Asymptotic Behaviors of Type-2 Algorithms and Induced Baire Topologies

Abstract We propose an alternative notion of asymptotic behaviors for the study of type2 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 ...

• Type: Conference
• Year: 2004
• Where: IFIP
• Authors: Chung-Chih Li
• Keywords: Communications, Complexity Theory, IFIPTCS 2004, Notion Explicit Type-2, Type-2 Complexity