America, P., J.J.M.M. Rutten, Solving reflexive domain equations in a category of complete metric spaces, Centrum voor Wiskunde en Informatica, Centre for Mathematics and Computer Science, Report CS-R8709, February 1987. Also appeared in Journal of Computer and System Sciences, 39(3):343-375, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....subset of) the real numbers. Metrics extend via the Hausdorff distance to distances between subsets, and it was possible for de Bakker and Zucker to give a denotational semantics to a wide range of language constructs in the realm of concurrent languages such as CSP. Later America and Rutten ( America Rutten 87] generalized the metric techniques to cover all the usual constructs in traditional programming language semantics, viz. products, sums (disjoint union) power set, and function space, with constants such as the one element data type, the Booleans and the natural numbers. Their approach allows ....

....we present a generalized version of Scott s inverse limit theorem and give sufficient conditions for when a functor has fixed points. These conditions gen eralize those of Scott (the original reference is [Scott 69] for a categorical exposition see [Smyth Plotkin 82] and America and Rutten ( America Rutten 87] Finally, we also discuss power set constructions, notions of compactness of elements and of domains. We relate to the Plotkin, Smyth, and Hoare power domains, and discuss the relation between the Egli Milner ordering and the Hausdorff distance, something Smyth has already done in a less ....

[Article contains additional citation context not shown here]

America, P., J.J.M.M. Rutten, Solving reflexive domain equa- tions in a category of complete metric spaces, Centrum voor Wiskunde en Informatica, Centre for Mathematics and Computer Science, Report CS-R8709, February 1987. Also appeared in Journal of Computer and System Sciences, 39(3):363-375, 1989.

....replacing the discrete distributions. Is in proven in the technical report [VR97b] that M 1 is well defined, preserves completeness and is locally nonexpansive. Thus the functor can be employed for the construction of semantical domains via metric domain equations a l a America Rutten (cf. [AR89, RT93]) Also, it is shown that a suitable functor based on M 1 possesses a final coalgebra. However, for M 1 to be a first class citizen in the world of categorical bisimulation it should preserve weak pullbacks. In this note only a partial result can be obtained for this. We are able to show that M 1 ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer Systems and Sciences, 39:343--375, 1989.

.... set Sflag of termination flags as ft; ntg: 4) Define the sets Sghist and Sphist of denotational histories as the solutions of the following recursive equations Sghist = Sstate (Sterm [ Sstate Theta Sghist) Sphist = Sstate (Sterm [ Sf lag Theta Sstate Theta Sphist) respectively (see [1] or [8] for the resolution of these equations) Denotational histories are thus essentially streams written as (ss 1 ; ss 2 ; ss 3 ; Delta Delta Delta) and (ff 1 ; ss 1 ; ff 2 ; ss 2 ; ff 3 ; ss 3 ; Delta Delta Delta) thanks to the Cartesian products. They are often rewritten in the ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343--375, 1989.

....hp n i n a Cauchy sequence in D , such that p n 2 D n for all n 2 IN. Then for p = lim n p n , the truncation of p for any k 2 IN is defined as follows: p[k] lim n p n [k] We show that D satisfies the required domain equation by constructing isometric embeddings. Categorical techniques of [1] have not been used as it is unclear how to define a functor to represent this construction; this is due to the fact that our metric is not defined inductively in correspondence with the inductively defined metric spaces. Theorem 4.8 D satisfies the domain equation: D = fp 0 g [ A Theta i ....

P.H.M.America and J.J.M.M.Rutten. Solving reflexive domain equations in a category of complete metric spaces, JCSS, 39, no.3, 1989.

....Zucker (with the missing half) is not a solution of the equation. The presence of the halves (or some other positive constant smaller than 1) is essential for De Bakker and Zucker s method for solving recursive equations and for the generalization of their method by Pierre America and Jan Rutten [AR89]. They are also a key ingredient of Rutten s above mentioned metric terminal coalgebra theorem. In this paper, we consider the following questions: What happens if we leave out the halves Do we still have terminal coalgebras We consider the equations X = P c (X) and X = P k (X) We show that ....

P. America and J.J.M.M. Rutten. Solving Reflexive Domain Equations in a Category of Complete Metric Spaces. Journal of Computer and System Sciences, 39(3):343--375, December 1989.

....constructs of the form succ( Theta) where Theta is a set of substitutions. The set of denotational histories, Sdhist is defined as the solution of the following recursive equation: Sdhist = Sterm [ Ssusp [ Ssubst Theta Sdhist) Shyp Theta Ssubst Theta Sdhist) Theta Sdhist (see [6] or [1] for the resolution of this equation) Histories are thus streams written as (e 1 ; e 2 ; e 3 ; Delta Delta Delta) thanks to the cartesian products. They are often rewritten in the simpler form e 1 :e 2 :e 3 : Delta Delta Delta to avoid the intricate use of brackets. However, the ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343--375, 1989.

....N a (metric) domain and M is the domain defined by the equation. Recursive domain equations are solved up to isomorphism, yielding instead the domain equation (or more accurately a domain isometry) M F(M ) The method of solving recursive domain equations over metric spaces comes from [7] [1] and [26] First some basic notions is introduced and then the functors used are given. More on domain equations can be found in e.g. 6] Definition 4.1 Let CUMS denote the category of all complete ultra metric spaces with non expansive functions as morphisms. a) A functor F : CUMS CUMS is ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer Systems and Sciences, 39:343--375, 1989.

....for Cauchy towers of spaces without using the classical categorical construction and how to find solutions of recursive domain equations inside Pnco(U ) 1 Introduction In the recent past metric spaces have often been used successfully in the semantics of concurrent programming languages. Since [3], where the technique of [12] for solving domain equations is adapted to the metric context, several categories of metric spaces have been introduced in the literature. Apart from technical differences, all the approaches follow a common pattern which guarantees the existence of categorical limits ....

....allow to introduce the notion of Cauchy towers of spaces (a sequence (Xn ; hf n ; g n i) n2IN is Cauchy if for each ffl 0 the ffi s of compositions of morphisms are eventually less than ffl) and it is proved that each Cauchy tower has a categorical limit. 4. Classes of functors (contracting [1, 2, 3, 13, 7] cut contracting [8] homcontracting [3] locally contracting [13, 11] are singled out that generate Cauchy towers when iteratively applied to an initial space. This allows to solve those domain equations which involve such functors. An important remark is that all the categories considered in ....

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P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39:343--375, 1989.

....or metric space X with typical elements x, x 0 , x 0 , The fact that there exists a unique (up to isometry) complete metric space A 1 being isometric 3 (see Definition A.11) to the complete metric space A Theta 1 2 Delta (1 A 1 ) follows from Theorem 4. 4 of America and Rutten s [AR89]. The proof of this theorem relies on Banach s theorem. The elements of the complete metric space A 1 can be viewed as nonempty and finite or infinite sequences over the action set A. For example, the element ha; hb; 0ii corresponds to the finite sequence ab and the infinite sequence a ....

....of the complete metric space A, the singleton metric space 1, and the operations 1 2 Delta, Theta, and P nk . Definition 2.1 The complete metric space (B 2) B is the unique complete metric space satisfying B = 1 P nk (A Theta 1 2 Delta B) Again we conclude from Theorem 4. 4 of [AR89] that such a complete metric space exists. The elements of the branching domain, the branching processes, can be viewed as labelled trees with the following three properties. First of all, the labelled trees are commutative, i.e. for all nodes of a tree, its subtrees are not ordered. For example, ....

P. America and J.J.M.M. Rutten. Solving Reflexive Domain Equations in a Category of Complete Metric Spaces. Journal of Computer and System Sciences, 39(3):343--375, December 1989.

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P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343-- 375, December 1989. References 31

....complete metric spaces, using as a metric on arrows the usual pointwise extension. A functor F on CMS is locally contracting if there exists ffl with 0 ffl 1 such that, for all D;E, the mapping FD;E is a contraction with factor ffl. In [RT93] it is shown (extending earlier results of [AR89]) that every locally contracting functor F has a unique fixed point which is both an initial F algebra and a final F coalgebra. A metric version of Theorem 5.1 is obtained by dropping both in the formulation of the theorem and in its proof the word order(ed) everywhere; considering in ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343--375, 1989. References 15

....is a new result. It is shown that locally contracting functors on the category of complete metric spaces (with non expansive mappings as arrows) have a final coalgebra. The proof is based on a theorem stating that such functors have fixed points. The latter theorem extends earlier results of [AR89] along the lines of [SP82] and is proved in full detail. For partial orders an initial algebra theorem and the so called limit colimit coincidence are well known (see [SP82] but, apparently, it was never proved in detail that (in CPO ) initial algebras and final coalgebras coincide. ....

...., which yields all metrically closed subsets. In [Bre92] domains are given suited for LTS s that satisfy even more general branching properties. 5.3 Fixed Points in CMS In this subsection, it will be shown that every locally contracting functor has a fixed point, thus proving Theorem 5.7. In [AR89], a similar theorem is proved: so called contracting functors on a category of complete metric spaces (with double arrows) have a fixed point (see also below) Here the results of [AR89] are generalized; in summary, a reconstruction of that paper is given along the lines of [SP82] and [Plo81a] A ....

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P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343--375, 1989.

....the specific way in which the semantic domains are constructed: It is defined in terms of their universal properties only. Traditionally, semantic domains have been constructed in a recursive manner by using sets with some additional structure, like partial orders or metric spaces. See, e.g. [SP82, Ken87, AR89]. See also [ArM82] for an early reference on final coalgebras of functors on sets. A construction of semantic domains in terms of sets with no additional structure occurs in [Acz88] however, a non standard set theory is used in which sets may be non wellfounded. In the same book, the final ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343-- 375, 1989.

....are equal if and only if they are bisimilar. It follows from a simple but very general argument that final coalgebras are fully abstract (see Aczel s final coalgebra model for nonwellfounded sets [Acz88] and also [RT93] We shall show that 2 it follows from general coalgebraic considerations [AR89,Bar93,RT93] that both our functors D and M 1 have a final coalgebra, which consequently are internally fully abstract with respect to (discrete and continuous) probabilistic bisimulation. Therefore these final coalgebras can be exploited as semantic domains for probabilistic bisimulation (an important ....

....for U 2 B(S) V 2 B(T ) The verification that fl(s; t) is well defined and mediating for ff(s) fi(t) is nontrivial but omitted for reasons of space. ut In the remainder of this section, we shall again use some general insights from the theory of coalgebras, this time by applying a result from [AR89,RT93]. In turns out, that we are only able to show the existence of a final coalgebra when we consider an adaptation of M 1 , say M 0 1 , which delivers Borel probability measures with so called compact support, i.e. measures that vanish outside a compact set. More precisely, for a metric space M , ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer Systems and Sciences, 39:343--375, 1989.

....for complete ultrametric spaces. 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (cf. e.g. the recent textbooks [Win93] and [BV95] Parts of their theory have been developed because of semantic necessity (see, e.g. SP82] and [AR89] Generalized ultrametric spaces provide a common framework for the study of both preorders and ordinary ultrametric spaces. A generalized ultrametric space consists of a set X together with a distance function X( Gamma; Gamma) X Theta X [0; 1] satisfying X(x; x) 0 and X(x; z) maxfX(x; ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences 39(3):343--375, 1989.

....sets [Acz88] and also [RT93] Here, final means that there exists a unique homomorphism from any coalgebra to the final one. Finality is to the world of coalgebras what initiality is to the world of algebras, cf. MG85] We shall show that it follows from general coalgebraic considerations [AR89, Bar93, RT93] that both our functors D and M 1 have a final coalgebra, which consequently are internally fully abstract with respect to (discrete and continuous) probabilistic bisimulation. Therefore these final coalgebras can be exploited as semantic domains for probabilistic bisimulation (an important ....

.... Gamma1 1 (E k ) E k Theta F k = Gamma1 2 (F k ) Thus M 1 ( 1 ) fl s;t ) ff(s) ff( 1 (s; t) and M 1 ( 2 ) fl s;t ) fi(t) fi( 2 (s; t) 2 In the remainder of this section, we shall again use some general insights from the theory of coalgebras, this time by applying a result from [AR89, RT93]. In turns out that we are only able to show the existence of a final coalgebra when we consider an adaptation of M 1 , say M 0 1 , which delivers Borel probability measures with so called compact support, i.e. measures that vanish outside a compact set. More precisely, for a metric space M , ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer Systems and Sciences, 39:343--375, 1989.

....also [RT93] Here, final means that there exists a unique homomorphism from any coalgebra to the final one. One can argue that finality is to the world of coalgebras what initiality is to the world of algebras, cf. MG85] We shall show that it follows from general coalgebraic considerations [AR89, Bar93, RT93] that both functors that are considered have a final coalgebra, which consequently is internally fully abstract with respect to (discrete and continuous) probabilistic bisimulation. Therefore these final coalgebras can be exploited as semantic domains for probabilistic bisimulation (an important ....

....more standard use of ordered structures, such as [Jon89, JP89] and [Eda95a, Eda95b] are twofold. Firstly, one can resort to the rich literature on standard measure theory for metric spaces (see, e.g. KV84] Secondly, we can use the recently developed coalgebraic theory on metric spaces [AR89, RT94], which seems to be better suited to describe (both ordinary and probabilistic) bisimulation than the corresponding theory for ordered spaces (cf. RT94] We shall see that the functor involved is locally contractive, from which it follows that it has a final coalgebra. Because of the coalgebraic ....

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P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer Systems and Sciences, 39:343--375, 1989.

.... h : 2 2 and t : 2 2 are the head and tail functions) supplied with distance function d 2 (v; w) 1 X i=0 d 2 (v i ; w i ) 2 i ; is a final (2 Theta Gamma) system: a proof is omitted but can be given using the techniques for the solution of metric domain equations of [AR89] and [RT93] Consequently, 2 ; t) is a dynamical system that is cofree on the metric space 2. Now define a colouring c : J 2 of J by c(x) ae 0 if x 2 [0; a] 1 if x 2 [b; 1] By the universal property of the cofree system (2 ; t) there exists a unique homomorphism c : J; f) ....

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343--375, December 1989.

....1996b) It also contains a full abstraction theorem with respect to the order enriched version of coalgebraic bisimulation. The last section is dedicated to final coalgebras in categories of metric spaces. It is shown that the categorical version of Banach s fixed point theorem introduced in (America and Rutten, 1989) yields a final coalgebra (canonically isomorphic to an initial algebra) Also, the above mentioned recent results on generalized notions of metric spaces, which reconcile the order theoretic with the metric theoretic approach, are discussed. 2. From Algebraic Induction to Coalgebraic ....

....X;Y , and for all pairs of parallel non expansive functions f; g : X Y between them. Theorem 7.2. Every locally contractive endofunctor F on the category of complete metric spaces and non expansive functions has a unique fixed point Fix(F ) F (Fix(F ) Proof. The proof is based on (America and Rutten, 1989), where a category Cms E of complete metric spaces is considered with as morphisms (a metric version of Smyth and D. Turi and J. Rutten 54 Plotkin s) embedding projection pairs. Cf x6.1. Then a notion of contractivity for endofunctors on that category is defined, and it is shown that ....

America, P. and Rutten, J. (1989). Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343--375.

No context found.

America, P., J.J.M.M. Rutten, Solving reflexive domain equations in a category of complete metric spaces, Centrum voor Wiskunde en Informatica, Centre for Mathematics and Computer Science, Report CS-R8709, February 1987. Also appeared in Journal of Computer and System Sciences, 39(3):343-375, 1989.

No context found.

P. America and J.J.M.M. Rutten, Solving Reflexive Domain Equations in a Category of Complete Metric Spaces, Centrum voor Wiskunde en Informatica, Report CS-R8709, 1987. Also in: Journal of Computer and System Science, vol. 39 (1989), pp. 343--375.

No context found.

P. America and J.J.M.M. Rutten, Solving Reflexive Domain Equations in a Category of Complete Metric Spaces, Centrum voor Wiskunde en Informatica, Report CS-R8709, 1987. Also in: Journal of Computer and System Science, vol. 39 (1989), pp. 343--375.

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P. America and J.J.M.M. Rutten, 1988. Solving reflexive domain equations in a category of complete metric spaces. Proc. Third Workshop on the Mathematical Foundations of Programming Language Semantics, Springer-Verlag Lecture Notes in Computer Science 298, pp. 254--288. To appear in JCSS.

No context found.

P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer and System Sciences, 39(3):343--375, December 1989.

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America P., Rutten J.J.M.M., Solving reflexive domain equations in a category of complete metric spaces, Journal of Computer and System Sciences, Vol 39, no. 3, 1989, pp. 343-375.

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