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.
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.