2026-09-21T13:49:52Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/20189 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Made available in DSpace on 2011-05-07T12:31:41Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9712398.pdf: 3856594 bytes, checksum: 9160445c25b2f9a33682363c6312eef0 (MD5) Previous issue date: 1996 Jockusch, Carl G., Jr. Parra, Carlos Mario 2011-05-07T12:31:41Z 2011-05-07T12:31:41Z 10000-01-01 1996 We study some of the complexity classes below P and, in particular, we concentrate on $AC\sp0\subseteq NC\sp1\subseteq L=$ LogSpace. We also study the nondeterministic classes $NAC\sp{i}$ and $NNC\sp{i},$ for $i\ge0,$ which are the counterparts to the more familiar class NP. In the final part of this work we characterize the so-called Steven's Class $SC=\bigcup\sb{i\ge1}Sc\sp{i}.$ "We start by proving that certain basic arithmetic operations such as Count, Multiplication, Multiple Addition, Sorting, etc. can be carried out in Uniform-$NC\sp1$ and that similar results hold in the class Uniform-$AC\sp0$ when dealing with sufficiently ""small"" numbers. The proofs are carried out by using algebraic characterizations of the previous classes as developed, for example, in (C14) and (CT2)." We continue with the classes $NAC\sp0\subseteq NNC\sp1\subseteq\cdots\subseteq NP$ introduced in (Ta2) and prove that, in fact, all of them coincide and therefore are equal to NPolyTime. Finally, we move further up and consider the complexity class SC as defined in (Co2). We introduce the notion of Extended k-Bounded Recursion on Notation $(E\sb{k}BRN)$ and prove that the class $SC\sp{k}$ equals the closure of the set of basic functions INITIAL (see for example, (CT2)), under composition, CRN and $E\sb{k}BRN.$ Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:42:13Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:18:20-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only 9780591198669 AAI9712398 (UMI)AAI9712398 http://hdl.handle.net/2142/20189 eng Copyright 1996 Parra, Carlos Mario Mathematics Uniformity and bounded arithmetic below P text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.