{"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 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