P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt et al., editors, Proc. category theory and computer science, volume 389 of LNCS, pages 357--365. Springer-Verlag, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 0 (R = fhx 1 ; y 1 i j R (x 1 ; y 1 )g [ fh x 2 ; y 2 i j R hev ffi Theta id; ev ffi Theta idi = f(f 1 ; f 2 ) j 8a 2 A R (f 1 (a) f 2 (a) g: Bisimulations have received a lot of attention in process theory (see e.g. 22] but also in coalgebra, see e.g. [1, 26, 27]. The greatest bisimulation on two coalgebras is usually written as . It captures behavioural indistinguishability. A standard result is that elements of the state spaces of two arbitrary T coalgebras are bisimilar if and only if they are mapped to the same element of the terminal T coalgebra. ....

P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, A. Poign'e, and D.E. Rydeheard, editors, Category Theory and Computer Science, number 389 in Lect. Notes Comp. Sci., pages 357--

....principle which is given directly by finality. We call it coiteration here to distinguish it from The research reported here was part of the NWO project ProMACS. c #2001 Published by Elsevier Science B. V. derived variants. The main tool for proving states equal is that of an F bisimulation [AM89], a categorical generalisation of notions of bisimulation used for di#erent concrete systems. But these basic principles are often too rigid to nicely cover given examples. Many functions into the carrier of a final coalgebra can be shown not to be coiterative and many statements about behavioural ....

Peter Aczel and Nax Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poigne, editors, Proceedings 3rd Conf. on Category Theory and Computer Science, CTCS'89, Manchester, UK, 5--8 Sept 1989, volume 389 of Lecture Notes in Computer Science, pages 357--365. Springer-Verlag, Berlin, 1989.

....graph ( a,f(a) a A) is a bisimulation from a to [36, Theorem 2.5] a morphism is essentially a functional bisimulation. When Dom a C Dom , a is a subcoalgebra of iff the identity relation on Dom a is a bisimulation from a to . The above categorical definition of bisimulation appeared in [1]. It has a chaxacterisation [18, 19] in terms of liftings of relations R C A x B to relations R T C TAx TB. This in turn was transformed in [10] to another characterisation of bisimulations that uses the idea of paths between functors, an idea introduced in [24, Section 6] A path is a ....

....and blank assigns a truth value to a state according to whether or not the screen is clear in that state. These functions combine into a coalgebra A , C over ) x (A x true, false ) with play = r o a, next = r o (r2 o a) and blank = r2 o (r2 o a) Let = data, l, bool , with [data] C, [1] = over and [bool] true,false . Define the type r to be (data 1) x (St x bool) Then (A, a) above is a r coalgebra. Now for any ground r term M of type St, let play(M) be the term rtr(M) of type data 1. The denotation [play(M) is the function x play( M] x) In particular, play(s) ....

Peter Aczel and Nax Mendler. A final coalgebra theorem. In D. H. Pitt et al., editors, Category Theory and Computer Science. Proceedings 1989.

....of the type P , we see that LTS are coalgebras for the functor . In the same way we get that PTS are coalgebras for the functor . The notions of a nondeterministic and probabilistic bisimulation from Def. 3.4 and Def. 3.5 can be generalized to arbitrary B coalgebras. Definition 6. 4 (cf. [AM89]) A bisimulation between two B coalgebras relation R such that there exists a B coalgebra operation #R : R BR making the projections P and # 2 : R Q homomorphisms from respectively. # BP BR The greatest bisimulation between two coalgebras is denoted ....

Peter Aczel and Nax Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poigne, editors, Proc. 3 rd CTCS, volume 389 of Lecture Notes in Computer Science, pages 357--365. Springer-Verlag, Berlin, 1989.

....that achieved this. Observational equivalence was defined as the relation which is in fact the largest bisimulation between processes. Equivalent processes were later dubbed bisimilar. A category theoretic definition of bisimulation relations between T co algebras (A,r) and (B, was given in [1]: a relation R C A x B is a bisimulation if there is a transition structure p: lrl Tlrl such that the projections A x B A and A x B B are coalgebraic morphisms from p to r and (see Section 5) Bisimilarity is the largest such relation, which always exists because the union of any ....

....iff its graph (a, f(a) a A is a bisimulation from c to [32, Theorem 2.5] a morphism is essentially a functional bisimulation. When Dom c C Dom, c is a subcoalgebra of iff the identity relation on Dom c is a bisimulation from to. The above categorical definition of bisimulation appeared in [1]. It has a characterisation in terms of liftings of relations [14, 15] For R C A x B, define a relation R T C TAx TB by induction on the formation of the polynomial functor T: R D = idD R Id = R R rXr2 = x,y) xRry and 2xr22y = x, y) xy U (2x, xry Rr = f,g) Vd D, ....

Peter Aczel and Nax Mendler. A final coalgebra theorem. In D. H. Pitt et al., editors, Category Theory and Computer Science. Proceedings 1989.

.... as independent concepts [JLY01] De Vink and Rutten [dVR99] argue that probabilistic transition systems are B coalgebras for a suitable functor B which further exhibits probabilistic bisimulations according to the famous definition by Larsen and Skou as B bisimulations in the categorical sense [AM89]. Moreover they show that the functor under consideration is well behaved in that it weakly preserves pullbacks and possesses a final system, two important properties for the structure theory of coalgebras. Some authors have specified basic composition operators for probabilistic transition ....

....desirable properties. To express them, we recall the following definitions: Definition 3.3 (initial final algebra (coalgebra, # bialgebra) An initial (final) object in Alg (Coalg B , # Bialg) is called an initial (final) # algebra (B coalgebra, # bialgebra) Definition 3. 4 (bisimulation [AM89]) A bisimulation between two B co algebras is a triple consisting of a set R, and two functions r 1 : R P and r 2 : R Q such that there exists a B coalgebra operation # : R BR making r 1 and r 2 homomorphisms from ## to respectively. When talking about a ....

Peter Aczel and Nax Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poigne, editors, Proc. 3 rd CTCS, volume 389 of Lecture Notes in Computer Science, pages 357--365. Springer-Verlag, Berlin, 1989.

....(as the above example of f demonstrates) The proof of the above theorem, as presented in Section IV below, uses a similar technique of iteration which M. Barr has used in [10] but our proof is technically more involved. Barr s paper has been inspired by that of P. Aczel and M. Mendler [1]. Coming back to # accessible functors F : Set Set, here our result extends to all covarieties V of F coalgebras: 1. # presentable objects of V are precisely the coalgebras of less than # elements, and 2. V is locally # presentable. Finally we relate the concepts of accessibility and ....

....endofunctor F of A (1) a coalgebra is # presentable in CoalgF i# its underlying object is # presentable in A and (2) the category CoalgF is # accessible. Remark The following proof, as mentioned in the introduction, uses techniques similar to those of Barr [10] and Aczel and Mendler [1], in particular, the statement ( below generalizes the small coalgebra lemma of [1] Proof I. Every coalgebra (A, #A ) with A A # is # presentable in CoalgF . The proof is completely analogous to that of Lemma III.2. II. Next we prove that the collection B CoalgF of all coalgebras with ....

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P. Aczel and M. Mendler. A final coalgebra theorem. Lect. Notes Comp. Sci. 389, Springer 1989, 357--365.

....if there exists (E; 2 CoAlg(T ) and a pair (f; g) of coalgebra morphisms, where f : C; E; and g : D; E; such that f(c) g(d) Some remarks concerning the de nition of behavioural equivalence are in order. Rutten [22] has studied bisimulation, as de ned by Aczel and Mendler [1], as the fundamental notion of equivalence. It is immediate that bisimilarity always implies behavioural equivalence. For functors preserving weak pullbacks, it can be shown that bisimilarity and behavioural equivalence actually coincide. For functors that do not have this property, such as TX = ....

P. Aczel and N. Mendler. A Final Coalgebra Theorem. In D. H. Pitt et al, editor, Category Theory and Computer Science, volume 389 of Lect. Notes in Comp. Sci., pages 357365. Springer, 1989.

.... ZFA (Zermelo Fraenkel set theory without the foundation axiom and with Aczel s axiom of anti foundation) It is likely, but we have not checked details, that use of the anti foundation can be avoided by applying other known results on the existence of final coalgebras of set functors (see [2] and [3] or [8] From another point of view, our choice of a functor for the semantics is such that a large number of semantic items have no intuitive meaning as processes , such as ( Act ) for example. This is technically harmless, even though a bit annoying. There are ways to cut down our ....

P. Aczel and N. Mendler, "A Final ,Coalgebra Theorem", in Lecture Notes in Computer Science 389, Category Theory and Computer Science, D.H. Pitt et al (eds), Springer-Verlag, 357-365.

....there exists a labelled Markov process S and zigzag morphisms f and g such that g T T Notice that if there is a zigzag morphism between two systems, they are bisimilar since the identity is a zigzag morphism. It is interesting to note that we can take a coalgebraic view of bisimulation [AM89, Rut95, dVR] as well. We can view a labelled Markov process as a coalgebra of a suitable functor; in fact it is a functor introduced by Giry [Gir81] in order to define a monad on Mes analogous to the powerset monad. From this point of view, bisimulation is a span of coalgebra homomorphisms. But if one checks ....

P. Aczel and N. Mendler. A final-coalgebra theorem. In Category Theory and Computer Science, Lecture Notes In Computer Science, pages 357--365, 1989.

....same speculative level we should also mention the possibility, suggested in [64] of incorporating fairness constraints in the models by moving from presheaves to sheaves. Finally, it is not clear to us how our approach relates to the abstract understanding of bisimulation provided by coalgebras [4, 114]. The hope is that the recent work and ongoing research of Turi and Plotkin [131, 132] will help provide the missing links. Recall that PomL is the category of (finite) pomsets over L. Appendix A Basic Definitions of Enriched Category Theory A.1 Enriched categories In this appendix we review ....

Peter Aczel and N. Mendler. A final coalgebra theorem. In D. H. Pitt et al., editor, Proceedings of CTCS '89, International Conference on Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science, pages 357--365, 1989.

....C = PProp, A; a) B; b) iff a; b are bisimilar in the Kripke frames A; B and (A; v; a) B; w; b) iff a; b are bisimilar in the Kripke models (A; v) and (B; w) 2. In case that F preserves weak pullbacks, A; a) B; b) iff a and b are related by a bisimulation in the sense of Aczel and Mendler [1]. The following gives an alternative characterisation of behavioural equivalence. Proposition 1.5. Suppose U : A Set creates colimits. A 1 ; v 1 ; a 1 ) A 2 ; v 2 ; a 2 ) iff there are (B; w) and morphisms f i : A i B such that C UA 1 Uf 1 v 1 UB w 6 oe Uf 2 UA 2 oe v ....

P. Aczel and N. Mendler. A final coalgebra theorem. In D. H. Pitt et al, editor, Category Theory and Computer Science, volume 389 of LNCS, pages 357--365. Springer, 1989.

....f(a; f(a) a 2 Ag is a bisimulation from to [17, Theorem 2.5] a morphism is essentially a functional bisimulation. When Dom Dom , is a subcoalgebra of i the identity relation on Dom is a bisimulation from to . 7 The above categorical de nition of bisimulation appeared in [1]. It has a characterisation [8, 9] in terms of liftings of relations R A B to relations R T TA TB. This in turn was transformed in [6] to another characterisation of bisimulations that uses the idea of paths between functors, an idea introduced in [12, Section 6] A path is a nite ....

....and Formulas Fix a set O of symbols called observable types, and a collection f[ o ] o 2 Og of non empty sets indexed by O . Members of [ o ] are called observable elements, or constants of type o. Example: O = fnum, bool, 1g, with [ num] f0; 1; g, bool] ftrue; falseg, [1]] f g. The set of types over O , or O types, is the smallest set T such that O T, St 2 T and (1) if 1 ; 2 2 T then 1 2 ; 1 2 2 T; 2) if 2 T and o 2 O , then o ) 2 T. A subtype of an O type is any type that occurs in the formation of . St is a type symbol that will ....

Peter Aczel and Nax Mendler. A Final Coalgebra Theorem. In D. H. Pitt et al., editors, Category Theory and Computer Science. Proceedings

....by M 1 as UMS functor is indicated. Keywords Ultrametric spaces, Borel measures, coalgebras, preservation of weak pullbacks 1 Introduction The notion of a coalgebra turns out to be one of the key concepts for an understanding of bisimulation from a category theoretical perspective. See, e.g. [AM89, RT93, Rut96, JR97]. In the coalgebraic approach bisimulation can be considered with respect to any functor. Of course, some functors are more appealing to use for the description of computational phenomena than others. Final coalgebra theorems provide conditions for a functor that guarantee the existence of ....

P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poigne, editors, Proc. Category Theory and Computer Science, pages 357--365. Lecture Notes in Computer Science 389, 1989.

....the environment. Then the type of push changes and in order to model the signature one needs the extended cartesian 6 functor G 0 Buf (Y; X) I ) X) I 1) List(N Y ) List(N X) There are two approaches in the literature to define bisimulations: Either in the Aczel Mendler style [1] or by exploiting relation lifting as suggested by Hermida and Jacobs [5] For the scope of this paper the choice in defining bisimulations is mainly a matter of taste, because for extended polynomial functors both approaches yield identical notions of bisimulation [14] I prefer to follow the ....

P. Aczel and P. F. Mendler. A final coalgebra theorem. In D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts, and A. Poign e, editors, Proceedings of the Conference on Category Theory and Computer Science, volume 389 of LNCS, pages 357--365, Berlin, September 1989. Springer.

....relations between coalgebras. Their importance for computer science applications had been realized long before coalgebras were introduced in this field. Intuitively, two states of a system are bisimilar, if they show the same behavior. The coalgebraic definition was introduced by Aczel and Mendler[AM89]: Definition 2.1. A bisimulation between coalgebras A and B is a binary relation R # A B, on which a coalgebra structure # : R # F (R) can be defined, making the projections #A : R # A and #B : R # B into homomorphisms. A #A ## R # ## # # # #A ## #B ## B #B ## F (A) ....

....of maps # i : A # n i A into the n i fold direct sum of A. However, this notion was too simple minded and, most of all, it was lacking any reasonable applications. The more useful category theoretical notion, using arbitrary Set functors as types, was considered by Aczel and Mendler[AM89] and Barr[Bar93] A comprehensive structure theory of universal coalgebra was formulated by J. Rutten in [Rut00] for type functors weakly preserving pullbacks . In [Gum99a] the theory was generalized and extended to work with arbitrary type functors. The structure theoretic e#ect of the (weak) ....

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P. Aczel and N. Mendler, A final coalgebra theorem, Proceedings category theory and computer science (D.H. Pitt et al, ed.), Lecture Notes in Computer Science, Springer, 1989, pp. 357--365.

....operator. Interestingly, naturality accounts exactly for the GSOS restrictions on the occurrences of variables in the rules. Any natural transformation of type (1) has the property that the coalgebraic behavioural equivalence associated to B (which in the above case coincides with bisimulation [2]) is a congruence with respect to the operators of the syntax #. This is a corollary of the more general fact that rules in the format (1) induce a denotational semantics which is adequate in the sense that it is fully abstract with respect to behavioural equivalence. The above result is ....

....paper is the following. Proposition 1.3 (See [24, 3] For a finitary (resp. accessible) endofunctor B on a locally finitely presentable (resp. accessible) category B, the forgetful functor B Coalg ## B has a right adjoint. # The above mentioned (coalgebraic) notion of bisimulation is due to [2]. In this paper, we will consider it in the following form: a B bisimulation between two coalgebras h : X ## BX and k : Y ## BY is a relation (i.e. equivalence class of monos) R # ## X Y between the carriers X and Y which lifts to the coalgebras in the sense that the diagram X R ## ## Y BX ....

P. Aczel and P. F. Mendler. A final coalgebra theorem. In D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts, and A. Poigne, editors, Proc. Category Theory and Computer Science, volume 389 of LNCS, pages 357--365. SpringerVerlag, 1989.

....coalgebra M # # = BM exists; it can be thought of as the solution to the domain equation X # = BX in the category at hand. Then it automatically gives a final object which incorporates a semantic algebra #M # #M . One can model bisimulation by spans of coalgebra maps (first done in [3]) With this, under mild conditions, one has that the semantics given by the final coalgebra is fully abstract and that there is a greatest bisimulation which is a congruence. These conditions are that kernel pairs exist and that weak kernel pairs are preserved by B. # This work has been done ....

P. Aczel and N. Mendler, A Final Coalgebra Theorem, in Proc. CTCS '89 (eds. D. H. Pitt, D. E. Ryeheard, P. Dybjer, A. M. Pitts and A. Poigne), LNCS Vol. 389, pp. 357--365, Berlin: Springer-Verlag, 1989.

....Terms and Formulas Types Fix a set O of symbols called observable types, and a collection f[ o ] o 2 O g of sets indexed by O . Members of [ o ] are observable elements, or constants, of type o. Example: O = fnum, bool, 1, 0g, with [ num] f0; 1; g, bool] ftrue; falseg, [1]] f0g, 0] The set of types over O , or O types, is the smallest set T such that O T , St 2 T and (1) if 1 ; 2 2 T then 1 2 ; 1 2 2 T ; 2) if 2 T and o 2 O , then o ) 2 T . A subtype of an O type is any type that occurs in the formation of . St is a type ....

....its graph f(a; f(a) a 2 Ag is a bisimulation from to [17, Theorem 2.5] a morphism is essential a functional bisimulation. When Dom Dom , is a subcoalgebra of i the identity relation on Dom is a bisimulation from to . The above categorial de nition of bisimulation appeared in [1]. It has a characterisation in terms of liftings of relations [5,6] For R A B, de ne a relation R T TA TB by induction on the formation of the polynomial functor T : R D = id D R Id = R R T1 T2 = f(x; y) 1 xR T1 1 y and 2 xR T2 2 yg R T1 T2 = f( 1 x; 1 y) ....

Aczel, P. and N. Mendler, A Final Coalgebra Theorem, in: D. H. Pitt et al., editors, Category Theory and Computer Science. Proceedings

....Maybe the most surprising and interesting one is the equivalence between finality and so called order strong extensionality, stating that two elements are ordered if and only if they are related by a so called ordered bisimulation. Order bisimulations generalize the F bisimulations of [AM89], which at their turn are categorical abstractions of the notion of bisimulation of [Par81, Mil89] In the present paper, the definition of ordered bisimulation from [Fio93] is used, which generalizes the original definition from [RT93] by the use of lax homomorphisms. The co induction theorem ....

....the following. Theorem 3.3 Let F : CPO CPO be a locally continuous functor and let (D; i Gamma1 ) be the (in CPO E ) initial F E algebra as described above. Then (D; i) is a final F P coalgebra in CPO P as well as a final F coalgebra in CPO . 2 4. Ordered F bisimulation In [AM89], a categorical generalization of the notion of bisimulation of [Par81, Mil89] has been given in terms of coalgebras of functors on a category of classes. In [RT93] this definition is extended to functors F on arbitrary categories, yielding the notion of F bisimulation. The order on hom sets in ....

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P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Ryeheard, P. Dybjer, A.M. Pitts, and A. Poign'e, editors, Proceedings Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science, pages 357--365, 1989.

....morphism, invariant, and bisimulation and discusses their properties. Coalgebras for higher order polynomial functors can be used to give semantics to class specifications that contain binary methods (like the example above) A first result is that the Aczel Mendler approach to define bisimulation [1] cannot be used for higher order polynomial functors. It yields a notion of bisimulation that is not closed under taking successor states (see Example 3.9 for details) The approach of Hermida and Jacobs [10] yields a bisimulation that corresponds to the intuitive notion of behavioral ....

....contained in some predicate give semantics to the infinitary modal operators always and its dual eventually; see [24] for details. Invariants are also used in coalgebraic refinement [12] There are two traditions to define the notions of bisimulation and invariant: Following Aczel and Mendler in [1], a bisimulation is the state space of a coalgebra that makes a certain diagram (of coalgebra morphisms) commute. Similarly one can define an invariant as the state space of a subcoalgebra. Rutten, Hennicker and Kurz follow this approach in [25] and [7] In the following I call this the ....

P. Aczel and P. F. Mendler. A final coalgebra theorem. In D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts, and A. Poign e, editors, Proceedings of the Conference on Category Theory and Computer Science, volume 389 of LNCS, pages 357--365, Berlin, September 1989. Springer.

....semantics) into the same process if and only if they are bisimilar. One of the advantage of working with final semantics is that there is a single coalgebraic notion of (possibly observational) equivalence which is parametric of the functor: it is the definition of F bisimulation as given in [AM89]. For a particular choice of the functor F , namely the one used in [Acz88] but see also [BZ82] F bisimulation coincides with bisimulation in the traditional sense, as was observed above. Also other equivalences, like for instance trace equivalence, can be obtained by instantiating ....

....all a 2 A and s; t 2 S with sR t, s a Gamma s 0 ) 9t 0 2 S; t a Gamma t 0 and s 0 R t 0 and t a Gamma t 0 ) 9s 0 2 S; s a Gamma s 0 and s 0 R t 0 : Next is defined as the union of all bisimulations and two states s and t are called bisimilar when s t. In [AM89] it was noticed that coalgebras can be used for a natural generalization of the above notion of bisimilarity: For every functor F on the category of classes, a relation on F coalgebras is defined, called F bisimulation. This definition is here (generalized to other categories and) repeated, and ....

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P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Ryeheard, P. Dybjer, A.M. Pitts, and A. Poign'e, editors, Proceedings category theory and computer science, Lecture Notes in Computer Science, pages 357--365, 1989.

....of this theory. Furthermore it is used for giving both an observational and a compositional semantics for the language CCS . The observational semantics is with respect to strong bisimulation; for the compositional semantics, an adhoc method is used rather than a general methodology. Later, in [AM89], more attention is given to final coalgebras in the category of (ordinary) sets. Moreover, the notion of (generalized) bisimulation of a functor is introduced there. In [Bar93] the results of [AM89] are expanded. The existence of a final coalgebra of the functor P f (A Theta ) is proved in ....

....the compositional semantics, an adhoc method is used rather than a general methodology. Later, in [AM89] more attention is given to final coalgebras in the category of (ordinary) sets. Moreover, the notion of (generalized) bisimulation of a functor is introduced there. In [Bar93] the results of [AM89] are expanded. The existence of a final coalgebra of the functor P f (A Theta ) is proved in the present paper using a theorem from [Bar93] In our previous paper [RT93] a first step is made towards a generalization of the above notions to a, say universal, semantics based on final ....

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P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Ryeheard, P. Dybjer, A.M. Pitts, and A. Poign'e, editors, Proceedings Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science, pages 357-- 365, 1989. References 55

....this tends to be the key ingredient in specifications of distributed systems whose observational contents (or parts thereof) are shared by different observers. In [14, 12] the notion of bisimulation was introduced in process calculi to capture this kind of observational equivalence. Later [1] gave a categorical definition of bisimulation which applies to arbitrary coalgebras (i.e. bisimulation acquired a shape ) Such a notion of T bisimulation, for a functor T, is defined as a span hR; r 1 ; r 2 i whose legs lift to T coalgebra morphisms, or, in other words, such that there is a ....

P. Aczel and N. Mendler. A final coalgebra theorem. In D. Pitt, D. Rydeheard, P. Dybjer, A. Pitts, and A. Poigne, editors, Proc. Category Theory and Computer Science, pages 357--365. Springer Lect. Notes Comp. Sci. (389), 1988.

....of L F . On the other hand, it should be weak enough so that bisimilar model world pairs should satisfy the the same formulas of L F . Another of our goals is to get generalizations of the characterization result. While practically every functor on sets has a final coalgebra (Aczel and Mendler [AM]) the more concrete a representation one has of the final coalgebra, the better. Final coalgebra theorems of various types may be found in the works cite above, and also in Barr [Bar93, Bar94] Moss and Danner[MD] and in Paulson [P] The results of this paper also give a final coalgebra: the ....

....= 2( x n y n ) 3 x n 3 y n p q y n 1 = 2 x n 3 x n p :q Then it is not hard to show by induction on n that x j= E x n and y j= E y n . Bisimulation A key ingredient in our reformulation of modal logic is the categorical notion of bisimulation of Aczel and Mendler [AM]. To get started, recall that we are considering the functor F : F given by F(A) P(A) Theta P(AtProp) We regard as a functor on the category of sets in a natural way: Given f : A B, we define Ff : F(A) F(B) by Ff(s; t) hff(c 0 ) c 0 2 cg; ti: It is easy to see that F preserves ....

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Aczel, Peter and Nax Mendler. 1989. A Final Coalgebra Theorem. In D. H. Pitt et al (eds.) Category Theory and Computer Science, 357--365. LNCS Heidelberg: Springer Verlag.

....the endofunctor F . 2.1.4 Four Important Properties A functor F is set based if for all classes C and all a 2 F (C) there is some set c C and some a 0 2 F (c)such that a = Fi)a 0 , where i is the inclusion of c in C. It is possible to provethatany set based functor has a final coalgebra (see [2]) A functor F is standard, if whenever f : A B is an inclusion, then F (f) F (A) F (B) is also an inclusion. If a functor F is standard then it is monotone as an operator on classes and hence it has a greatest fixed point F . Since F (F ) F , the pair F = F # id) is a coalgebra, where ....

P. Aczel and N. Mendler. A Final Coalgebra Theorem. In Category Theory and Computer Science, Lecture Notes in Computer Science 389, ed Pitt et al. 1989.

....results of Section 2 are instances of basic facts from universal algebra [Rut96, JR97] Coalgebras and final coalgebras have been around in the literature already for quite some time now. But it was not until the formulation of a general notion of coalgebraic bisimulation, by Aczel and Mendler in [AM89], generalising Park s [Par81] and Milner s [Mil80] definition of strong bisimulation for concurrent processes, that coalgebra could be really put to work . Notably, one needs the notion of coalgebraic bisimulation to formulate a general principle of coinduction, and it is coinduction which ....

P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Ryeheard, P. Dybjer, A. M. Pitts, and A. Poigne, editors, Proceedings category theory and computer science, number 389 in Lecture Notes in Computer Science, pages 357--365, 1989. 44

....with S Theta A Theta S given by (s; a; s 0 ) 2 iff (a; s 0 ) 2 ff(s) is clearly a transition system. See [Rut96] for more details. One of the advantages of the coalgebraic view on transition systems is the existence of a general definition of F bisimulation, for any functor F (cf. [AM89]) For instance, applying that definition to the functor L above yields the standard notion of strong bisimulation. In general, the coalgebraic theory gives a generic approach to the definition and description of bisimulation: First define or characterize the transition systems one is interested ....

....as studied, e.g. by Jones and Plotkin [JP89] and by Edalat [Eda94] are twofold. Firstly, one can resort to the rich literature for standard measure theory on metric spaces. Secondly, we can apply the recently developed theory on coalgebraic bisimulation and final coalgebras in the metric setting [AM89,RT94]. Notably, we shall see that M 1 is locally contractive, from which it follows that it has a final coalgebra. Because of the coalgebraic definition of bisimulation, we thus obtain an internally fully abstract domain. Such a full abstractness result has been lacking so far in the literature. In ....

P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt et al., editor, Proc. Category Theory and Computer Science, pages 357--365. LNCS 389, 1989.

....1, B 1 = B, C 1 = 0, A 2 = A, B 2 = 0 and C 2 = 1. 2.4 Bisimulations and mongruences Bisimulations and mongruences are relations and predicates on carriers of coalgebras which are suitably closed under the coalgebra operations. One can describe these notions in terms of the functor involved, see [1], or [15] Here we shall describe these notions concretely for functors ( as above. 2.3. Definition. Let T : Sets Sets be the above polynomial functor ( and let c: U T (U) and d: V T (V ) be two coalgebras of this functor. i) A relation R U Theta V is called a bisimulation (on these ....

P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, A. Poign'e, and D.E. Rydeheard, editors, Category Theory and Computer Science, number 389 in Lect. Notes Comp. Sci., pages 357--365. Springer, Berlin, 1989.

.... view and the viewpoint adopted here coincide for initial algebras and final coalgebras are canonically isomorphic (see e.g. Fre90, Smy91, Fre92, FP92, Fio94] The mysterious definition of bisimulation in our example is an instance of an abstract notion of bisimulation on a coalgebra (taken from [AM89]) motivated by concurrency theory. This has two important methodological consequences. 1. We are able to provide bisimulations for recursive data types by characterising the abstract notion for concrete examples; this is done for (finite and or infinite) trees in Section 1, and for natural ....

....data type. Our approach differs from his in a couple of points. First, we do not just concentrate on categories of domains but rather study the coinduction principle for recursive data types in arbitrary categories (to encompass domains we develop an order enriched theory) Second, as we follow [AM89], we consider bisimulations on recursive data types as relations (within the category of data types) with closure properties solely determined by the action of the data type constructor. Recently, in [HJ95] Hermida and Jacobs have provided an abstract analysis of induction and coinduction in the ....

[Article contains additional citation context not shown here]

P. Aczel and P.F. Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poign'e, editors, Category Theory and Computer Science, volume 389 of Springer-Verlag, pages 357--365, September 1989.

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P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt et al., editors, Proc. category theory and computer science, volume 389 of LNCS, pages 357--365. Springer-Verlag, 1989.

No context found.

Peter Aczel and Nax Mendler. A final co-algebra theorem. In D. H. Pitt et al., editors, Category Theory and Computer Science Lecture Notes in Computer Science, 389:357--365 Berlin, 1989. Springer-Verlag.

No context found.

Peter Aczel and Max Mendler. A Final Coalgebra Theorem. In Category Theory and Computer Science, Lecture Notes in Computer Science 389, eds. D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts, and A. Poign'e, pages 357--365. September 1989. 32

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In Proc. CTCS'89, LNCS 389, pages 357--365, Berlin, 1989. Springer-Verlag.

No context found.

Peter Aczel and N. Mendler. A Final Coalgebra Theorem. In D. Pitt and D. Ryeheard and P. Dybjer and A. Pitts and A. Poigne, editor, Proceedings Category Theory and Computer Science, volume 389 of LNCS, pages 357--365. Springer, 1989.

No context found.

P. Aczel and P. F. Mendler. A final coalgebra theorem. In D. H. Pitt, D. E. Rydeheard, P. Dybjer, A. M. Pitts, and A. Poign e, editors, Proc. Category Theory and Computer Science, volume 389 of LNCS, pages 357--365. SpringerVerlag, 1989.

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Peter Aczel and Nax Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poigne, editors, Proceedings 3rd Conf. on Category Theory and Computer Science, CTCS'89, Manchester, UK, 5--8 Sept 1989, volume 389 of Lecture Notes in Computer Science, pages 357--365. Springer-Verlag, Berlin, 1989.

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In D. H. Pitt et al, editor, Category Theory and Computer Science, volume 389 of LNCS, pages 357--365, Berlin, 1989. Springer.

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In Proc. CTCS'89, LNCS 389, pages 357--365, 1989. Springer-Verlag.

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In D. H. Pitt et al, editor, Category Theory and Computer Science, volume 389 of LNCS, pages 357--365, Berlin, 1989. Springer.

No context found.

P. Aczel and N. Mendler, A final coalgebra theorem, Proceedings category theory and computer science (D.H. Pitt et al., eds.), Lecture Notes in Computer Science, Springer, 1989, pp. 357--365.

No context found.

Peter Aczel and Nax Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poigne, editors, Proc. 3 rd CTCS, volume 389 of LNCS, pages 357--365. Springer, 1989. 22

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Ryeheard, P. Dybjer, A. M. Pitts, and A. Poigne, editors, Proceedings category theory and computer science, number 389 in Lecture Notes in Computer Science, pages 357--365. Springer-Verlag, 1989.

No context found.

P. Aczel and N. Mendler. A Final Coalgebra Theorem. In D. H. Pitt et al, editor, Category Theory and Computer Science, volume 389 of Lect. Notes in Comp. Sci., pages 357365. Springer, 1989.

No context found.

Peter Aczel and Nax Mendler. A final coalgebra theorem. In D.H. Pitt, D.E. Rydeheard, P. Dybjer, A.M. Pitts, and A. Poigne, editors, Proc. 3 rd CTCS, volume 389 of Lecture Notes in Computer Science, pages 357--365. Springer Verlag, 1989.

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In D. H. Pitt et al, editor, Category Theory and Computer Science, volume 389 of LNCS. Springer, 1989.

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In D.H. Pitt, A. Poign'e, and D.E. Rydeheard, editors, Category Theory and Computer Science, number 389 in Lect. Notes Comp. Sci., pages 357--

No context found.

P.Aczel, N.Mendler. A Final Coalgebra Theorem, in Category Theory and Computer Science, D.H.Pitt et al. eds., Springer LNCS 389, 1989, 357--365.

No context found.

P. Aczel and N. Mendler. A final coalgebra theorem. In Proc. CTCS'89, LNCS 389, pages 357--365, Berlin, 1989. Springer-Verlag.

No context found.

Peter Aczel and Nax Mendler. "A Final Coalgebra Theorem", in Lecture Notes in Computer Science 389, 357-365.

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