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ON THE UNDECIDABILITY OF THE IDENTITY CORRESPONDENCE PROBLEM AND ITS APPLICATIONS FOR WORD AND MATRIX SEMIGROUPS

Bell

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Abstract

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several questions for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.

Acceptance Date Jul 21, 2010
Publication Date Dec 1, 2010
Journal International Journal of Foundations of Computer Science
Print ISSN 0129-0541
Publisher World Scientific Publishing
Pages 963 - 978
DOI https://doi.org/10.1142/S0129054110007660
Publisher URL https://www.worldscientific.com/doi/pdf/10.1142/S0129054110007660?download=true

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