Topology via logic
Semantics of programming languages: structures and techniques
Semantics of programming languages: structures and techniques
Algebraic posets, algebraic cpo's and models of concurrency
Topology and category theory in computer science
Totally bounded spaces and compact ordered spaces as domains of computation
Topology and category theory in computer science
Handbook of logic in computer science (vol. 1)
The Smythe-completion of a quasi-uniform space
Semantics of programming languages and model theory
Semi-metrics, closure spaces and digital topology
Selected papers of the workshop on Topology and completion in semantics
Continuity spaces: reconciling domains and metric spaces
MFPS '94 Proceedings of the tenth conference on Mathematical foundations of programming semantics
Generalized metric spaces: completion, topology, and power domains via the Yoneda embedding
Theoretical Computer Science
Quasi Uniformities: Reconciling Domains with Metric Spaces
Proceedings of the 3rd Workshop on Mathematical Foundations of Programming Language Semantics
Semantics and axiomatics of a simple recursive language.
STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
A characterization of partial metrizability: domains are quantifiable
Theoretical Computer Science - Topology in computer science
Fundamental study: Complete and directed complete Ω-categories
Theoretical Computer Science
Static space–times naturally lead to quasi-pseudometrics
Theoretical Computer Science
The space of formal balls and models of quasi-metric spaces
Mathematical Structures in Computer Science
A quantitative computational model for complete partial metric spaces via formal balls†
Mathematical Structures in Computer Science
Enriched Categories and Quasi-uniform Spaces
Electronic Notes in Theoretical Computer Science (ENTCS)
Simulation hemi-metrics between infinite-state stochastic games
FOSSACS'08/ETAPS'08 Proceedings of the Theory and practice of software, 11th international conference on Foundations of software science and computational structures
Domain theoretic characterisations of quasi-metric completeness in terms of formal balls†
Mathematical Structures in Computer Science
The bicompletion of fuzzy quasi-metric spaces
Fuzzy Sets and Systems
The formal ball model for -categories
Mathematical Structures in Computer Science
Sequence spaces and asymmetric norms in the theory of computational complexity
Mathematical and Computer Modelling: An International Journal
ICFCA'12 Proceedings of the 10th international conference on Formal Concept Analysis
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Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. Flagg and Kopperman, Theoret. Comput. Sci. 177 (1) (1997) 111-138; Bonsangue et al., Theoret. Comput. Sci. 193 (1998) 1-51; Symth, Quasi-Uniformities: Reconciling Domains with Metric Spaces, Lectures Notes in Computer Science, vol. 298, Springer, Berlin, 1987, pp. 236-253; Wagner, Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, July 1994, Technical Report CMU-CS-94-159). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of (Bonsangue et al., Theoret. Comput. Sci 193 (1998) 1-51) which finds its roots in work by Lawvere (Lawvere, Rend. Sem. Mat. Fis. Milano 43 (1973) 135-166; cf. also Wagner, Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, July 1994, Technical Report CMU-CS-94-159) and which is related to early work by Stoltenberg (e.g. Stoltenberg, Proc. London. Math. Soc. (3) 17 (1967) 226-240; Stoltenberg, Proc. Amer. Math. Soc. 18 (1967) 864-867 and Ferrer and Gregori, Proc. London Math. Soc. (3) 49 (1984) 36), and the Smyth completion (Smyth, Quasi-Uniformities: Reconciling Domains with Metric Spaces, Lecture Notes in Computer Science, vol. 298, Springer, Berlin, 1987, pp. 236-253; Smyth, In: G.M. Reed, A.W. Roscoe, R.F. Wachter (Eds.), Topology and Category Theory in Computer Science, Oxford University Press, Oxford, 1991, pp. 207-229; Smyth, J. London Math. Soc. 49 (1994) 385-400; Snderhauf, In: M. Droste, Y. Gurevich (Eds.), Semantics of Programming Languages and Model Theory, Algebra, Logic and Applications, vol. 5, Gordon and Breach, London, 1993, pp. 189-212; Snderhauf, Acta Math. Hungar. 69 (1995) 47-54). A net-version of the Yoneda completion, complementing the net-version of the Smyth completion (Snderhauf, Acta. Math. Hungar. 69 (1995) 47-54), is given and a comparison between the two types of completion is presented. The following open question is raised in Bonsangue et al. (Theoret. Comput. Sci. 193 (1998) 1-51): "An interesting question is to characterize the family of generalized metric spaces for which [the Yoneda] completion is idempotent (it contains at least all ordinary metric spaces)". We show that the largest class of quasi-metric spaces idempotent under the Yoneda completion is precisely the class of Smyth-completabe spaces. A similar result has been obtained independently by Flagg and Snderhauf in Flagg and Snderhauf (preprint, available at: ftp://theory.doc.ic.ac.uk/theory/papers/Sunderhauf/eicqf.ps). 2 We present an entirely new proof of the result via concrete standard techniques and compare this approach with the more abstract categorical machinery of Flagg and Snderhauf (preprint, available at: ftp://theory.doc.ic.ac.uk/theory/papers/Sunderhauf/eicqf.ps). Our proof is based on a new characterization of Smyth-completability of quasi-metric spaces in terms of sequences, which considerably simplifies prior characterizations for quasi-uniform spaces (e.g. Snderhauf, In: M. Droste, Y. Gurevich (Eds.), Semantics of Programming Languages and Model Theory, Algebra, Logic and Applications, vol. 5, Gordon and Breach, London, 1993, pp. 189-212; Sunderhauf, Acta Math. Hungar. 69 (1995) 715-720) We also show that the ideal completion, and hence the Yoneda completion and the Smyth completion, are not sequentially adequate in general. The study of the properties of total boundedness, precompactness, hereditary precompactness and compactness is motivated and we analyze the preservation of these properties under the two kinds of completion in the possible absence of idempotency.