Abstract
We propose an alternative notion of asymptotic behaviors for the study of type-2 computational complexity. Since the classical asymptotic notion (for all but finitely many) is not acceptable in type-2 context, we alter the notion of “small sets” from “finiteness” to topological “compactness” for type-2 complexity theory. A natural reference for type-2 computations is the standard Baire topology. However, we point out some serious drawbacks of this and introduce an alternative topology for describing compact sets. Following our notion explicit type-2 complexity classes can be defined in terms of resource bounds. We show that such complexity classes are recursively representable; namely, every complexity class has a programming system. We also prove type-2 analogs of Rabin’s Theorem, Recursive Relatedness Theorem, and Gap Theorem to provide evidence that our notion of type-2 asymptotic is workable. We speculate that our investigation will give rise to a possible approach in examining the complexity structure at type-2 along the line of the classical complexity theory.
A full version with detailed proofs of the theorems in this paper is available at http://hal.lamar.edu/∼licc/T2Asy/Full.T2AsyTCS2004.pdf
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Li, CC. (2004). Asymptotic Behaviors of Type-2 Algorithms and Induced Baire Topologies. In: Levy, JJ., Mayr, E.W., Mitchell, J.C. (eds) Exploring New Frontiers of Theoretical Informatics. IFIP International Federation for Information Processing, vol 155. Springer, Boston, MA. https://doi.org/10.1007/1-4020-8141-3_36
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DOI: https://doi.org/10.1007/1-4020-8141-3_36
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