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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 28 Nov 2023 17:22:08 +0000Resource-Bounded Category and Measure in Exponential Complexity Classes
https://resolver.caltech.edu/CaltechCSTR:1987.5243-tr-87
Authors: Lutz, Jack H.
Year: 1987
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 curtain two-person
infinite games called resource-bounded Banach-Maxur 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
fl 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 i n ESPACE have high space-bounded Kolmogorov
complexity.https://authors.library.caltech.edu/records/qny92-v6h14