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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 · +p given on l p by 𝐱 +p =[Σ n=0 ((x n 0) p )] 1/p , where 𝐱:=(x n ) nω . We also obtain some results on completeness and compactness of these spaces.

68Q25Analysis of algorithms and problem complexity
46A45Sequence spaces
54E15Uniform structures and generalizations
54C35Function spaces (general topology)