{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20189"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20189","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Uniformity and bounded arithmetic below P","abstract":"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","abstract_html":"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","abstract_has_math":false,"creators":["Parra, Carlos Mario"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Jockusch, Carl G., Jr."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:31:41Z","date_published":"2011-05-07T12:31:41Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1996 Parra, Carlos Mario"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591198669","AAI9712398","(UMI)AAI9712398"],"render_values":[{"text":"9780591198669","href":null,"code":true},{"text":"AAI9712398","href":null,"code":true},{"text":"(UMI)AAI9712398","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20189","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Jockusch, Carl G., Jr."]},{"key":"dc:creator","label":"Author","values":["Parra, Carlos Mario"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:31:41Z","10000-01-01","1996"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1996 Parra, Carlos Mario"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591198669","AAI9712398","(UMI)AAI9712398","http://hdl.handle.net/2142/20189"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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","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"]},{"key":"dc:title","label":"Title","values":["Uniformity and bounded arithmetic below P"]}]}],"canonical_facts":{"dc:contributor":["Jockusch, Carl G., Jr."],"dc:creator":["Parra, Carlos Mario"],"dc:date":["2011-05-07T12:31:41Z","10000-01-01","1996"],"dc:description":["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","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"],"dc:identifier":["9780591198669","AAI9712398","(UMI)AAI9712398","http://hdl.handle.net/2142/20189"],"dc:language":["eng"],"dc:rights":["Copyright 1996 Parra, Carlos Mario"],"dc:subject":["Mathematics"],"dc:title":["Uniformity and bounded arithmetic below P"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}