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Coalgebras for Binary Methods: Properties of Bisimulations and Invariants

Published online by Cambridge University Press:  15 April 2002

Hendrik Tews
Hendrik Tews*
Affiliation:
Institut für Theoretische Informatik, TU Dresden, D-01062 Dresden, Germany; e-mail: tews@tcs.inf.tu-dresden.de
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Abstract

Coalgebras for endofunctors ${\mathcal C}\rightarrow{\mathcalC}$ can be used to model classes of object-oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors ${\mathcal C}^{op} \times {\mathcalC}\rightarrow{\mathcal C}$ . This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation and invariants for coalgebras of extended polynomial functors and proves many standard results.

Keywords

Binary methodcoalgebrabisimulationinvariantobject-orientation.

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 1: Coalgebraic Methods in Computer Science , January 2001 , pp. 83 - 111
Copyright
© EDP Sciences, 2001

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References

P. Aczel and P.F. Mendler, A final coalgebra theorem, in Proc. of the Conference on Category Theory and Computer Science, edited by D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts and A. Poigné. Springer, Lecture Notes in Comput. Sci. 389 (1989) 357-365.
H.P. Barendregt, Lambda calculi with types, edited by S. Abramsky, D.M. Gabbay and T.S.E. Maibaum. Oxford Science Publications, Handb. Log. Comput. Sci. 2 (1992).
Bruce, K., Cardelli, L. and Castagna, G., The Hopkins Object Group, edited by G.T. Leavens and B. Pierce, On binary methods. Theory and Practice of Object Systems 1 (1995) 221-242.
C. Cîrstea, A coalgebraic equational approach to specifying observational structures, in Coalgebraic Methods in Computer Science '99, edited by B. Jacobs and J. Rutten. Elsevier, Amsterdam, Electron. Notes Theor. Comput. Sci. 19 (1999).
Goguen, J. and Malcolm, G., A hidden agenda. Theoret. Comput. Sci. 245 (2000) 55-101. CrossRef
J. Gosling, B. Joy and G. Steele, The Java Language Specification. Addison-Wesley (1996).
R. Hennicker and A. Kurz, (Ω,Ξ)-Logic: On the algebraic extension of coalgebraic specifications, in Coalgebraic Methods in Computer Science '99, edited by B. Jacobs and J. Rutten. Elsevier, Electron. Notes Theor. Comput. Sci. 19 (1999) 195-212.
U. Hensel, Definition and Proof Principles for Data and Processes, Ph.D. Thesis. University of Dresden, Germany (1999).
U. Hensel, M. Huisman, B. Jacobs and H. Tews, Reasoning about classes in object-oriented languages: Logical models and tools, in European Symposium on Programming, edited by Ch. Hankin. Springer, Berlin, Lecture Notes in Comput. Sci. 1381 (1998) 105-121.
C. Hermida and B. Jacobs, Structural induction and coinduction in a fibrational setting. Inform. and Comput. (1998) 107-152.
B. Jacobs, Objects and classes, co-algebraically, in Object-Orientation with Parallelism and Peristence, edited by B. Freitag, C.B. Jones, C. Lengauer and H.-J. Schek. Kluwer Acad. Publ. (1996) 83-103.
B. Jacobs, Invariants, bisimulations and the correctness of coalgebraic refinements, in Algebraic Methodology and Software Technology, edited by M. Johnson. Springer, Berlin, Lecture Notes in Comput. Sci. 1349 (1997) 276-291.
B. Jacobs, Categorical Logic and Type Theory. North Holland, Elsevier, Stud. Logic Found. Math. 141 (1999).
Jacobs, B. and Rutten, J., A tutorial on (co)algebras and (co)induction. EATCS Bull. 62 (1997) 222-259.
Kawahara, Y. and Mori, M., A small final coalgebra theorem. Theoret. Comput. Sci. 233 (2000) 129-145. CrossRef
X. Leroy, D. Doligez, J. Garrigue, D. Rémy and J. Vouillon, The Objective Caml system, release 3.01, March 2001. Available at URL http://caml.inria.fr/ocaml/htmlman/.
B. Meyer, Eiffel: The Language. Prentice Hall (1992).
R. Milner, Communication and Concurrency. Prentice Hall (1989).
S. Owre, S. Rajan, J.M. Rushby, N. Shankar and M. Srivas, PVS: Combining specification, proof checking, and model checking, in Computer Aided Verification, edited by R. Alur and T.A. Henzinger. Springer, Berlin, Lecture Notes in Comput. Sci. 1102 (1996) 411-414.
E. Poll and J. Zwanenburg, From algebras and coalgebras to dialgebras, in Coalgebraic Methods in Computer Science '01, edited by A. Corradini, M. Lenisa and U. Montanari. Elsevier, Amsterdam, Electron. Notes Theor. Comput. Sci. 44 (2001).
H. Reichel, Behavioural validity of conditional equations in abstract data types, in Contributions to General Algebra 3. Teubne, (1985); in Proc. of the Vienna Conference (June 21-24, 1984).
Reichel, H., An approach to object semantics based on terminal co-algebras. Math. Structure Comput. Sci. 5 (1995) 129-152. CrossRef
G. Rosu, Hidden Logic, Ph.D. Thesis. University of California at San Diego (2000).
Rothe, J., Tews, H. and Jacobs, B., The coalgebraic class specification language CCSL. J. Universal Comput. Sci. 7 (2001) 175-193.
Rutten, J.J.M.M., Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249 (2000) 3-80. CrossRef
B. Stroustrup, The C++ Programming Language: Third Edition. Addison-Wesley Publishing Co., Reading, Mass. (1997).
H. Tews, Coalgebras for binary methods, in Coalgebraic Methods in Computer Science '00, edited by H. Reichel. Elsevier, Amsterdam, Electron. Notes Theor. Comput. Sci. 33 (2000).