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Sequence spaces and asymmetric norms in the theory of computational complexity. (English) Zbl 1063.68057
Summary: In 1995, M. Schellekens [Proc. MFPS 11, Electronic Notes in Theoretical Computer Science 1, 211–232 (1995; Zbl 0910.68135)] introduced the complexity (quasi-metric) space as a part of the development of a topological foundation for the complexity analysis of algorithms. Recently, S. Romaguera and M. Schellekens [Topology Appl. 98, 311–322 (1999; Zbl 0941.54028)] have obtained several quasi-metric properties of the complexity space which are interesting from a computational point of view, via the analysis of the so-called dual complexity space.
Here, we extend the notion of the dual complexity space to the $$p$$-dual case, with $$p > 1$$, in order to include some other kinds of exponential time algorithms in this study. We show that the dual $$p$$-complexity space is isometrically isomorphic to the positive cone of $$l_p$$ endowed with the asymmetric norm $$\|\cdot\|_{+p}$$ given on $$l_p$$ by $$\| \mathbf x\|_{+p} = [{\Sigma}_{n=0}^{\infty}((x_n \vee 0)^p)]^{1/p}$$, where $$\mathbf x := (x_n)_{n\in {\omega}}$$. We also obtain some results on completeness and compactness of these spaces.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 46A45 Sequence spaces (including Köthe sequence spaces) 54E15 Uniform structures and generalizations 54C35 Function spaces in general topology
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##### References:
 [1] Fletcher, P.; Lindgren, W.F., Quasi-uniform spaces, (1982), Marcel Dekker New York · Zbl 0402.54024 [2] Künzi, H.P.A., Nonsymmetric topology, (), 303-338 · Zbl 0888.54029 [3] Schellekens, M., The smyth completion: A common foundation for denonational semantics and complexity analysis, (), 211-232 [4] Romaguera, S.; Schellekens, M., Quasi-metric properties of complexity spaces, Topology appl., 98, 311-322, (1999) · Zbl 0941.54028 [5] Romaguera, S.; Schellekens, M., Duality and quasi-normability for complexity spaces, Appl. gen. topology, 3, 91-112, (2002) · Zbl 1022.54018 [6] Aho, A.V.; Hopcroft, J.E.; Ullman, J.D., Data structures and algorithms, (1983), Addison-Wesley Boston, MA · Zbl 0307.68053 [7] Romaguera, S.; Schellekens, M., Cauchy filters and strong completeness of quasi-uniform spaces, Rostock. math. kolloq., 54, 69-79, (2000) · Zbl 0961.54022 [8] Ferrer, J.; Gregori, V.; Alegre, C., Quasi-uniform structures in linear lattices, Rocky mountain J. math., 23, 877-884, (1993) · Zbl 0803.46007 [9] Romaguera, S.; Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. approx. theory, 103, 292-301, (2000) · Zbl 0980.41029 [10] Keimel, K.; Roth, W., Ordered cones and approximation, (1992), Springer-Verlag Berlin · Zbl 0752.41033 [11] Romaguera, S.; Schellekens, M., On the structure of the dual complexity space: the general case, Extracta math., 13, 249-253, (1998) · Zbl 1006.54039 [12] Matthews, S.G., Partial metric topology, (), 183-197 · Zbl 0911.54025 [13] Smyth, M.B., Quasi-uniformities: reconciling domains with metric spaces, (), 236-253 · Zbl 0668.54018 [14] Smyth, M.B., (), 207-229 [15] Künzi, H.P.A.; Schellekens, M., The yoneda-completion of a quasi-metric space, Theoretical comp. sci., 278, 159-194, (2002) · Zbl 1025.54014 [16] Schellekens, M., (), 337-348 [17] Reilly, I.L.; Subhramanyam, P.V.; Vamanamurthy, M.K., Cauchy sequences in quasi-pseudo-metric spaces, Monatsh. math., 93, 127-140, (1982) · Zbl 0472.54018 [18] Grothendieck, A., Critères de compacité dans LES espaces fonctionnels généraux, Amer. math. J., 74, 168-186, (1952) · Zbl 0046.11702 [19] Asanov, M.O.; Velichko, N.V., Compact sets in C_{p}(X), Comm. math. univ. carolinae, 22, 255-266, (1981) · Zbl 0491.54011 [20] Romaguera, S.; Schellekens, M., The quasi-metric of complexity convergence, Quaestiones math., 23, 359-374, (2000) · Zbl 0965.54028
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