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....by Hennessy [14] We showed that the model can capture the notion of admissibility but full abstractness must fail in this framework. In producing here a fully abstract model for fair bisimilarity we extend the framework developed by Aczel [1, 3] and then also investigated by J. Rutten in [23, 25] and J. Rutten and D. Turi in [24, 26] Since we work within a non standard set theory, we have collected our set theoretic assumptions in Appendix A, where we also review some of the technical aspects of Aczel s approach to modeling processes as hypersets. 2 Transition Systems as Coalgebras A ....

J.J.M.M. Rutten. A Structural Co-Induction Theorem. Technical Report, CWI, CS-R9346 1993, Amsterdam. 36

....conditions under which the operational notion of bisimulation and the denotational notion of final semantics coincide and provided a coinduction principle for recursive data types. Further study of the relationship between order strong extensionality and finality in Cppo can be found in [Rut93]. There, final coalgebras of Cpo endofunctors mapping bistrict lax kernel pairs to weak right strict lax kernel pairs are characterised as order strongly extensional dense epis. In view of the ideas exposed throughout, one of the more important open issues is whether the theory of data types can ....

J. Rutten. A structural co-induction theorem. In Proc. of the 9 th Conference on the Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science. Springer-Verlag, 1993. To appear.

....by Hennessy [14] We showed that the model can capture the notion of admissibility but full abstractness must fail in this framework. In producing here a fully abstract model for fair bisimilarity we extend the framework developed by Aczel [1, 3] and then also investigated by J. Rutten in [23, 25] and J. Rutten and D. Turi in [24, 26] Since we work within a non standard set theory, we have collected our set theoretic assumptions in Appendix A, where we also review some of the technical aspects of Aczel s approach to modeling processes as hypersets. 2 Transition Systems as Coalgebras A ....

J.J.M.M. Rutten. A Structural Co-Induction Theorem. Technical Report, CWI, CS-R9346 1993, Amsterdam.

....R would now contain (u 1 ; u 2 ) if and only if u 1 6= implies u 2 6= and (u 1 ; u 2 ) 2 R, and u 2 6= implies u 1 6= and (u 1 ; u 2 ) 2 R. With these changes Theorem 4.2 remains valid as stated except that x vD x 0 is replaced by x = x 0 . We refer the reader to [22] and to [27, 4] for further discussion and applications of this kind of co induction principle. 5 Parameterized recursive domains The results in this paper exploit the fact that various simple domain constructors have well behaved actions on relations. The solution of recursive domain equations with parameters ....

J. J. M. M. Rutten, A Structural Co-induction Theorem. In: Proc. Math. Foundations of Programming Language Semantics, New Orleans, 1993.

....applicative bisimulation, used by Abramsky (1990) in connection with the lazy lambda calculus. We refer the reader to (Pitts, 1994) for further discussion and applications of this co inductive characterization of the partial order on recursively defined cppos. See also (Fiore, 1993) and (Rutten, 1993); and (Paulson, 1993) for related applications. The existence of a simulation (i.e. a relation satisfying (37) containing two elements of a recursively defined cppo establishes that the order relation holds between them. Thus equality of elements d and d 0 can be established by exhibiting two ....

....unitary admissible action on relations that a particular construction possesses may well not be a consequence of its functoriality properties. Indeed the very notion of relation may take us outside the category of domains (as is the case for the notion of relation used in Sect. 5, for example) Rutten (1993) and Fiore (1993) develop results analogous to Theorem 6.12, but where relations are subobjects in the category and the notion of (bi)simulation is phrased purely in terms of (order enriched) categorical properties of the functor. Such an approach has the advantage of relaxing the conditions on a ....

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Rutten, J. J. M. M. (1993), A structural co-induction theorem, in "Proc. 9th Int. Conf. on the Math. Foundations of Programming Language Semantics, New Orleans" (Brookes, S. et al, Eds), Lecture Notes in Computer Science, Vol. 802, pp. 83--102, Springer-Verlag, Berlin.

....to the intermediate semantics. This is proved by structural induction. Second, the intermediate semantics is related to (an extension of) the operational semantics by means of the linearize operator and the semantify operators. This relation is proved by a co inductive argument (see Section 7 of [Rut93]) Like the denotational semantics the intermediate semantics uses the space IP as codomain. This requires that the transition system employs semantic actions rather than syntactic ones. Clause (9) of Definition 2.6 then obtains the form [x; #] 2 [# (x) #] As a consequence, denotations ....

J.J.M.M. Rutten. A Structural Co-Induction Theorem. In S. Brookes, M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Proceedings of the 9th International Conference on Mathematical Foundations of Programming Semantics, volume 802 of Lecture Notes in Computer Science, pages 83--102, New Orleans, April 1993. Springer-Verlag.

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J. Rutten, A Structural Coinduction Theorem, Proc. MFPS '93, Springer LNCS 802 (1993) 83-102

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