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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.

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
Full Text: DOI
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