C., Prehn, S., Wagner, K.R.: The Raise Specification Language. Prentice Hall, Englewood Cliffs (1992)
Goguen, J., Burstall, R.: Institutions: Abstract model theory for specification and programming. J. ACM 39, 95–146 (1992)
Goguen, J., Malcolm, G.: A hidden agenda. Theoret. Comput. Sci. 245, 55–101 (2000)
Goguen, J., Rosu, G.: Institution morphisms. Formal aspects of computing 13, 274–307 (2002)
Goguen, J., Tracz, W.: An implementation-oriented semantics for module composition. In: Foundations of Component-Based Systems, Cambridge, pp. 231–263 (2000)
Kubiak, R., Borzyszkowski, A., Sokolowski, S.: A set-theoretic model for a typed polymorphic lambda calculus — a contribution to MetaSoft. In: Bloomfield, R.E., Jones, R.B., Marshall, L.S. (eds.) VDM 1988. LNCS, vol. 328, pp. 267–298. Springer, Heidelberg (1988)
Kurz, A.: Logics for coalgebras and applications to computer science, Ph.D. thesis, Universität München (2000)
Meseguer, J.: General logics, Logic Colloquium, vol. 87, pp. 275–329. North-Holland, Amsterdam (1989)
Moerdijk, I., Palmgren, E.: Wellfounded trees in categories. Ann. Pure Appl. Logic 104, 189–218 (2000)
Moggi, E.: Categories of partial morphisms and the λ p -calculus. In: Poigné, A., Pitt, D.H., Rydeheard, D.E., Abramsky, S. (eds.) Category Theory and Computer Programming. LNCS, vol. 240, pp. 242–251. Springer, Heidelberg (1986)
Mossakowski, T.: Representations, hierarchies and graphs of institutions, Ph.D. thesis, Universität Bremen (1996), also: Logos-Verlag (2001)
Mossakowski, T.: Foundations of heterogeneous specification. In: Wirsing, M., Pattinson, D., Hennicker, R. (eds.) WADT 2003. LNCS, vol. 2755, pp. 359–375. Springer, Heidelberg (2003)
Mossakowski, T.: heterogeneous specification. Language summary (2004), http://www.tzi.de/cofi/hetcasl
Mosses, P.D. (ed.): CASL Reference Manual. LNCS, vol. 2960. Springer, Heidelberg (2004)
Nielsen, M., Pletat, U.: Polymorphism in an institutional framework, Tech. report, Technical University of Denmark (1986)
Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)
Pepper, P.: Transforming algebraic specifications – lessons learnt from an example. In: Constructing Programs from Specifications, pp. 1–27. Elsevier, Amsterdam (1991)
Peyton-Jones, S. (ed.): Haskell 1998 language and libraries — the revised report, Cambridge (2003), also: J. Funct. Programming 13 (2003)
Salibra, A., Scollo, G.: A soft stairway to institutions. In: Bidoit, M., Choppy, C. (eds.) Abstract Data Types 1991 and COMPASS 1991. LNCS, vol. 655, pp. 310–329. Springer, Heidelberg (1993)
Sannella, D., Tarlecki, A.: Specifications in an arbitrary institution. Information and Computation 76, 165–210 (1988)
Schröder, L.: The prologue: categorical syntax and semantics of the partial λ-calculus, available as, http://www.informatik.uni-bremen.de/~lschrode/hascasl/plam.ps
Schröder, L., Mossakowski, T.: Towards integrated specification and development of functional programs. In: Kirchner, H., Ringeissen, C. (eds.) AMAST 2002. LNCS, vol. 2422, pp. 99–116. Springer, Heidelberg (2002)
Schröder, L., Mossakowski, T., Maeder, C.: Integrated functional specification and programming. Language summary, available under, http://www.informatik.uni-bremen.de/agbkb/forschung/formal_methods/CoFI/HasCASL
Tarlecki, A.: Quasi-varieties in abstract algebraic institutions. J. Comput. System Sci. 33, 333–360 (1986)
Wenzel, M.: Type classes and overloading in higher-order logic. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275, pp. 307–322. Springer, Heidelberg (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schröder, L., Mossakowski, T., Lüth, C. (2005). Type Class Polymorphism in an Institutional Framework. In: Fiadeiro, J.L., Mosses, P.D., Orejas, F. (eds) Recent Trends in Algebraic Development Techniques. WADT 2004. Lecture Notes in Computer Science, vol 3423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31959-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-31959-7_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25327-3
Online ISBN: 978-3-540-31959-7
eBook Packages: Computer ScienceComputer Science (R0)