Towards computing distances between programs via Scott domains
This paper introduces an approach to defining and computing distances between programs via continuous generalized distance functions ρ: A×A→D, where A and D are directed complete partial orders with the induced Scott topology, A is a semantic domain, and D is a domain representing distances (usually, some version of interval numbers). A continuous distance function ρ can define a To topology on a nontrivial domain A only if the axiom ∃0 ε D.∀x ε A.ρ(x,x)=0 does not hold. Hence, the notion of relaxed metric is introduced for domains — the axiom ρ(x,x)=0 is eliminated, but the axiom ρ(x,y)=ρ(y,x) and a version of the triangle inequality tailored for the domain D remain.
The paper constructs continuous relaxed metrics yielding the Scott topology for all continuous Scott domains with countable bases. This construction is closely related to partial metrics of Matthews and valuation spaces of O'Neill, but it describes a wider class of domains in a more intuitive way from the computational point of view.
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- 1.Bukatin M.A., Scott J.S. Towards Computing Distances between Programs via Domains: a Symmetric Continuous Generalized Metric for Scott Topology on Continuous Scott Domains with Countable Bases. Available via URL http://www.cs.brandeis.edu/∼bukatin/dist-new.ps.gz, December 1996.Google Scholar
- 2.Edalat A. Domain theory and integration. Theoretical Computer Science, 151 (1995) 163–193.Google Scholar
- 3.Hoofman R. Continuous information systems. Information and Computation, 105 (1993) 42–71.Google Scholar
- 4.Kopperman R.D., Flagg R.C. The asymmetric topology of computer science. In S. Brooks et al., eds., Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science, 802, 544–553, Springer, 1993.Google Scholar
- 5.Kunzi H.P.A., Vajner V. Weighted quasi-metrics. In S. Andima et al., eds., Proc. 8th Summer Conference on General Topology and Applications, Annals of the New York Academy of Sciences, 728, 64–77, New York, 1994.Google Scholar
- 6.Matthews S.G. An extensional treatment of lazy data flow deadlock. Theoretical Computer Science, 151 (1995), 195–205.Google Scholar
- 7.Matthews S.G. Partial metric topology. In S. Andima et al., eds., Proc. 8th Summer Conference on General Topology and Applications, Annals of the New York Academy of Sciences, 728, 183–197, New York, 1994.Google Scholar
- 8.O'Neill S.J. Partial metrics, valuations and domain theory. In S. Andima et al., eds., Proc. 11th Summer Conference on General Topology and Applications, Annals of the New York Academy of Sciences, 806, 304–315, New York, 1997.Google Scholar
- 9.Smyth M.B. Quasi-uniformities: reconciling domains and metric spaces. In M. Main et al., eds., Mathematical Foundations of Programming Language Semantics, Lecture Notes in Computer Science, 298, 236–253, Springer, 1988.Google Scholar
- 10.Stoy J.E. Denotational Semantics: The Scott-Strachey Approach to Programming Language Semantics. MIT Press, Cambridge, Massachusetts, 1977.Google Scholar
- 11.Vickers S. Matthews Metrics. Unpublished notes, Imperial College, UK, 1987.Google Scholar