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Resource-bounded category and measure in exponential complexity classes

Citation

Lutz, Jack H. (1987) Resource-bounded category and measure in exponential complexity classes. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:03192012-101000956

Abstract

This thesis presents resource-bounded category and resource bounded measure - two new tools for computational complexity theory - and some applications of these tools to the structure theory of exponential complexity classes.

Resource-bounded category, a complexity-theoretic version of the classical Baire category method, identifies certain subsets of PSPACE, E, ESPACE, and other complexity classes as meager. These meager sets are shown to form a nontrivial ideal of "small" subsets of the complexity class. The meager sets are also (almost) characterized in terms of certain two-person infinite games called resource-bounded Banach-Mazur games.

Similarly, resource-bounded measure, a complexity-theoretic version of Lebesgue measure theory, identifies the measure 0 subsets of E, ESPACE, and other complexity classes, and these too are shown to form nontrivial ideals of "small" subsets. A resource-bounded extension of the classical Kolmogorov zero-one law is also proven. This shows that measurable sets of complexity-theoretic interest either have measure 0 or are the complements of sets of measure 0.

Resource-bounded category and measure are then applied to the investigation of uniform versus nonuniform complexity. In particular, Kannan's theorem that ESPACE P/Poly is extended by showing that P/ Poly ∩ ESPACE is only a meager, measure 0 subset of ESPACE. A theorem of Huynh is extended similarly by showing that all but a meager, measure 0 subset of the languages in ESPACE have high space-bounded Kolmogorov complexity.

These tools are also combined with a new hierarchy of exponential time complexity classes to refine known relationships between nonuniform complexity and time complexity.

In the last part of the thesis, known properties of hard languages are extended. In particular, recent results of Schoning and Huynh state that any language L which is ≤^P_m -hard for E or ≤^P_T -hard for ESPACE cannot be feasibly approximated (i.e., its symmetric difference with any feasible language has exponential density). It is proven here that this conclusion in fact holds unless only a meager subset of E is ≤^P_m -reducible to L and only a meager, measure 0 subset of ESPACE is ≤^(PSPACE)_m -reducible to L. (It is conjectured, but not proven, that this result is actually stronger than those of Schoning and Huynh.) This suggests a new lower bound method which may be useful in interesting cases.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Kechris, Alexander S. (advisor)
  • Abu-Mostafa, Yaser S. (advisor)
Thesis Committee:
  • Levin, Leonid
  • Luxemburg, W. A. J.
  • Fuller, Brock
Defense Date:18 May 1987
Record Number:CaltechTHESIS:03192012-101000956
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:03192012-101000956
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6854
Collection:CaltechTHESIS
Deposited By: Benjamin Perez
Deposited On:19 Mar 2012 18:50
Last Modified:26 Dec 2012 04:40

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