On deciding whether a monoid is a free monoid or is a group
Acta Informatica
Multiplicative equations over commuting matrices
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Decision problems for semi-Thue systems with a few rules
Theoretical Computer Science - Insightful theory
On the membership of invertible diagonal and scalar matrices
Theoretical Computer Science
On undecidability bounds for matrix decision problems
Theoretical Computer Science
Reachability problems in quaternion matrix and rotation semigroups
Information and Computation
Post Correspondence Problem and Small Dimensional Matrices
DLT '09 Proceedings of the 13th International Conference on Developments in Language Theory
Periodic and infinite traces in matrix semigroups
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
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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.