{"id":360848,"date":"2017-02-02T09:54:56","date_gmt":"2017-02-02T17:54:56","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=360848"},"modified":"2018-10-16T20:06:16","modified_gmt":"2018-10-17T03:06:16","slug":"word-problem-classes-semigroups","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/word-problem-classes-semigroups\/","title":{"rendered":"The Word Problem for Some Classes of Semigroups"},"content":{"rendered":"<p>The word problem for finite semigroups is the following decision problem: given some number n of word pairs (u<sub>1<\/sub>,v<sub>1<\/sub>), &#8230;, (u<sub>n<\/sub>,v<sub>n<\/sub>) and an additional word pair (u<sub>0<\/sub>,v<sub>0<\/sub>), decide whether the n equations u<sub>1<\/sub>=v<sub>1<\/sub>,&#8230;, u<sub>n<\/sub>=v<sub>n<\/sub> impliy the additional equation u<sub>0<\/sub>=v<sub>0<\/sub> in all finte semigroups. We prove that the word problem for finite semigroups is undecidable.<\/p>\n<p>In fact, the undecidability result holds for a particular premise \u00a0\u00a0E \u00a0 =\u00a0 (u<sub>1<\/sub>=v<sub>1<\/sub> and &#8230; and u<sub>n<\/sub>=v<sub>n<\/sub>). Furthermore, this particular E can be chosen so that the following classes K1 and K2 of word pairs are recursively inseparable:<br \/>\nK1 is the class of word pairs (u<sub>0<\/sub>,v<sub>0<\/sub>) such that E implies u<sub>0<\/sub>=v<sub>0<\/sub> in every periodic semigroup.<br \/>\nK2 is the class of word pairs (u<sub>0<\/sub>,v<sub>0<\/sub>) such that E fails to imply u<sub>0<\/sub>=v<sub>0<\/sub> in some finite semigroup.<br \/>\nThe paper contains some additional undecidability results.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The word problem for finite semigroups is the following decision problem: given some number n of word pairs (u1,v1), &#8230;, (un,vn) and an additional word pair (u0,v0), decide whether the n equations u1=v1,&#8230;, un=vn impliy the additional equation u0=v0 in all finte semigroups. We prove that the word problem for finite semigroups is undecidable. In [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Algebra and Logic (Russian)","msr_number":"","msr_organization":"","msr_pages_string":"25-35","msr_page_range_start":"25","msr_page_range_end":"35","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"1966-07-08","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/08\/08.htm","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13560],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-360848","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-programming-languages-software-engineering","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"1966-07-08","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"25-35","msr_chapter":"","msr_isbn":"","msr_journal":"Algebra and Logic (Russian)","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/08\/08.htm","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/08\/08.htm","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/08\/08.htm"}],"msr-author-ordering":[{"type":"user_nicename","value":"gurevich","user_id":31929,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=gurevich"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"article","related_content":[],"_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/360848","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":1,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/360848\/revisions"}],"predecessor-version":[{"id":521927,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/360848\/revisions\/521927"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=360848"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=360848"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=360848"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=360848"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=360848"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=360848"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=360848"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=360848"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=360848"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=360848"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=360848"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=360848"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=360848"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}