Smyth, M.B., Quasi-uniformities: reconciling domains with metric spaces, Proceedings of the 3'rd Workshop on the Mathematical Foundations of Programming Language Semantics, Tulane, April 1987, Lecture Notes in Computer Science, Vol. 298, Springer Verlag, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the Yoneda completion of a quasi metric space H. P. Kunzi and M. P. Schellekens Abstract Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. FK97] BvBR9 8] [Smy89] and [Wag94] We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98] which finds its roots in work by Lawvere ( Law73] cf. also [Wag94] and which is related to early work by Stoltenberg (e.g. Sto67] Sto67a] and [FG84] and the ....

.... focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98] which finds its roots in work by Lawvere ( Law73] cf. also [Wag94] and which is related to early work by Stoltenberg (e.g. Sto67] Sto67a] and [FG84] and the Smyth completion ([Smy89], Smy91] Smy94] Sun93] and [Sun95] A net version of the Yoneda completion, complementing the net version of the Smyth completion ( Sun95] is given and a comparison between the two types of completion is presented. The following open question is raised in [BvBR98] An interesting ....

[Article contains additional citation context not shown here]

M. B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, LNCS 298, Springer Verlag, 1987, 236-253.

....# d(x, y) r for all x X and r 0. It is clear, that d is a quasi metric if and only if T (d) is a T 0 topology. A quasi (pseudo)metric d on X is called bicomplete [7] if d is a complete (pseudo)metric on X. In this case, X, d) is said to be a bicomplete quasi (pseudo)metric space. In [27] and [28] Smyth presented a topological framework for denotational semantic based on the theory of complete (and totally bounded) quasi uniform and quasi metric spaces. Sunderhauf continued this work in the setting of topological quasi uniform spaces [29] Kunzi characterized in [12] both Smyth ....

M.B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, in: Proc. MFPS 3, LNCS 298, M. Main et al. editors, Springer, Berlin 1988, 236-253.

....if the uniform space (X, is complete, where is the coarsest uniformity on X finer than 1 . A(n extended) quasi metric space (X, d) is said to be bicomplete if (X, d ) is a bicomplete quasi uniform space, equivalently if (X, d ) is a complete (extended) metric space. In [14] and [15] Smyth presented a topological framework for denotational semantic based on the theory of complete (and totally bounded) quasi uniform and quasi metric spaces. Sunderhauf continued this work in the setting of topological quasi uniform spaces [17] Kunzi characterized in [4] both Smyth ....

M.B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, in: Proc. MFPS 3, LNCS 298, M. Main et al. editors, Springer, Berlin 1988, 236253.

....in [Sun91] as a category extending the quasi uniform spaces ( FL82] which provides a suitable basis on which to develop the Smyth completion. We recall that the Smyth completion provides a topological foundation for Denotational Semantics, and in particular for the well know cpo completion (e.g. [Smy89], Smy92] and [Sun91] The S completable (topological) quasi uniform spaces have been defined in [Sun91] as the (topological) quasi uniform spaces of which the Smyth completion is again a quasi uniform space; a condition which in general is violated as indicated in [Sun91] Apart from forming a ....

M. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, LNCS 298 (Springer Verlag, 1987) 236-253.

....partial order are induced by a partial metric. For # algebraic dcpo s the Lawson topology is induced by the associated metric. The partial metrization of general domains improves prior approaches in two ways: The partial metric is guaranteed to capture the Scott topology as opposed to e.g. [Smy87], BvBR95] FS96] and [FK97] which in general yield a coarser topology. Partial metric spaces are Smyth completable and hence their Smyth completion reduces to the standard bicompletion. This type of simplification is advocated in [Smy91] Our results extend [Smy91] s scope of application ....

.... This class includes the Baire partial metric spaces of [Mat94] as well as the complexity spaces of [Sch95] cf. also [RS96] It also incorporates the Scott Domains, whether they be represented as totally bounded quasi metric spaces, as in [Smy91] or via 0 1 valued quasi metrics (e.g. [Smy87] or [BvBR95] and the interval domain ( EEP97] To analyze partial metrizability, we study the slightly more general class of quasiuniform semilattices. These structures are defined to be semilattices equipped with a quasi uniformity with respect to which the semilattice operation is ....

M. B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, LNCS 298, Springer Verlag, 1987, 236-253.

....have fixed points in this setting. The conditions generalize those already known from partial order semantics and metric semantics. Concerning connections to other work than what we have referred to extensively in the body of this thesis we wish to mention the following. Smyth ( Smyth 87] Smyth 88] and [Smyth 92] has found a common ground for pre order and metric semantics by combining the order properties and the distance properties in quasi metric spaces. These spaces are precisely the [0,c] categories with the extra requirement of symmetric separatedness, i.e. Sh( 0,O]max) d(x,y) A ....

Smyth, M.B., Quasi-uniformities: reconciling domains with metric spaces, Proceedings of the 3'rd Workshop on the Mathematical Foundations of Programming Language Semantics, Tulane, April 1987, Lecture Notes in Computer Science, Vol. 298, Springer Verlag, 1988.

....relative to which a function f on a metric space X is continuous. On the other hand the operator T need not have a unique fixed point in which case there can be no metric on D making T a contraction. Nevertheless, there is a common generalization of these two results to be found in [10, 11, 14] obtained by replacing metrics by quasi metrics or, more generally, by generalized ultrametrics. These notions have proven to be useful in programming language semantics and in particular in solving recursive domain equations, see [3, 4, 10, 11] In this note, we want to consider them from the ....

....respectively general (not necessarily definite) logic programs. In a sense, generalized ultrametrics and quasi metrics combine, in a single concept, the advantages of order with those of distance. 2 Generalized Ultrametrics and Quasi metrics 2. 1 Definitions and Examples Following [10, 11, 14, 15] we make: Definition 2.1 Let X be a non empty set. A generalized ultrametric on X is a map X( Gamma; Gamma) X Theta X Gamma [0; 1] satisfying: 1: X(x; x) 0 2: X(x; y) maxfX(x; z) X(z; y)g. A quasi metric on X is a generalized ultrametric which satisfies: 3: if X(x; y) X(y; ....

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M. B. Smyth, Quasi Uniformities: Reconciling Domains with Metric Spaces. In: M. Main, A. Melton, M. Mislove and D. Schmidt (Eds.), Mathematical Foundations of Programming Language Semantics. Lecture Notes in Computer Science, Vol. 298, Springer-Verlag, 1987, pp. 236-253. 10

....into convergent sequences, and to preserve their limits. Fixpoint equations are solved by constructing convergent sequences, and taking their limits. Each of the two approaches has its advantages and drawbacks. It should be mentioned that M. Smyth has recently proposed a generalization, cf. [Smy87], in an attempt to have the best of both worlds. In the programming languages practice recursion may appear on two levels: one may wish to define recursively types and or their elements. Recursive Types The class of all sets has a natural structure of a cppo: it is ordered (by the inclusion ....

M. B. Smyth. Quasi-uniformities: Reconciling domains with metric spaces. LNCS vol. 298, 1987.

....spaces were introduced by Lawvere [Law73] as an illustration of the thesis that fundamental structures are categories. The present work is inspired by Lawvere s enriched categorical view of generalized metric spaces [Law73] as well as the more topological view of Smyth on quasi metric spaces [Smy87, Smy91] It is based on the work [Rut95] in which some of the basic theory of generalized ultrametric spaces has Department of Computer Science, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands, email: marcello cs.vu.nl. y School of Computer Science, McGill ....

....topology. Both topologies are defined in two ways: by giving the open sets and by a closure operator. For both topologies the two alternative definitions are shown to coincide. Our definition of the generalized Alexandroff topology in terms of open sets is similar to the ones given by Smyth [Smy87, Smy91] and Flagg and Kopperman [FK95] A definition of a generalized Scott topology in terms of open sets similar to ours is briefly mentioned by Smyth in [Smy87] The definitions of the topologies in terms of closure operators are new. The key observation first made by Lawvere [Law73, ....

[Article contains additional citation context not shown here]

M.B. Smyth. Quasi uniformities: reconciling domains with metric spaces. In M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Proceedings of the 3rd MFPS, volume 298 of Lecture Notes in Computer Science, pages 236--253, New Orleans, 1987. Springer-Verlag.

....contraction mapping theorem are made in this context, too, see [Rut96] for a discussion and references. Therefore, it has been a question of some considerable interest in the recent past to unify the ordertheoretic and metric approaches to the semantics of imperative programming languages, see [Rut96, Smy87]. This work culminates, perhaps, in the quasi metric fixed point theorem of Rutten [Rut96] earlier formulated in terms of quasi uniformities by Smyth [Smy87] which contains the Kleene and Banach theorems as special cases. This paper has two main objectives, and the first of these is as follows. ....

.... in the recent past to unify the ordertheoretic and metric approaches to the semantics of imperative programming languages, see [Rut96, Smy87] This work culminates, perhaps, in the quasi metric fixed point theorem of Rutten [Rut96] earlier formulated in terms of quasi uniformities by Smyth [Smy87], which contains the Kleene and Banach theorems as special cases. This paper has two main objectives, and the first of these is as follows. For computability purposes, it is desirable to eliminate where possible the need to work transfinitely in obtaining models of disjunctive programs such as ....

M.B. Smyth, Quasi Uniformities: Reconciling Domains with Metric Spaces. In: M. Main, A. Melton, M. Mislove and D. Schmidt (Eds.), Mathematical Foundations of Programming Language Semantics. Lecture Notes in Computer Science, Vol. 198, Springer-Verlag, Berlin, 1987, pp. 236--253.

....J.J.M.M. Rutten CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands email: jan.rutten cwi.nl Abstract Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995) Combining Lawvere s (1973) enrichedcategorical and Smyth (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized ultrametric spaces. Restricted to the special cases of preorders and ordinary ultrametric spaces, these constructions yield, respectively: 1. ....

....the powerdomain of compact subsets. 8 Related work The thesis that fundamental structures are categories has been the main motivation for Lawvere in his study of generalized metric spaces as enriched categories [Law73] Lawvere s work together with the more topological perspective of Smyth [Smy87] have been our main source of inspiration for the present paper which continues the work of Rutten [Rut95] Generalized ultrametric spaces are a special instance of Lawvere s V categories. The non symmetric ultrametric for [0; 1] is also described and studied in his paper. The notion of forward ....

[Article contains additional citation context not shown here]

M.B. Smyth. Quasi uniformities: reconciling domains with metric spaces. In M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Proceedings of the 3rd Workshop on Mathematical Foundations of Programming Language Semantics, volume 298 of Lecture Notes in Computer Science, pages 236--253, New Orleans, 1987. Springer-Verlag.

....email: franck di.unipi.it J.J.M.M. Rutten CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands email: jan.rutten cwi.nl Abstract Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973) Combining Lawvere s (1973) enriched categorical and Smyth (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion ....

....the powerdomain of compact subsets. 34 8 Related work The thesis that fundamental structures are categories has been the main motivation for Lawvere in his study of generalized metric spaces as enriched categories [Law73] Lawvere s work together with the more topological perspective of Smyth [Smy88] have been our main source of inspiration for the present paper which continues the work of Rutten [Rut95] Generalized metric spaces are a special instance of Lawvere s V categories. The non symmetric metric for [0; 1] is also described and studied in his paper. The notion of forward Cauchy ....

[Article contains additional citation context not shown here]

M.B. Smyth. Quasi uniformities: reconciling domains with metric spaces. In M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Proceedings of the 3rd Workshop on Mathematical Foundations of Programming Language Semantics, volume 298 of Lecture Notes in Computer Science, pages 236--253, New Orleans, 1988. Springer-Verlag. 37

....the Yoneda completion of a quasi metric space H. P. Kunzi and M. P. Schellekens Abstract Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. FK97] BvBR9 8] [Smy89] and [Wag94] We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98] which finds its roots in work by Lawvere ( Law73] cf. also [Wag94] and which is related to early work by Stoltenberg (e.g. Sto67] Sto67a] and [FG84] and the ....

.... focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98] which finds its roots in work by Lawvere ( Law73] cf. also [Wag94] and which is related to early work by Stoltenberg (e.g. Sto67] Sto67a] and [FG84] and the Smyth completion ([Smy89], Smy91] Smy94] Sun93] and [Sun95] A net version of the Yoneda completion, complementing the net version of the Smyth completion ( Sun95] is given and a comparison between the two types of completion is presented. The following open question is raised in [BvBR98] An interesting ....

[Article contains additional citation context not shown here]

M. B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, LNCS 298, Springer Verlag, 1987, 236-253.

....COMPLETENESS IN TERMS OF NETS: THE GENERAL CASE PHILIPP S UNDERHAUF Abstract. Smyth completeness is the appropriate notion of completeness for quasi uniform spaces carrying an additional topology to serve as domains of computation [2, 3]. The goal of this paper is to provide a better understanding of Smyth completeness by giving a characterization in terms of nets. We develop the notion of computational Cauchy net and an appropriate notion of strong convergence to get the result that a space is Smyth complete if and only if ....

....as an approximate order, the entourages being increasingly better approximants for U = T U . This and motivation from domain theory leads to the investigation of quasi uniform spaces with an additional topology which may, but need not coincide with the induced one. Mike Smyth introduced in [2] the following axioms for an additional topology T on a quasi uniform space (X; U ) A1) 8O 2 T 8x 2 X: x 2 O = 9U 2 U : x OE U O) A2) The set f U j U 2 Ug is a base for U (where U = f(x; y) j x OE U O = y 2 Og) INT) 8U 2 U9V 2 U : OE U OE V ffi OE V . Date: May 6, 1996. 1991 ....

[Article contains additional citation context not shown here]

M.B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, Third Workshop on Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science, vol. 298, Springer Verlag, 1988, pp. 236--253.

....of posets, it seems advantageous to consider a different topology with the completed spaces. We introduce Smyth completion as tool to automatically end up with the right topology after completing. 1 Introduction This paper is part of the ongoing foundational work on quantitative domain theory [Smy88,BvBR95,Rut96,FWS96,Sun94,Wag94], which refines ordinary domain theory by replacing the qualitative notion of approximation by a quantitative notion of degree of approximation (cf. the introduction of [FWS96] We investigate the generalization of ideal completion of posets for quantitative domains suggested in [BvBR95] and ....

....quasi metric on the space for example, it does not in the poset case, where the latter is the Alexandroff topology. It was Mike Smyth, who first realized that, accordingly, one has to consider a different topology on the space; he also introduced a quantitative version of the Scott topology [Smy88]. Hence the objects of our studies are topological V continuity spaces, quantitative domains carrying an additional topology. We define Smyth completeness for these and construct the completion, which is idempotent. Moreover, Smyth completion of a quantitative domain with its Alexandroff topology ....

[Article contains additional citation context not shown here]

M.B. Smyth. Quasi-uniformities: Reconciling domains with metric spaces. In Third Workshop on Mathematical Foundations of Programming Language Semantics, volume 298 of Lecture Notes in Computer Science, pages 236--253. Springer Verlag, 1988.

....some of the results on computability of topological spaces by working in PER(P) with topological objects instead of just countably based countable T 0 spaces 3. 1 Metric Spaces Computability in metric spaces has been studied extensively, see for example Edalat [12] Wagner [35] Smyth [31], and America and Rutter [3] Every 14 metric space (M; d) with the topology induced by the metric is a T 0 space; in fact, it is a normal Hausdorff space. It is countably based if, and only if, it is separable, i.e. it contains a countable dense set. The topology on a separable metric space ....

M.B. Smyth, "Quasi-uniformities: reconciling domains with metric spaces", Proceedings of the 3rd Workshop on the Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science, vol. 298, Springer-Verlag, 1988.

.... O 0 O 0 B: 3 These linking axioms are: A1) 8O 2 T 8x 2 X: x 2 O = 9U 2 U : x OE U O) A2) There is a base for U consisting of entourages U such that U(x) is T closed for all x 2 X. A3) 8U 2 U9V 2 U8O 2 T : O OE V [O]U . The first two of these were introduced by Mike Smyth in [Smy88, Smy94], the last is from [Sun93] Note that the entourage V in (A3) may be chosen such that V U (just intersect with U ) We employ the notation V C U ( V U 8O 2 T : O OE V [O]U and will frequently make use of the fact that V C U implies [O]V int T [O]U for all O 2 T . The ....

....the notation V C U ( V U 8O 2 T : O OE V [O]U and will frequently make use of the fact that V C U implies [O]V int T [O]U for all O 2 T . The interpolation property (INT) 8U 2 U9V 2 U : OE U OE V ffi OE V is a consequence of (A3) Sun93, Proposition 1] It was introduced in [Smy88] and proves useful in various situations. The appropriate notion of completeness for these spaces was defined in [Smy94] The definitions read as follows: Definition 1 A topological quasi uniform space is a triple (X; U ; T ) where (X; U) is a quasi uniform space and T is a topology on X ....

[Article contains additional citation context not shown here]

M.B. Smyth. Quasi-uniformities: Reconciling domains with metric spaces. In Third Workshop on Mathematical Foundations of 14 Programming Language Semantics, volume 298 of Lecture Notes in Computer Science, pages 236--253. Springer Verlag, 1988.

No context found.

M.B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, in: Proc. MFPS 3, LNCS 298, M. Main et al. editors, Springer, Berlin 1988, 236-253.

No context found.

Smyth, M.B., Quasi-uniformities: reconciling domains with metric spaces, Proceedings of the 3'rd Workshop on the Mathematical Foundations of Programming Language Semantics, Tulane, April 1987, Lecture Notes in Computer Science, Vol. 298, Springer Verlag, 1988.

No context found.

M.B. Smyth, "Quasi-uniformities: reconciling domains with metric spaces", Proceedings of the 3rd Workshop on the Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science, vol. 298, Springer-Verlag, 1988.

No context found.

M.B. Smyth, "Quasi-uniformities: reconciling domains with metric spaces", Proceedings of the 3rd Workshop on the Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science, vol. 298, Springer-Verlag, 1988.

No context found.

M.B. Smyth. Quasi uniformities: reconciling domains with metric spaces. In M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Proceedings of the 3rd Workshop on Mathematical Foundations of Programming Language Semantics, volume 298 of Lecture Notes in Computer Science, pages 236--253, New Orleans, 1987. Springer-Verlag.

No context found.

M. B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, LNCS 298, Springer Verlag, 1987, 236-253.

No context found.

M. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, LNCS 298, Springer Verlag, 1987.

No context found.

M. B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, LNCS 298, Springer Verlag, 1987, 236-253.

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