{"id":156126,"date":"2000-01-01T00:00:00","date_gmt":"2000-01-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/on-the-undecidability-of-second-order-unification\/"},"modified":"2018-10-16T20:21:32","modified_gmt":"2018-10-17T03:21:32","slug":"on-the-undecidability-of-second-order-unification","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/on-the-undecidability-of-second-order-unification\/","title":{"rendered":"On the Undecidability of Second-Order Unification"},"content":{"rendered":"<p>There is a close relationship between word unification and second-order unification. This similarity has been exploited, for instance, in order to prove decidability of monadic second-order unification and decidability of linear second-order unification when no second-order variable occurs more than twice. The attempt to prove the second result for (nonlinear) second-order unification failed and led instead to a natural reduction from simultaneous rigid <em>E<\/em>-unification to this second-order unification. This reduction is the first main result of this paper, and it is the starting point for proving some novel results about the undecidability of second-order unification presented in the rest of the paper. We prove that second-order unification is <em>undecidable<\/em> in the following three cases: (1) each second-order variable occurs at most twice and there are only two second-order variables; (2) there is only one second-order variable and it is unary; (3) the following conditions (i)\u2013(iv) hold for some fixed integer <em>n<\/em>: (i) the arguments of all second-order variables are ground terms of size <<em>n<\/em>, (ii) the arity of all second-order variables is <<em>n<\/em>, (iii) the number of occurrences of second-order variables is \u2a7d5, (iv) there is either a single second-order variable or there are two second-order variables and no first-order variables.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>There is a close relationship between word unification and second-order unification. This similarity has been exploited, for instance, in order to prove decidability of monadic second-order unification and decidability of linear second-order unification when no second-order variable occurs more than twice. The attempt to prove the second result for (nonlinear) second-order unification failed and led [&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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-156126","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"","msr_edition":"Information and 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