Flagg, B., R. Kopperman, Continuity spaces: reconciling domains and metric spaces, Copies of slides for MFPS-X, 1994.  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] ....

R. C. Flagg, R. Kopperman, Continuity Spaces: reconciling domains and metric spaces, Theoretical Computer Science, 177(1), 1997, 111-138.

....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 from the context of 2 3 SFP ....

....opposed to the adjective qualitative which indicates the traditional order theoretic approach. The terminology Quantitative Domain Theory was coined in [FSW96] At this point several foundations exist. The more abstract approaches include the Yoneda completion ( BvBR95] the continuity spaces ([FK97]) and the topological quasi uniform spaces ( Sun93] These approaches are essentially equivalent (cf. FS96] and [KS97] and lead to complex completions, involving non idempotency or subtle relations between two topologies and a quasi uniformity. Moreover, they involve generalized metrics which ....

R. C. Flagg, R. Kopperman, Continuity Spaces: reconciling domains and metric spaces, Theoretical Computer Science, 177(1), 1997, 111-138.

....categories, and an analysis of his work using the suitable logic, viz. the logic of [0, c] should also be interesting. Flagg and Kopperman have also worked towards a unification of metric spaces and do mains for semantics (see for instance [Flagg Kopperman 93] Flagg Kopperman 93b] and [Flagg Kopperman 94] They have, independently from the theory of enriched cate gories, developed a closely related concept. Their continuity spaces are categories enriched over what they call value quantales. A value quantale is almost like a commutative unital quantale, but with an additional property that makes ....

Flagg, B., R. Kopperman, Continuity spaces: reconciling do- mains and metric spaces, Copies of slides for MFPS-X, 1994.

....basic notions of functional analysis, domain theory, algebraic and continuous data types as e.g. in (respectively) 22] 23] and [15] We present the basic ingredients of the approach to semantics via continuity spaces but leaving the interested reader to fill in the gaps in the literature (q.v. [12, 11, 13]) 2 The Lazy Reals One of the most widely used representation of the reals in both practical and theoretical computer science is the signed digit representation (q.v. 4, 14, 18, 26, 16] The reason comes from its simplicity, computability of the arithmetic operations, and easy implementation ....

....Turing machines. 16] proposes a modified real RAM model which ends up being polynomially equivalent to the Turing model of [26] 3 Continuity Spaces We quickly survey the main results and definitions of continuity spaces needed in Section 4. The reader interested in more details may consult [12, 11, 13] The bottom element of a complete lattice V is denoted by 0, the top element by 1 and for A a subset of V , the least upper bound of A is denoted by sup A and the greatest lower bound of A by inf A. Definition 3.1 A value distributive lattice is a completely distributive lattice V satisfying the ....

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Robert C. Flagg and Ralph Kopperman. Continuity spaces: Reconciling domains and metric spaces -- Part I. Submitted to Theoretical Computer Science. 10

....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, Law86] is that, intuitively, one may ....

....if, for x in X, x 2 V ) 9ffl 0; B ffl (x) V: The set of all gA open subsets of X is denoted by O gA (X) For instance, for every x in X, the ffl ball B ffl (x) is a gA open set. The pair hX; O gA (X)i can be shown to be a topological space with fB ffl (x) j ffl 0 and x 2 Xg as basis (cf. FK95] Every topology O(X) on a set X induces a preorder on X called the specialization preorder: for any x and y in X, x O y if and only if 8V 2 O(X) x 2 V ) y 2 V . The specialization preorder on a gum X induced by its generalized Alexandroff topology coincides with the preorder underlying X: ....

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B. Flagg and R. Kopperman. Continuity spaces: reconciling domains and metric spaces. To appear, 1995.

....particular functors to classify the domains. The linear and branching domains are both turned into a quasimetric space which induces a preorder and hence a category. Lately, there is a growing interest in quasimetric spaces. See, e.g. Wagner s thesis [Wag94] and Flagg and Kopperman s [FK94]. The quasimetrics are obtained from the metrics the d omains are endowed with by dropping one half of the Hausdorff metric. The morphisms of the branching domain can be seen as simulations and the morphisms of the linear domains can be viewed simply as inclusion functions. The linearize operator ....

B. Flagg and R. Kopperman. Continuity Spaces: reconciling domains and metric spaces - part I. Draft, May 1994.

....ultrametric spaces, which is the main motivation for the present study. Our sources of inspiration are the work of Lawvere on V categories and generalized metric spaces [Law73] and the work by Smyth on quasi metric spaces [Smy91] and we have been influenced by recent work of Flagg and Kopperman [FK95] and Wagner [Wag94] The present paper continues earlier work [Rut95] in which some of the basic theory of generalized ultrametric spaces has been developed. The guiding principle throughout is Lawvere s view of ultrametric spaces as [0; 1] categories , by which they are structures that are ....

....ffl (x) o: The set of all gA open subsets of X is denoted by O gA (X) For instance, for every x 2 X the ffl ball B ffl (x) is a generalized Alexandroff open set. The pair hX; O gA (X)i can be shown to be topological space with B ffl (x) for every ffl 0 and x 2 X , as basic open sets (cf. FK95] For a subset V of X we write cl A (V ) for the closure of V in the generalized Alexandroff topology. Proposition 6.3 For every subset V of a gum X, cl A (V ) R A ffi ae A (V ) Proof: It follows from the characterization (11) of R A ffi ae A that it is sufficient to prove cl A (V ) ....

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B. Flagg and R. Kopperman. Continuity spaces: reconciling domains and metric spaces. To appear, 1995.

....categories, and an analysis of his work using the suitable logic, viz. the logic of [0; 1] should also be interesting. Flagg and Kopperman have also worked towards a unification of metric spaces and domains for semantics (see for instance [Flagg Kopperman 93] Flagg Kopperman 93b] and [Flagg Kopperman 94] They have, independently from the theory of enriched categories, developed a closely related concept. Their continuity spaces are categories enriched over what they call value quantales. A value quantale is almost like a commutative unital quantale, but with an additional property that makes ....

Flagg, B., R. Kopperman, Continuity spaces: reconciling domains and metric spaces, Copies of slides for MFPS-X, 1994.

....metric spaces, which is the main motivation for the present study. Our sources of inspiration are the work of Lawvere on V categories and generalized metric spaces [Law73] and the work by Smyth on quasi metric spaces [Smy91] and we have been influenced by recent work of Flagg and Kopperman [FK95] and Wagner [Wag94] The present paper continues earlier work [Rut95] in which part of the theory of generalized metric spaces has been developed. The guiding principle throughout is Lawvere s view of metric spaces as [0; 1] categories , by which they are structures that are formally similar to ....

....x 2 o ) 9ffl 0; B ffl (x) o: The set of all gA open subsets of X is denoted by O gA (X) For instance, for every x 2 X the ffl ball B ffl (x) is a gA open set. The pair hX; O gA (X)i can be shown to be topological space with B ffl (x) for every ffl 0 and x 2 X , as basic open sets (cf. FK95] For a subset V of X we write cl A (V ) for the closure of V in the generalized Alexandroff topology. Proposition 6.3 For every subset V of a gms X, cl A (V ) R A ffi ae A (V ) 17 Proof: It follows from the characterization (10) of R A ffi ae A that it is sufficient to prove cl A ....

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B. Flagg and R. Kopperman. Continuity spaces: reconciling domains and metric spaces. To appear in Theoretical Computer Science, 1995.

....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] ....

R. C. Flagg, R. Kopperman, Continuity Spaces: reconciling domains and metric spaces, Theoretical Computer Science, 177(1), 1997, 111-138.

....into account, Banach s theorem holds also in this setting and the algebraic compactness theorem for Cms carries over to the category Cqms of complete quasimetric spaces and functions which are both non expansive and continuous. A further generalization is achieved in (Wagner, 1994) see also (Flagg and Kopperman, 1997)) where structures parametric in a quantale Omega are studied. In particular, generalized notions of Cauchy sequence and limit are given which, at a higher level, are used to show that the standard constructions of final coalgebras (alias initial algebras) D. Turi and J. Rutten 6 in Cms and ....

....working in the category of pointed complete quasi metric spaces and strict continuous and non expansive functions, one would still have the same result, thus generalizing Theorem 6.1. Generalized Metric Spaces. Smyth s notion of Cauchy sequence for quasimetric spaces has been generalized both in (Flagg and Kopperman, 1997) and in K.R. Wagner s thesis (Wagner, 1994) Wagner s notion of limit is made parametric in a quantale Omega Gamma for Omega equal to the two elements lattice , it specializes to the standard notion of an chain in a partial order; for Omega equal to [0; 1] it specializes to Smyth s Cauchy ....

Flagg, B. and Kopperman, R. (1997). Continuity spaces: Reconciling domains and metric spaces.

....classify domains. The linear and branching domain are both turned into a generalized metric space. Generalized metric spaces were already studied by Lawvere [Law73] Lately, there is a growing interest in generalized metric spaces (see, e.g. Wagner s thesis [Wag94] Flagg and Kopperman s [FK94] and Rutten s [Rut95] The generalized metrics are obtained from the metrics the domains are endowed with by dropping one half of the Hausdorff metric [Hau14] Generalized metric spaces induce very simple categories, namely preorders. The morphisms of the branching domain can be seen as ....

B. Flagg and R. Kopperman. Continuity Spaces: reconciling domains and metric spaces - part I. Draft, May 1994.

....metric spaces, which is the main motivation for the present study. Our sources of inspiration are the work of Lawvere on V categories and generalized metric spaces [Law73] and the work by Smyth on quasi metric spaces [Smy91] and we have been influenced by recent work of Flagg and Kopperman [FK] and Wagner [Wag94] The present paper continues earlier work [Rut96a] in which part of the theory of generalized metric spaces has been developed. The guiding principle throughout is Lawvere s view of metric spaces as [0; 1] categories , by which they are structures that are formally similar to ....

....2 o ) 9ffl 0; B ffl (x) o: The set of all gA open subsets of X is denoted by O gA (X) For instance, for every x 2 X the ffl ball B ffl (x) is a gA open set. The pair hX; O gA (X)i can be shown to be a topological space with B ffl (x) for every ffl 0 and x 2 X , as basic open sets (cf. FK] For a subset V of X we write cl A (V ) for the closure of V in the generalized Alexandroff topology. Proposition 6.2 For every subset V of a gms X, cl A (V ) R A ffi ae A (V ) Proof: For all subset V of X and x in X , x 2 cl A (V ) 8o 2 O gA (X) x 2 o ) o V 6= 8ffl 0; B ....

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B. Flagg and R. Kopperman. Continuity spaces: reconciling domains and metric spaces. To appear in Theoretical Computer Science.

....particular functors to classify the domains. The linear and branching domains are both turned into a quasimetric space which induces a preorder and hence a category. Lately, there is a growing interest in quasimetric spaces. See, e.g. Wagner s thesis [Wag94] and Flagg and Kopperman s [FK94]. The quasimetrics are obtained from the metrics the d omains are endowed with by dropping one half of the Hausdorff metric. The morphisms of the branching domain can be seen as simulations and the morphisms of the linear domains can be viewed simply as inclusion functions. The linearize operator ....

B. Flagg and R. Kopperman. Continuity Spaces: reconciling domains and metric spaces - part I. Draft, May 1994.

....function space construction is introduced. The latter seems more appropriate for the uniform case and for certain applications in programming semantics. The constructions are well behaved with regard to uniform local boundedness and completability. Introduction This paper is part of the program [Smy88, Smy91, Sun95b, FK96] of developing a quantitative version of domain theory, that is to provide the theory of quasi uniformities or related structures with constructions and notions from (ordinary) domain theory [AJ94] Specifically, we consider the question of supplying suitable function space constructors, a ....

R. C. Flagg and R. Kopperman. Continuity spaces: Reconciling domains and metric spaces --- part I. Theoretical Computer Science, 1996. To appear.

....are defined in two ways: by specifying the open sets and by a closure operator. These 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 [FK94] A definition of a generalized Scott topology in terms of open sets similar to ours is briefly mentioned by Smyth in [Smy87] The definition of the generalized Alexandroff topology in terms of a closure operator already appears in [Law73, Law86, Ken87] New is the definition of the generalized ....

.... topology for complete partial orders, the generalized Scott topology encodes all information about order, convergence, and continuity (cf. Smy91] The generalized Alexandroff topology only encodes the information about order, just like the ordinary Alexandroff topology for preorders (cf. Smy87, FK94] The paper is organized as follows. Section 2 and 4 give some basic definitions and facts on gms s. The Yoneda lemma and the generalized Alexandroff topology are discussed in Section 3, while the generalized Scott topology is presented in Section 5. Finally, in Section 6 some related work is ....

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B. Flagg and R. Kopperman. Continuity Spaces: reconciling domains and metric spaces, November 1994. To appear in Theoretical Computer Science.

....categories, of which the hom functor X( Gamma; Gamma) does not take values in the category of sets but in the (category of) real numbers. Recently, some of Lawvere s ideas have been further pursued with the aim of unifying traditional order theoretic and metric domain theory, in [14] [6], and [12] The latter paper deals, more specifically, with the aforementioned generalized metric spaces. As was mentioned in the introduction, in the literature several techniques have been proposed for solving metric domain equations. All of them use Alessi et al. embedding projection pairs: ....

B. Flagg and R. Kopperman. Continuity spaces: reconciling domains and metric spaces. To appear, 1995.

....this work in x6. This paper builds on [27] and much of our notation and terminology is as established in [27] Acknowledgement . The author thanks an anonymous referee for comments which improved the presentation of this paper at certain points. In particular, he thanks the referee for bringing [17, 18] to his attention. These two papers are concerned with continuity spaces, which are a common refinement of metric and ordered spaces, and therefore contain ideas close in spirit to those in this paper although there is no actual overlap between our work and that of [17, 18] 2 Quasi metric Spaces ....

....the referee for bringing [17, 18] to his attention. These two papers are concerned with continuity spaces, which are a common refinement of metric and ordered spaces, and therefore contain ideas close in spirit to those in this paper although there is no actual overlap between our work and that of [17, 18]. 2 Quasi metric Spaces As stated in the Introduction, we find it convenient to give here a brief summary of the facts we need concerning quasi metric spaces; in this respect we shall follow [7, 23, 24, 29] Definition 1 Let X be a non empty set. A quasi metric on X is a map from X Theta X ....

R.C. Flagg and R. Kopperman, Continuity Spaces: Reconciling Domains and Metric Spaces, Theoretical Computer Science, to appear.

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Flagg, B., R. Kopperman, Continuity spaces: reconciling domains and metric spaces, Copies of slides for MFPS-X, 1994.

No context found.

B. Flagg and R. Kopperman. Continuity spaces: reconciling domains and metric spaces. To appear, 1995.

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