{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/68190"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/68190","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Groups and Simple Languages","abstract":"With any finitely generated presentation (pi) = of a group G, one can associate a formal language WP(,0)((pi)), the reduced word problem of (pi), consisting of all words on the generators and their inverses which are equal to the identity of G but have no proper prefix equal to the identity. A general problem, then, is to determine what the nature of the reduced word problem implies about the structure of the group G, or, conversely, how the properties of G affect WP(,0)((pi)).","abstract_html":"With any finitely generated presentation (pi) = of a group G, one can associate a formal language WP(,0)((pi)), the reduced word problem of (pi), consisting of all words on the generators and their inverses which are equal to the identity of G but have no proper prefix equal to the identity. A general problem, then, is to determine what the nature of the reduced word problem implies about the structure of the group G, or, conversely, how the properties of G affect WP(,0)((pi)).","abstract_has_math":false,"creators":["Haring-Smith, Robert Henry"],"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":"2014-12-14T13:09:52Z","date_published":"2014-12-14T13:09:52Z","updated_at":"2026-07-22T22:25:58Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8203480"],"render_values":[{"text":"(UMI)AAI8203480","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/68190","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Haring-Smith, Robert Henry"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-14T13:09:52Z","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/2142/68190","(UMI)AAI8203480"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["With any finitely generated presentation (pi) = of a group G, one can associate a formal language WP(,0)((pi)), the reduced word problem of (pi), consisting of all words on the generators and their inverses which are equal to the identity of G but have no proper prefix equal to the identity. A general problem, then, is to determine what the nature of the reduced word problem implies about the structure of the group G, or, conversely, how the properties of G affect WP(,0)((pi)).","The simple languages form a class of prefix-free languages properly contained in the class of context-free languages. Simple languages can be accepted by one-state deterministic pushdown automata which read an input symbol at every step in a computation. The main results of the thesis are:","Theorem. A finitely generated presentation (pi) of a group G has a simple reduced word problem if and only if there are only a finite number of simple closed paths passing through each vertex in the Cayley diagram of (pi).","Theorem. A group G has a presentation with a simple reduced word problem if and only if G is the free product of a finitely generated free group and a finite number of finite groups.","Made available in DSpace on 2014-12-14T13:09:52Z (GMT). No. of bitstreams: 1 8203480.pdf: 2129536 bytes, checksum: 470dcb09d22e97b91c16b509fb2ca804 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68368 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","U of I Only","77 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["Groups and Simple Languages"]}]}],"canonical_facts":{"dc:creator":["Haring-Smith, Robert Henry"],"dc:date":["2014-12-14T13:09:52Z","10000-01-01","1981"],"dc:description":["With any finitely generated presentation (pi) = of a group G, one can associate a formal language WP(,0)((pi)), the reduced word problem of (pi), consisting of all words on the generators and their inverses which are equal to the identity of G but have no proper prefix equal to the identity. A general problem, then, is to determine what the nature of the reduced word problem implies about the structure of the group G, or, conversely, how the properties of G affect WP(,0)((pi)).","The simple languages form a class of prefix-free languages properly contained in the class of context-free languages. Simple languages can be accepted by one-state deterministic pushdown automata which read an input symbol at every step in a computation. The main results of the thesis are:","Theorem. A finitely generated presentation (pi) of a group G has a simple reduced word problem if and only if there are only a finite number of simple closed paths passing through each vertex in the Cayley diagram of (pi).","Theorem. A group G has a presentation with a simple reduced word problem if and only if G is the free product of a finitely generated free group and a finite number of finite groups.","Made available in DSpace on 2014-12-14T13:09:52Z (GMT). No. of bitstreams: 1 8203480.pdf: 2129536 bytes, checksum: 470dcb09d22e97b91c16b509fb2ca804 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68368 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","U of I Only","77 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/68190","(UMI)AAI8203480"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Groups and Simple Languages"],"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"}