Complexity spaces as quantitative domains of computation

Abstract We study domain theoretic properties of complexity spaces. Although the so-called complexity space is not a domain for the usual pointwise order, we show that, however, each pointed complexity space is an ω-continuous domain for which the complexity quasi-metric induces the Scott topology, and the supremum metric induces the Lawson topology. Hence, each pointed complexity space is both a quantifiable domain in the sense of M. Schellekens and a quantitative domain in the sense of P. Waszkiewicz, via the partial metric induced by the complexity quasi-metric.

[1]  Keye Martin,et al.  The Measurement Process in Domain Theory , 2000, ICALP.

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

[3]  Pawel Waszkiewicz,et al.  Quantitative Continuous Domains , 2003, Appl. Categorical Struct..

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

[5]  Óscar Valero,et al.  An Application of Generalized Complexity Spaces to Denotational Semantics via the Domain of Words , 2009, LATA.

[6]  Neil D. Jones,et al.  Computability and complexity - from a programming perspective , 1997, Foundations of computing series.

[7]  Pawel Waszkiewicz.  Partial metrisability of continuous posets , 2006, Math. Struct. Comput. Sci..

[8]  Salvador Romaguera,et al.  Duality and quasi-normability for complexity spaces , 2002 .

[9]  Abbas Edalat,et al.  A Computational Model for Metric Spaces , 1998, Theor. Comput. Sci..

[10]  Salvador Romaguera,et al.  Sequence spaces and asymmetric norms in the theory of computational complexity , 2002 .

[11]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[12]  Michael B. Smyth,et al.  The constructive maximal point space and partial metrizability , 2006, Ann. Pure Appl. Log..

[13]  Michel P. Schellekens,et al.  A characterization of partial metrizability: domains are quantifiable , 2003, Theor. Comput. Sci..

[14]  Simon John O'Neill.  A fundamental study into the theory and application of the partial metric spaces , 1998 .

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

[16]  Óscar Valero,et al.  The complexity space of partial functions: a connection between complexity analysis and denotational semantics , 2011, Int. J. Comput. Math..

[17]  L. M. García-Raffi,et al.  Applications of the complexity space to the General Probabilistic Divide and Conquer Algorithms , 2008 .

[18]  Michel P. Schellekens,et al.  The correspondence between partial metrics and semivaluations , 2004, Theor. Comput. Sci..

[19]  Salvador Romaguera,et al.  On the structure of the space of complexity partial functions , 2008, Int. J. Comput. Math..

[20]  Óscar Valero,et al.  A quantitative computational model for complete partial metric spaces via formal balls† , 2009, Mathematical Structures in Computer Science.

[21]  Reinhold Heckmann,et al.  Approximation of Metric Spaces by Partial Metric Spaces , 1999, Appl. Categorical Struct..