M. P. Fiore. A coinduction principle for recursive data types based on bisimulation. Information and Computation, 127(2):186--198, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....one (dynamical) system and another, expressing that if one system can do a move, then the other can do a similar move. Simulations are heavily used for transition systems and automata (see e.g. 11] especially for refinement proofs. Also, they are studied in modal logic [2] domain theory [12,5], category theory [16] using spans, following earlier, unpublished work of Claudio Hermida on modules) Here we study simulations in a purely coalgebraic context, starting from a new, elementary notion of ordering on a functor, and using familiar techniques based on relation lifting or ....

....consists of a collection of preorders #X # F (X) for each set X, in such a way that F (f) F (X) F (Y ) preserves the order, for each function f : X Y . Preorderedness seems to be the minimal requirement that one wishes to impose on such orders in the current setting. Often, like in [12,5], notions of simulation are studied in an ordered setting, where the functor F acts on some category of dcpos. In that case each X and F (X) is a dcpo and thus automatically carries on order. Our approach is minimal in a sense, because it only requires an order on the images F (X) of F , and not ....

M.P. Fiore. A coinduction principle for recursive data types based on bisimulation. Inf. & Comp., 127(2):186--198, 1996.

....and so formalising our intuition about Cocont being a category of domains. Further we use these results to provide a domain theoretical understanding of open map bisimulation by means of relational structures [94, 100] and induction coinduction principles for recursively defined domains as in [99, 31]. In particular we define the notion of intensional relation in Cocont and give a domain theoretical characterisation of strong bisimulation for arbitrary trees. The results presented in this chapter are part of a joint paper with Marcelo Fiore INTRODUCTION 13 and Glynn Winskel [21] that appeared ....

....that include Cocont which is the 2 categorical equivalent of Prof . We develop a domain theoretical approach to open map bisimulation using relational structures [94, 100] and induction coinduction principles for recursively defined domains and coinduction properties based on bisimulation [99, 31]. For technical reasons (so as to have less coherence conditions to worry about) we shall state our results as holding for 2 categories. Thus, we prefer in this chapter the 2 category Cocont over Prof (cf. Chapter 4) and consider the interpretation of the type theory of Section 4.5 in Cocont. We ....

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Marcelo P. Fiore. A coinduction principle for recursive data types based on bisimulation. Information and Computation, 127(2):186--198, 1996.

....of recursion we take final coalgebras, corresponding to greatest fixed points and the finite delay operator is simply obtained as an initial algebra corresponding to a least fixed point of the process equation (3.2) given above. This is analogous to work on finite and infinite data types, cf. e.g. [38]. Finally, the categorical relationships between the di#erent models and the general theory of bisimulation from open maps reduce the problem of relating the two semantics to finding an open map within the category of generalised transition systems. As already mentioned, our approach is closely ....

Marcelo P. Fiore. A coinduction principle for recursive data types based on bisimulation. Information and Computation, 127, 1996.

....need improvement. 8.1. Source of Examples The proof plans make no claim to deal with cases where there are terms with no value, so such theorems were excluded from consideration. These theorems were identi ed by inspection. Examples of coinduction were drawn from the literature, in particular [16,24,26,31]. Examples are relatively few in number, the same one or two theorems appearing in almost every paper on the subject. Recent research is beginning to produce a larger corpus but this has tended to concentrate on problems outside the domain chosen for consideration (i.e. lazy functional programs) ....

M. Fiore, A coinduction principle for recursive data types based on bisimulation, in: Proceedings of the Eight IEEE Symposium on Logic in Computer Science, 1993, pp. 110-119.

....are not enough bisimulations to equate processes with the same final semantics. One can instead develop a general theory for locally ordered categories (better: CPPO enriched ones) One can define simulations (or ordered bisimulations) as spans of lax and and oplax coalgebra maps; it is shown in [4] that these include partial bisimulations [1] also known as prebisimulations) There is then a new version of the general theory but now with the conditions that insertors exist and that weak insertors are preserved. Unfortunately, the weak preservation condition fails. This motivates us to give ....

M. P. Fiore, A Coinduction Principle for Recursive Data Types Based on Bisimulation, in Information and Computation, Vol. 127, No. 2, pp. 186--198, 1996.

....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 (Section 5) is presented as and named after a dualization of the structural induction theorem of [Plo81] but see also [LS81] which is repeated here in the Appendix. Part of ....

....such that R 1 A oe 2 R fi ff fi F (R) F ( 1 ) F (A) oe F ( 2 ) F (R) That is, 2 is a homomorphism of coalgebras (satisfying F ( 2 ) ffi fi = ff ffi 2 ) and 1 is a so called lax homomorphism: it satisfies F ( 1 ) ffi fi ff ffi 1 . 2 The above definition is from [Fio93] and generalizes an earlier definition of ordered bisimulation given in [RT93] which required the existence of two coalgebra mappings fi 1 ; fi 2 : R F (R) such that fi 1 fi 2 and both 1 and 2 are coalgebra homomorphisms. The latter can be seen to be a special instance of the definition ....

M. Fiore. A coinduction principle for recursive data types based on bisimulation. In Proceedings of the Eighth IEEE Symposium on Logic In Computer Science, 1993.

....in terms of coalgebras. The former is the same as the final semantics for trace equivalence used in the present paper. The latter is a reformulation in terms of final semantics of Abramsky s observational semantics for his lazy lambda calculus; it is given in an order enriched setting. See [Fio93, Rut93, Pit92] for related work in order enriched categories. The idea of deriving compositional models from observational semantics based on transition system specifications is already described in [DG87] and [Bad87] A more general construction is given in [Rut92] which is the starting point for the present ....

M. Fiore. A coinduction principle for recursive data types based on bisimulation. In Proceedings of the Eighth IEEE Symposium on Logic In Computer Science, 1993.

....are functions on the natural numbers, and other examples are functions which act upon co recursively defined data structures, circular lists up to bisimulation. Implicit definitions of programs are very natural for functions whose arguments could include co recursively defined data structures [BM96, Fio96, MD96, Par83, MT91, Pit94, Gor95]. 1 1.1 Fixed Points Many programming languages are naturally characterized in terms of the least fixed point of a recursive operator. The least fixed point of a recursive operators is computable. Consider a simple example of the factorial function. f(0) 1 f(n 1) n 1) f(n) This is ....

M. Fiore. A coinduction principle for recursive data types based on bisimulation. Information and Computation, pages 186--198, 1996.

.... final coalgebras, open maps [Acz88, Abr91, HM85, RT94, JNG94] Bisimilarity has also been advocated outside concurrency theory; for instance, co induction principles based on bisimilarity have been proposed to reason about equality between elements of recursively defined domains and data types [Fio93, Pit94]. We first consider bisimilarity on standard labelled transition systems: Their transitions are of the form P Gamma Q, where P and Q are called processes, and label is drawn from some alphabet of actions. In such systems, bisimilarity, abbreviated , is defined as the largest symmetric ....

M. Fiore. A coinduction principle for recursive data types based on bisimulation. In 8th LICS Conf. IEEE Computer Society Press, 1993.

....we give a condition under which the operational notion of simulation coincides with the denotational notion of nal semantics. 1 Introduction Coinduction is a principle for reasoning about potentially in nite or circular elements of recursive data types, like streams, processes or exact reals [14,5,7]. Typically, one takes data types to be objects of some category of domains , and type constructors to be endofunctors on that category. Recursive data types can be modelled as xed points, or invariants, of the endofunctors. Of course, the sort of thing one may want to prove about elements of a ....

....A morphism of F coalgebras (A; B; is a map f : A B in C such that f = F (f) In particular, if i : F (X) X is an invariant of F , then (X; i 1 ) is an F coalgebra. In this paper we will be concerned with invariants which are nal in the category of F coalgebras. In [5], given an endofunctor F : C C, an F bisimulation between two F coalgebras is a relation in C between their carriers whose projections are both coalgebra maps. This generalizes Aczel and Mendler s de ntion of bisimulations for endofunctors on Set [2] The coinduction principle says that the ....

M. Fiore. (1996) A coinduction principle for recursive data types based on bisimulation. Information and computation, 127(2), 186-198.

....specifications which appear not to be easily expressible in coalgebraic terms are discussed. Introduction Coinductive definitions and coinduction proof principles are a natural tool for defining and reasoning on infinite and circular objects, such as streams, exact reals, processes. See e.g. [Mil83, Coq94, HL95, BM96, Fio96, Len96, Pit96, Rut96, HJ98, HLMP98, Len98] for various approaches to infinite objects based on coinduction. Many of such objects and concepts arise in connection with a maximal fixed point construction of some kind. One of the advantages offered by the coinductive approach with respect to others based on domain theory or metric semantics, ....

M.Fiore. A Coinduction Principle for Recursive Data Types Based on Bisimulation, Information and Computation 127, 1996, 186--198.

....sets; BM96] a recent textbook on non wellfounded sets and circularity; and [MD97] where corecursion is further studied in that context. Other categorical approaches to bisimulation include [Abr91] on a domain for bisimulation; WN95] on categories of transition systems; Pit94] [Fio96], and [Pit96] on mixed inductioncoinduction principles on domains in terms of relational properties; HJ96] on functors on categories of relations; JNW96] on a characterization of bisimulation in terms of open maps and presheaves. In [BV96] a metric domain for bisimulation can be found. ....

M. P. Fiore. A coinduction principle for recursive data types based on bisimulation. Information and Computation, 127(2):186--198, 1996.

....need improvement. 8.1. Source of Examples The proof plans make no claim to deal with cases where there are terms with no value, so such theorems were excluded from consideration. These theorems were identified by inspection. Examples of coinduction were drawn from the literature, in particular [16], 22] 24] and [26] Examples are relatively few in number, the same one or two theorems appearing in almost every paper on the subject. Recent research is beginning to produce a larger corpus but this has tended to concentrate on problems outside the domain chosen for consideration (i.e. lazy ....

M. Fiore, A coinduction principle for recursive data types based on bisimulation, in: Proceedings of the Eight IEEE Symposium on Logic in Computer Science, 1993, pp. 110--119.

....of lazy functional languages. Paulson has also done work providing a theory for coinduction within HOL [22] Other work has been done applying coinduction to Input Output Effects [14] Object oriented Languages [16] and generally to recursively defined domains [25] and over recursive datatypes [11]. Several theorem provers have capabilities for coinductive proof although they all require user interaction. Perhaps the most work has been done in Isabelle for which a special package has been developed for coinductive definitions [24] and in which Milner and Tofte s work has been reproduced ....

Fiore, M, P. A Coinduction Principle for Recursive Data Types Based on Bisimulation. In Proc 8th Annual Symposium on Logic in Computer Science, Montreal, pages 110--119, IEEE Computer Society Press, Washington.

....science. The partial computable functions are naturally those defined via least fixed points on recursive operators. Patterson and Costello 1996) have recently proposed using combinations of computable greatest and least fixed points as a programming constructs. Pitts (1994) Milner (1991) Fiore (1996) and others have argued for the importance of corecursion, or greatest fixed points. Pitts, in (1994) stresses the need for mixing recursion and corecursion, and suggests that C op Theta C is an important candidate for these investigations. Our algebra is the posetal case of his suggestion. We ....

Fiore, M. 1996. A coinduction principle for recursive data types based on bisimulation. Information and Computation 186--198.

....of lazy functional languages. Paulson has also done work providing a theory for coinduction within HOL [21] Other work has been done applying coinduction to input output effects [13] Object oriented languages [14] and generally to recursively defined domains [23] and over recursive datatypes [10]. Several theorem provers have capabilities for coinductive proof although they all require user interaction. Perhaps the most work has been done in Isabelle for which a special package has been developed for coinductive definitions [22] and in which Milner and Tofte s work has been reproduced ....

M. Fiore. A coinduction principle for recursive data types based on bisimulation. In Proceedings of the Eight IEEE Symposium on Logic in Computer Science, pages 110--119, 1993.

....: Delta ae R ffl For z : Z TZ and R 2 R(Z) z : R ae TR (R) coit(z) R ae Delta Bisimulation. Define a T # bisimulation to be an R 2 R(D) such that ud : R ae TR (R) Then, Delta is a T # bisimulation. Moreover, free pseudo algebras satisfy the following coinduction property (cf. [23, 24, 7, 12]) Delta = W fR 2 R(D) j R is a T # bisimulationg : 6.2 Mixed variance case To treat the mixed variance case we consider involutory 2 categories (viz. 2 categories which are self dual via an involution see [6] Formally, these are 2 categories K equipped with a pseudo functor O : K ....

M. Fiore. A coinduction principle for recursive data types based on bisimulation. Inform. and Comput., 127(2):186--198, 1996.

....2 5.6 Coinduction Dana Scott suggested that we should also consider a characterization of the partial unit interval via co Peano axioms based on coinduction and coiteration. Although we don t have such a characterization yet, a coinduction principle related to the ideas of Smyth [32] and Fiore [13] immediately follows by considering the bifree T algebra. A bisimulation on the partial unit interval is a binary relation I Theta I such that x y implies that left(x) left(y) and pred a (x) pred a (y) for a 2 fR; Lg. We say that x and y are bisimilar if they are related by some ....

....a (x) pred a (y) for a 2 fR; Lg. We say that x and y are bisimilar if they are related by some bisimulation. Proposition 34 (Coinduction) If x; y 2 I are bisimilar then x = y. Proof. Let x and y be bisimilar partial numbers. Then bin(x) and bin(y) are bisimilar trees. Hence bin(x) bin(y) by [13]. Therefore x = y because bin is split mono. 2 Of course, we can replace equalities by inequalities thus obtaining the notion of a simulation and a more general coinduction principle for establishing inequalities. 6 Applications to the programming language Real PCF Real PCF [9] is an extension ....

M.P. Fiore. A coinduction principle for recursive data types based on bisimulation. Information and Computation, 127(2):186--198, 1996.

.... universal property given by the compactness axiom, he established a mixed induction coinduction property of abstract relations on recursive domains in Cppo (the category of pointed cpos and strict continuous functions) Abstract category theoretic accounts of these issues can be found in [Fio93, HJ95] In type theory. In [CP92] Crole and Pitts introduced a higher order typed predicate logic for fixed point computations. This was done by exploiting Moggi s treatment of computations using monads [Mog91] and by introducing the key notion of fixpoint object . Fixpoint objects were partly ....

M.P. Fiore. A coinduction principle for recursive data types based on bisimulation. In 8 LICS Conf. IEEE, Computer Society Press, 1993. (Full version to appear in Information and Computation special issue for LICS93).

....Algebraic compactness is a universal property due to Freyd [7] that provides canonical interpretations of recursive domains. In this section we show this property for so called Kcats; these may be seen as a 2 categorical analogue 6 of cppos ( complete pointed partial orders) Following [5], our approach is to obtain the result from the Local Characterisation and Limit Colimit Coincidence Theorems, together with the Basic Lemma [28] Recall that the Basic Lemma provides conditions under which an initial algebra (and hence a fixed point, by a lemma due to Lambek) of an endofunctor ....

....all R ; R 2 R(D) Define a TR bisimulation to be an R 2 R(D) such that unfold : R ae TR ( Delta; R) Clearly, Delta is a TR bisimulation. Moreover, if TR (R; Delta) TR ( Delta; Delta) 3) for all R 2 R(D) then free pseudo dialgebras satisfy the following coinduction property (c.f. [21, 22, 5, 11]) Delta = fR 2 R(D) j R is a TR bisimulationg : Notice that the requirement (3) is vacuous when T is essentially covariant ; that is, when it factors through an endofunctor on fK j Rg via the projection fK j Rg Theta fK j Rg fK j Rg. 13 7 Open map bisimulation We provide ....

M. P. Fiore. A Coinduction Principle for Recursive Data Types Based on Bisimulation. Inf. & Comp., 127:186--198, 1996.

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M. P. Fiore. A coinduction principle for recursive data types based on bisimulation. Information and Computation, 127(2):186--198, 1996.

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Fiore, M.: A coinduction principle for recursive data types based on bisimulation. In: Proceedings Eighth Symposium on Logic in Computer Science, IEEE (1993)

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M.P. Fiore. A coinduction principle for recursive data types based on bisimulation. Inf. & Comp., 127(2):186--198, 1996.

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M. Fiore. A coinduction principle for recursive data types based on bisimulation. In Proc. 8th LICS Conf. IEEE Computer Society Press, 1993. 32

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Fiore, M. (1996b). A coinduction principle for recursive data types based on bisimulation.

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