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 Schöning and Huynh state that any language L which is ≤Pm-hard for E or ≤PT-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 ≤Pm-reducible to L and only a meager, measure 0 subset of ESPACE is ≤PSPACEm-reducible to L. (It is conjectured, but not proven, that this result is actually stronger than those of Schöning and Huynh.) This suggests a new lower bound method which may be useful in interesting cases.
