K.R. Wagner, Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, Carnegie Mellon University, Technical Report CMU-CS-94-159, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... of Smith ( Smy87] have been our main source of inspiration for the present paper, which continues the work of Rutten ( Rut95] and is part of the work of Bonsangue, van Breugel and Rutten ( BBR95] Recently, we have been influenced also by the work of Flagg and Kopperman [FK95] and Wagner [Wag94] Generalized ultrametric spaces are a special instance of Lawvere s V categories. The nonsymmetric ultrametric for [0; 1] is also described and studied in his paper. The notion of forward Cauchy for a non symmetric metric space is from [Smy87] as well as the notion of limit. A purely ....

....nonsymmetric ultrametric for [0; 1] is also described and studied in his paper. The notion of forward Cauchy for a non symmetric metric space is from [Smy87] as well as the notion of limit. A purely enriched categorical definition of forward Cauchy sequences and of limits can be found in Wagner s [Wag94, Wag95] The notion of finiteness and algebraicity for a generalized ultrametric spaces are from [Rut95] Clearly we are working in the tradition of domain theory and metric spaces, for which Plotkin s [Plo83] and Engelking s [Eng89] have been our respective main source of information. The ....

K.R. Wagner. Solving recursive domain equations with enriched categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994. Technical report CMU-CS-94-159.

....SNW93, WN94] using category theory in 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 ....

K.R. Wagner. Solving Recursive Domain Equations with Enriched Categories. PhD thesis, Carnegie Mellon University, Pittsburgh, June 1994. Draft.

....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 formally similar to ....

....ultrametric for [0; 1] is also described and studied in his paper. The notion of forward Cauchy sequence for a non symmetric metric space is from [Smy87] as well as the notion of limit. A purely enriched categorical definition of forward Cauchy sequences and of limits can be found in Wagner s [Wag94, Wag95] The notion of finiteness and algebraicity for a generalized ultrametric space are from [Rut95] Clearly we are working in the tradition of domain theory, for which Plotkin s [Plo83] has been our main source of information. Completion and topology of non symmetric metric spaces have been ....

K.R. Wagner. Solving recursive domain equations with enriched categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994. Technical report CMU-CS-94-159.

....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 (ordinary) ....

....metric for [0; 1] is also described and studied in his paper. The notion of forward Cauchy sequence for a non symmetric metric space is from [Smy88] as well as the notion of limit. A purely enriched categorical definition of forward Cauchy sequences and of limits can be found in Wagner s [Wag94, Wag95, Rut95] In [Rut95] and [Rut96] the definitions of forward limit and backward limit are shown to be special instances of the enriched categorical notions of weighted limit and weighted colimit. The notion of finiteness and algebraicity for a generalized metric space are from [Rut95] ....

K.R. Wagner. Solving recursive domain equations with enriched categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994. Technical report CMU-CS-94-159.

....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 ....

K.R. Wagner. Solving Recursive Domain Equations with Enriched Categories. PhD thesis, Carnegie Mellon University, 1994.

....and generalize 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 ....

K.R. Wagner, Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, Carnegie Mellon University, Technical Report CMU-CS-94-159, 1994.

....these differences are taken 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 ....

....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 sequence. Wagner use the parameter Omega to ....

Wagner, K. (1994). Solving recursive domain equations with enriched categories. PhD thesis, Carnegie Mellon University, Pittsburgh. Technical report CMU-CS-94-159.

....theory in particular functors to 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 ....

K.R. Wagner. Solving Recursive Domain Equations with Enriched Categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994.

....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 (ordinary) ....

....metric for [0; 1] is also described and studied in his paper. The notion of forward Cauchy sequence for a non symmetric metric space is from [Smy88] as well as the notion of limit. A purely enriched categorical definition of forward Cauchy sequences and of limits can be found in Wagner s [Wag94, Wag95] In [Rut96a] and [Rut96b] the definitions of forward limit and backward limit are shown to be special instances of the enriched categorical notions of weighted limit and weighted colimit. The notions of finiteness and algebraicity for a generalized metric space are from [Rut96a] Clearly ....

K.R. Wagner. Solving recursive domain equations with enriched categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994. Technical report CMU-CS-94-159.

....SNW93, WN94] using category theory in 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 ....

K.R. Wagner. Solving Recursive Domain Equations with Enriched Categories. PhD thesis, Carnegie Mellon University, Pittsburgh, June 1994. Draft.

....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, BBR95, Rut95, FW95, Wag94], which refines ordinary do Supported by the Deutsche Forschungsgemeinschaft. y Contact Author. Department of Mathematics and Statistics, University of Southern Maine, Portland ME 04103, phone: 207) 780 4576, fax: 207) 780 4933 main theory by replacing the qualitative notion of ....

K.R. Wagner. Solving Recursive Domain Equations with Enriched Categories. PhD thesis, Carnegie Mellon University, 1994.

.... recursive domain equations (like our lift algebra equation) can be carried out in the metric setting. Analogous results also have been obtained by Flagg and Kopperman [17] who use the different setting of quantales. Perhaps the most penetrating results so far have been obtained by Wagner [59] who has shown that the domain theoretic and metric space approaches can be understood as instances of a common theme. This theme is to regard the categories CPO and MET as enriched categories. For CPO, the enrichment is over the two point lattice, while for MET, it is over the quantale (R op ; ....

....recursive domain equations within the metric space world. Further work along this line has been done by Flagg and Kopperman [18] as well as Alessi, Baldan, Bell e and Rutten [5] But the most extensive results along the lines of synthesizing domain theory and metric spaces are due to Wagner [59]. Our discussion of power domains focused on presenting them first in their original algebraic formulation, and then from a topological view. There are topological analogues to these constructs, which have evolved from the original Vietoris hyperspaces. These constructs have received new interest ....

Wagner, K., Solving Recursive Domains Equations With Enriched Categories, Ph.D. Thesis, Carnegie-Mellon University (1994).

....5.9 again, we can conclude that f (x) O gS y. By Proposition 5.8, Y (f (x) y) 0. 2 6 Related work In this paper we have presented two topologies for gms s. The main contribution of our paper is the reconciliation of the enriched categorical approach of Lawvere [Law73, Law86] cf. Ken87, Wag94, Wag95] and the topological approach of Smyth [Smy87, Smy91] cf. FK94] The present paper continues the work of Rutten [Rut95] and is part of [BBR95] In the latter paper, besides the topologies presented in this paper also completion and powerdomains for generalized ultrametric spaces are ....

....presented in this paper also completion and powerdomains for generalized ultrametric spaces are defined by means of the Yoneda embedding. The basic definitions and facts on ordered spaces, metric spaces and topology, and gms s are taken from [GHK 80, Plo83] Eng89, Smy92] and [Smy91, Wag94, Rut95] respectively. Smyth [Smy91] and Flagg and Kopperman [FK94] have represented algebraic complete partial orders by another gms than the one given in the introduction. The distance function they use is in general not two valued. In that case, the generalized Alexandroff topology reconciles ....

K.R. Wagner. Solving Recursive Domain Equations with Enriched Categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994.

....as generalized 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 ....

K. R. Wagner. Solving recursive domain equations with enriched categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994. Technical report CMU-CS-94-159.

.... inverse limit theorem, which subsumes both Scott s original proof in the partial order setting (see e.g. Lambek Scott 86] Chapter 18) and America and Rutten s proof in the metric setting ( America Rutten 87] Much of the background to this note is elaborated in the authors PhD thesis, Wagner 94] although the formulation of convergence has been streamlined in this document. The first step in establishing the theory is to use Lawvere s insight ( Lawvere 73] that the notion of an enriched category is a unifying concept for among many other structures, generalized) metric spaces and ....

....from A op Omega A to Omega . Proof : Easy. 2 The adjunction a Omega a a ( in Omega lifts as well, implying that Omega Gamma CAT is a monoidal closed category. Proposition 1. 14 For every Omega category A it holds that A Omega a [A; see e.g. Eilenberg Kelly 66] 2 We remarked in [Wagner 94] that Omega Gamma CAT is Cartesian closed if and only if Omega is a complete Heyting algebra with Omega as meet. This is by no means a new observation. Interestingly for domain theory, the category Omega Gamma CAT also has finite coproducts. Proposition 1.15 The coproduct X Y for two Omega ....

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Wagner, K.R., Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, Carnegie Mellon University, Technical Report CMU-CS-94-159, July 1994.

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K.R. Wagner, Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, Carnegie Mellon University, Technical Report CMU-CS-94-159, 1994.

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K.R. Wagner, Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, Carnegie Mellon University, Technical Report CMU-CS-94-159, 1994.

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K.R. Wagner. Solving recursive domain equations with enriched categories. PhD thesis, Carnegie Mellon University, Pittsburgh, July 1994. Technical report CMUCS -94-159.

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