{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/68181"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/68181","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"P-Genericity for Recursively Enumerable Sets","abstract":"U of I Only","abstract_html":"U of I Only","abstract_has_math":false,"creators":["Ingrassia, Michael Anthony"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-14T13:09:49Z","date_published":"2014-12-14T13:09:49Z","updated_at":"2026-07-22T22:25:58Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8114437"],"render_values":[{"text":"(UMI)AAI8114437","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/68181","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Ingrassia, Michael Anthony"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-14T13:09:49Z","10000-01-01","1981"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/68181","(UMI)AAI8114437"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["U of I Only","162 p.","1-generic sets possess all properties which can be obtained through "sufficiently simple" Kleene-Post constructions, but no recursively enumerable (r.e.) set can be 1-generic. P-genericity is the result of a generalization of 1-genericity, and p-generic sets have many of the properties which can be obtained through finite injury priority arguments. Specifically, A is p-generic if for every 1-quantifier property P true of A, P follows from a finite set of true negative information about A and a coinfinite r.e. set of true positive information. Because we allow the set of positive information in the definition to be r.e., we can prove that r.e. p-generic sets exist. We thus may consider the notion of p-genericity only for r.e. sets in the dissertation.","A 1-quantifier property of the form ((FOR ALL)x)P(x,X) is equivalent to an r.e. list of statements of the form C (L-HOOK) X (--->) D (INTERSECT) X (NOT=) (SLASHCIRC). Call it an (m,n) property if the cardinality of C is bounded by m and the cardinality of D bounded by n for statements on the list. Then we may refine the notion of p-genericity by calling A (m,n) p-generic if A is p-generic for the class of (m,n) properties. A is (<(INFIN),n) p-generic if for every m A is (m,n) p-generic. We allow both m and n to take on the values "<(INFIN)" and "(INFIN)". In chapter II we show that (1,n) p-genericity is equivalent to (<(INFIN),n) p-genericity. We also prove the following implications between properties, and show that no arrows can be reversed.","(<(INFIN),0) (<---) . . . (<---) (<(INFIN),n) (<---) (<(INFIN),n+1) (<---) . . . (<---) (<(INFIN),<(INFIN)) (<---) (<(INFIN),(INFIN))","(UPARR) (UPARR) (UPARR) (UPARR) (UPARR)","((INFIN),0) (<---) . . . (<---) ((INFIN),n) (<---) ((INFIN),n+1) (<---) . . . (<---) ((INFIN),<(INFIN)) (<---) ((INFIN),(INFIN))","(p-generic)","For r.e. sets, (<(INFIN),0) p-genericity is the same as simplicity, and ((INFIN),0) p-genericity is the same as hypersimplicity. (1,(INFIN)) p-genericity is a strong form of non-autoreducibility. Hyperhypersimplicity does not fit in a nice way into the above diagram, since p-generic sets need not be hyperhypersimple. All maximal sets are (<(INFIN),1) p-generic, although they need not be ((INFIN),2) p-generic.","((INFIN),<(INFIN)) p-generic sets occur in all nonzero r.e. degrees, but by a result of R. E. Ladner (<(INFIN),(INFIN)) p-generic sets do not occur in all nonzero r.e. degrees. In chapter V we exhibit a complete p-generic set, and in chapter VI we show that the nonzero r.e. degrees containing p-generic sets are dense in the ordering of r.e. degress, but not trivially. The proofs are infinite injury arguments of technical interest.","Made available in DSpace on 2014-12-14T13:09:49Z (GMT). No. of bitstreams: 1 8114437.pdf: 4843276 bytes, checksum: 8468ba6adcbedf378dc5f723c3f309b1 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68359 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["P-Genericity for Recursively Enumerable Sets"]}]}],"canonical_facts":{"dc:creator":["Ingrassia, Michael Anthony"],"dc:date":["2014-12-14T13:09:49Z","10000-01-01","1981"],"dc:description":["U of I Only","162 p.","1-generic sets possess all properties which can be obtained through "sufficiently simple" Kleene-Post constructions, but no recursively enumerable (r.e.) set can be 1-generic. P-genericity is the result of a generalization of 1-genericity, and p-generic sets have many of the properties which can be obtained through finite injury priority arguments. Specifically, A is p-generic if for every 1-quantifier property P true of A, P follows from a finite set of true negative information about A and a coinfinite r.e. set of true positive information. Because we allow the set of positive information in the definition to be r.e., we can prove that r.e. p-generic sets exist. We thus may consider the notion of p-genericity only for r.e. sets in the dissertation.","A 1-quantifier property of the form ((FOR ALL)x)P(x,X) is equivalent to an r.e. list of statements of the form C (L-HOOK) X (--->) D (INTERSECT) X (NOT=) (SLASHCIRC). Call it an (m,n) property if the cardinality of C is bounded by m and the cardinality of D bounded by n for statements on the list. Then we may refine the notion of p-genericity by calling A (m,n) p-generic if A is p-generic for the class of (m,n) properties. A is (<(INFIN),n) p-generic if for every m A is (m,n) p-generic. We allow both m and n to take on the values "<(INFIN)" and "(INFIN)". In chapter II we show that (1,n) p-genericity is equivalent to (<(INFIN),n) p-genericity. We also prove the following implications between properties, and show that no arrows can be reversed.","(<(INFIN),0) (<---) . . . (<---) (<(INFIN),n) (<---) (<(INFIN),n+1) (<---) . . . (<---) (<(INFIN),<(INFIN)) (<---) (<(INFIN),(INFIN))","(UPARR) (UPARR) (UPARR) (UPARR) (UPARR)","((INFIN),0) (<---) . . . (<---) ((INFIN),n) (<---) ((INFIN),n+1) (<---) . . . (<---) ((INFIN),<(INFIN)) (<---) ((INFIN),(INFIN))","(p-generic)","For r.e. sets, (<(INFIN),0) p-genericity is the same as simplicity, and ((INFIN),0) p-genericity is the same as hypersimplicity. (1,(INFIN)) p-genericity is a strong form of non-autoreducibility. Hyperhypersimplicity does not fit in a nice way into the above diagram, since p-generic sets need not be hyperhypersimple. All maximal sets are (<(INFIN),1) p-generic, although they need not be ((INFIN),2) p-generic.","((INFIN),<(INFIN)) p-generic sets occur in all nonzero r.e. degrees, but by a result of R. E. Ladner (<(INFIN),(INFIN)) p-generic sets do not occur in all nonzero r.e. degrees. In chapter V we exhibit a complete p-generic set, and in chapter VI we show that the nonzero r.e. degrees containing p-generic sets are dense in the ordering of r.e. degress, but not trivially. The proofs are infinite injury arguments of technical interest.","Made available in DSpace on 2014-12-14T13:09:49Z (GMT). No. of bitstreams: 1 8114437.pdf: 4843276 bytes, checksum: 8468ba6adcbedf378dc5f723c3f309b1 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68359 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/68181","(UMI)AAI8114437"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["P-Genericity for Recursively Enumerable Sets"],"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":"2026-07-22T22:25:58Z"}