Citation
Lutz, Jack H. (1987) Resourcebounded category and measure in exponential complexity classes. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:03192012101000956
Abstract
This thesis presents resourcebounded 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.
Resourcebounded category, a complexitytheoretic 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 twoperson infinite games called resourcebounded BanachMazur games.
Similarly, resourcebounded measure, a complexitytheoretic 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 resourcebounded extension of the classical Kolmogorov zeroone law is also proven. This shows that measurable sets of complexitytheoretic interest either have measure 0 or are the complements of sets of measure 0.
Resourcebounded 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 spacebounded 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): 

Thesis Committee: 

Defense Date:  18 May 1987 
Record Number:  CaltechTHESIS:03192012101000956 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:03192012101000956 
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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