Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-18T08:24:01.792Z Has data issue: false hasContentIssue false
  • English
  • Français

Behavioural reasoning for conditional equations†

Published online by Cambridge University Press:  01 October 2007

MANUEL A. MARTINS  and
DON PIGOZZI
MANUEL A. MARTINS
Affiliation:
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal Email: martins@mat.ua.pt
DON PIGOZZI
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA 50011, U.S.A. Email: dpigozzi@iastate.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Object-oriented (OO) programming techniques can be applied to equational specification logics by distinguishing visible data from hidden data (that is, by distinguishing the output of methods from the objects to which the methods apply), and then focusing on the behavioural equivalence of hidden data in the sense introduced by H. Reichel in 1984. Equational specification logics structured in this way are called hidden equational logics, HELs. The central problem is how to extend the specification of a given HEL to a specification of behavioural equivalence in a computationally effective way. S. Buss and G. Roşu showed in 2000 that this is not possible in general, but much work has been done on the partial specification of behavioural equivalence for a wide class of HELs. The OO connection suggests the use of coalgebraic methods, and J. Goguen and his collaborators have developed coinductive processes that depend on an appropriate choice of a cobasis, which is a special set of contexts that generates a subset of the behavioural equivalence relation. In this paper the theoretical aspects of coinduction are investigated, specifically its role as a supplement to standard equational logic for determining behavioural equivalence. Various forms of coinduction are explored. A simple characterisation is given of those HELs that are behaviourally specifiable. Those sets of conditional equations that constitute a complete, finite cobasis for a HEL are characterised in terms of the HEL's specification. Behavioural equivalence, in the form of logical equivalence, is also an important concept for single-sorted logics, for example, sentential logics such as the classical propositional logic. The paper is an application of the methods developed through the extensive work that has been done in this area on HELs, and to a broader class of logics that encompasses both sentential logics and HELs.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 5 , October 2007 , pp. 1075 - 1113
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berreged, N., Bouhoula, A. and Rusinowitch, M. (1998) Observational proofs with critical contexts. In: Fundamental Approaches to Software Engineering. Springer-Verlag Lecture Notes in Computer Science 1382 38–53.CrossRefGoogle Scholar
Bidoit, M. and Hennicker, R. (1994) Proving behavioural theorems with standard first-order logic. In: Levi, G. et al. . (eds.) Proceedings: Algebraic and logic programming. 4th international conference, ALP'94. Springer-Verlag Lecture Notes in Computer Science 850 41–58.Google Scholar
Bidoit, M. and Hennicker, R. (1996) Behavioural theories and the proof of behavioural properties. Theoretical Computer Science 165 (1)3–55.CrossRefGoogle Scholar
Bidoit, M. and Hennicker, R. (1999) Observer complete definitions are behaviourally coherent. In: Proc. OBJ/CafeOBJ/Maude Workshop at Formal Methods'99, Toulouse, France 83–94.Google Scholar
Bidoit, M., Hennicker, R. and Wirsing, M. (1995) Behavioural and abstractor specifications. Sci. Comput. Program. 25 (2-3)149–186.CrossRefGoogle Scholar
Blok, W. J. and Pigozzi, D. (1989) Algebraizable logics. Mem. Am. Math. Soc. 396.Google Scholar
Bouhoula, A. and Rusinowitch, M. (2002) Observational proofs by rewriting. Theoretical Computer Science 275 675–698.CrossRefGoogle Scholar
Buss, S. and Roşu, G. (2000) Incompleteness of behavioral logics. In: Reichel, H. (ed.) Coalgebraic Methods in Computer Science: CMCS 2000. Electronic Notes in Theoretical Computer Science 33.Google Scholar
Diaconescu, R. and Futatsugi, K. (1998) CafeOBJ report: The language, proof techniques, and methodologies for object-oriented algebraic specification, AMAST series in Computing 6, World Scientific.CrossRefGoogle Scholar
Diaconescu, R. and Futatsugi, K. (2000) Behavioural coherence in object-oriented algebraic specification. Journal of Universal Computer Science 6 (1)74–96.Google Scholar
Ehrig, H. and Mahr, B. (1985) Fundamentals of algebraic specification 1: Equations and initial semantics, EATCS Monographs on Theoretical Computer Science, Springer-Verlag.CrossRefGoogle Scholar
Fiadeiro, J. and Sernadas, A. (1988) Structuring theories on consequence. In: Recent trends in data type specification. Specification of abstract data types. Selected Papers, 5th Workshop, Gullane/UK 1987. Springer-Verlag Lecture Notes in Computer Science 332 44–72.CrossRefGoogle Scholar
Font, J., Jansana, R. and Pigozzi, D. (2003) A survey of abstract algebraic logic. Studia Logica, 74 13–97.CrossRefGoogle Scholar
Goguen, J. and Malcolm, G. (1999) Hidden coinduction: Behavioural correctness proofs for objects. Mathematical Structures in Computer Science 9 (3)287–319.CrossRefGoogle Scholar
Goguen, J. and Malcolm, G. (2000) A hidden agenda. Theoretical Computer Science 245 (1)55–101.CrossRefGoogle Scholar
Goguen, J., Lin, K. and Roşu, G. (2002) Conditional circular coinductive rewriting with case analysis. In: 16th International Workshop, WADT 2002, Frauenchiemsee, Germany. Springer-Verlag Lecture Notes in Computer Science 2755 216–232.CrossRefGoogle Scholar
Gorbunov, V. (1998) Algebraic theory of quasivarieties, (Siberian School of Algebra and Logic), Consultants Bureau, Plenum Publishing Corporation (translated from the Russian).Google Scholar
Hennicker, R. (1997) Structural specifications with behavioural operators: semantics, proof methods and applications, Habilitationsschrift Institut für Informatik, Ludwig-Maximilians-Universität München.Google Scholar
Hennicker, R. and Bidoit, M. (1999) Observational logic. In: Proc. AMAST'98, 7th International Conference on Algebraic Methodology and Software Technology. Springer-Verlag Lecture Notes in Computer Science 1548 263–277.CrossRefGoogle Scholar
Hofmann, M. and Sannella, D. (1996) On behavioural abstraction and behavioural satisfaction in higher-order logic. Theoretical Computer Science 167 (1-2)3–45.CrossRefGoogle Scholar
Leavens, G. and Pigozzi, D. (2002) Equational reasoning with subtypes, Iowa State University, Technical Report TR #02-07 (available at ftp://ftp.cs.iastate.edu/pub/techreports/TR02-07/TR.pdf.)Google Scholar
Lin, K., Goguen, J. and Roşu, G. (2000) Circular coinductive rewriting. In: Proceedings, Automated Software Engineering'00, Grenoble, France, IEEE Press 123–131.Google Scholar
Meseguer, J. (1989) General logics. In: Proc. Logic colloq.'87, Granada/Spain. Stud. Logic Found. Math. 129 275–329.CrossRefGoogle Scholar
Martins, M. (2004) Behavioral reasoning in generalized hidden logics, Ph.D. thesis, University of Lisbon.Google Scholar
Pigozzi, D. (2001) Abstract algebraic logic. In: Hazewinkel, M. (ed.) Encyclopedia of Mathematics, Supplement III, Kluwer Academic Publishers 2–13.Google Scholar
Reichel, H. (1985) Behavioural validity of conditional equations in abstract data types. In: Contributions to general algebra 3, (Conference Proceedings, Vienna 1984), 301–324.Google Scholar
Roşu, G. (2000) Hidden logic, Ph.D. thesis, University of California, San Diego.Google Scholar
Roşu, G. and Goguen, J. (2000) Hidden congruence deduction. In: Caferra, R. and Alzer, G. (eds.) Automated Deduction in Classical and Non-Classical Logics. Springer-Verlag Lecture Notes in Artificial Intelligence 1761 252–267.Google Scholar
Roşu, G. and Goguen, J. (2001) Circular coinduction. In: Proceedings, International Joint Conference on Automated Reasoning, IJCAR'01, Siena.Google Scholar
Wójcicki, R. (1988) Theory of logical calculi. Basic theory of consequence operations, Synthese Library 199, Reidel.CrossRefGoogle Scholar