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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 28 Nov 2023 18:47:56 +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-v6h14The tile assembly model is intrinsically universal
https://resolver.caltech.edu/CaltechAUTHORS:20130529-100843467
Authors: Doty, David; Lutz, Jack H.; Patitz, Matthew J.; Schweller, Robert T.; Summers, Scott M.; Woods, Damien
Year: 2012
DOI: 10.1109/FOCS.2012.76
We prove that the abstract Tile Assembly Model (aTAM) of nanoscale self-assembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is "intrinsic" in the sense that the self-assembly process carried out by U is exactly that carried out by T, with each tile of T represented by an m × m "super tile" of U. Our construction works for the full aTAM at any temperature, and it faithfully simulates the deterministic or nondeterministic behavior of each T. Our construction succeeds by solving an analog of the cell differentiation problem in developmental biology: Each super tile of U, starting with those in the seed assembly, carries the "genome" of the simulated system T. At each location of a potential super tile in the self-assembly of U, a decision is made whether and how to express this genome, i.e., whether to generate a super tile and, if so, which tile of T it will represent. This decision must be achieved using asynchronous communication under incomplete information, but it achieves the correct global outcome(s).https://authors.library.caltech.edu/records/4mhkx-r2152