The complexity space of a valued linearly ordered set

By a valued linearly ordered set (a VLOS for short), we mean a pair (X, ϕ )s uch that X is a linearly ordered set and ϕ is a strictly increasing (= positive monotone) nonnegative real valued function. Clearly, any VLOS is a valuation space. Each VLOS (X, ϕ) generates a linear weightable quasi-metric dϕ on X whose conjugate is order preserving. We show that the Smyth completion of (X, dϕ )a lso admits the structure of a VLOS. On the other hand, M. Schellekens introduced in 1995, the theory of complexity spaces to develop a topological foundation for the complexity analysis of programs. Here, we introduce the so-called complexity space of a VLOS (X, ϕ) and discuss some of its properties. In particular, we show that it is weightable and preserves Smyth completeness of (X, dϕ). We apply this complexity approach to the measurement of real numbers and discuss some advantages of our methods with respect to those that use the classical Baire metric.

[1]  Michel P. Schellekens,et al.  The Smyth completion: a common foundation for denotational semantics and complexity analysis , 1995, MFPS.

[2]  Fletcher.  Quasi-Uniform Spaces , 1982 .

[3]  Salvador Romaguera,et al.  The quasi-metric of complexity convergence , 2000 .

[4]  S. G. Matthews,et al.  Partial Metric Topology , 1994 .

[5]  Robert Lowen,et al.  Handbook of the History of General Topology , 1997 .

[6]  E. A. Sánchez-Pérez,et al.  Bicompleting weightable quasi-metric spaces and partial metric spaces , 2002 .

[7]  Michael B. Smyth,et al.  Totally bounded spaces and compact ordered spaces as domains of computation , 1991 .

[8]  P. V. Subrahmanyam,et al.  Cauchy sequences in quasi-pseudo-metric spaces , 1982 .

[9]  Salvador Romaguera,et al.  Quasi-metric properties of complexity spaces , 1999 .

[10]  Michel Schellekens.  On Upper Weightable Spaces , 1996 .

[11]  M. J. Saegrove.  On bitopological spaces , 1971 .

[12]  Salvador Romaguera,et al.  Weightable quasi-metric semigroups and semilattices , 2001, MFCSIT.

[13]  Anna Di Concilio.  Spazi quasimetrici e topologie ad essi associate , 1971 .

[14]  Michael B. Smyth,et al.  Quasi Uniformities: Reconciling Domains with Metric Spaces , 1987, MFPS.

[15]  Hans-Peter A. Künzi,et al.  Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology , 2001 .

[16]  A. W. Roscoe,et al.  Topology and category theory in computer science , 1991 .