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Title: On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups
Authors: Bell, Paul C.
Potapov, Igor
Keywords: Combinatorics on Words
Group problem
Post's correspondence problem
Matrix semigroups
Undecidability
Issue Date: 2010
Publisher: © World Scientific Publishing
Citation: BELL, P. and POTAPOV, I., 2010. On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups. International Journal of Foundations of Computer Science, 21 (6), pp.963-978.
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 nite 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 nitely 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 nite set of matrices generates a group is also undecidable. We also answer several questions for matrices over di erent number elds. 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.
Description: Electronic version of an article published as in the International Journal of Foundations of Computer Science [© World Scientific Publishing Company]: http://www.worldscientific.com/doi/abs/10.1142/S0129054110007660
Version: Accepted for publication
DOI: 10.1142/S0129054110007660
URI: https://dspace.lboro.ac.uk/2134/12042
Publisher Link: http://dx.doi.org/10.1142/S0129054110007660
ISSN: 0129-0541
Appears in Collections:Published Articles (Computer Science)

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