{"id":360824,"date":"2017-02-02T09:43:57","date_gmt":"2017-02-02T17:43:57","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=360824"},"modified":"2018-10-16T20:06:08","modified_gmt":"2018-10-17T03:06:08","slug":"decision-problem-logic-predicates-operations","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/decision-problem-logic-predicates-operations\/","title":{"rendered":"The Decision Problem for Logic of Predicates and Operations"},"content":{"rendered":"<p>The article consists of two chapters. In the first part of the first chapter, the author rediscovers well-partial-orderings and well-quasi-orderings, which he calls tight partial orders and tight quasi orders, and develops a theory of such orderings. (In this connection, it may be appropriate to point out Joseph B. Kruskal&#8217;s article &#8220;The theory of well-quasi-ordering: A frequently discoverred concept&#8221; in J. Comb. Theory A, vol. 13 (1972), 297-305.) To understand the idea behind the term &#8220;tight&#8221;, think of a boot: you cannot move your foot far down or sidewise &#8212; only up. This is similar to tight partial orders where infinite sequences have no infinite descending subsequences, no infinite antichains, but always have infinite ascending subsequences.<\/p>\n<p>In the second part of the first chapter, the author applies the theory of tight orders to prove a classifiability theorem for prefix-vocabulary classes of first-order logic. The main part of the classifiability theorem is that the partial order of prefix-vocabulary classes (ordered by inclusion) is tight. But there is an additional useful part of the classifiability theorem, about the form of the minimal classes outside a downward closed collection, e.g. the minimal classes that are undecidable in one way or another.<\/p>\n<p>In the second chapter, the author completes the decision problem for (the prefix-vocabulary fragments of) pure logic of predicates and functions, though the treatment of the most difficult decidable class is deferred to <a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"http:\/\/web.eecs.umich.edu\/~gurevich\/annotated.htm#18\">18<span class=\"sr-only\"> (opens in new tab)<\/span><\/a>. In particular, the classes [\u2200<sup>2<\/sup>,(0,1),(1)] and [\u2200<sup>2<\/sup>,(1),(0,1)] are proved to be conservative reduction classes. (This abstract is written in January 2006.)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The article consists of two chapters. In the first part of the first chapter, the author rediscovers well-partial-orderings and well-quasi-orderings, which he calls tight partial orders and tight quasi orders, and develops a theory of such orderings. (In this connection, it may be appropriate to point out Joseph B. Kruskal&#8217;s article &#8220;The theory of well-quasi-ordering: [&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":"","msr_number":"","msr_organization":"","msr_pages_string":"Pages 284-308 of the Russian original Pages - 160-174 of the English translation","msr_page_range_start":"","msr_page_range_end":"","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":"1969-04-08","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/web.eecs.umich.edu\/~gurevich\/Opera\/13.pdf","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-360824","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":"1969-04-08","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"Pages 284-308 of the Russian original Pages - 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