Polynomial-time algorithm for the orbit problem
Journal of the ACM (JACM)
Cornerstones of undecidability
Cornerstones of undecidability
Concrete Math
From post systems to the reachability problems for matrix semigroups and multicounter automata
DLT'04 Proceedings of the 8th international conference on Developments in Language Theory
A Note on the Emptiness of Semigroup Intersections
Fundamenta Informaticae
Reachability problems in quaternion matrix and rotation semigroups
Information and Computation
The Identity Correspondence Problem and Its Applications
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
A Note on the Emptiness of Semigroup Intersections
Fundamenta Informaticae
Reachability problems in quaternion matrix and rotation semigroups
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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In this paper, we consider decidability questions that are related to the membership problem in matrix semigroups. In particular, we consider the membership of a given invertible diagonal matrix in a matrix semigroup and then a scalar matrix, which has a separate geometric interpretation. Both problems have been open for any dimensions and are shown to be undecidable in dimension 4 with integral matrices by a reduction of the Post Correspondence Problem (PCP). Although the idea of PCP reduction is standard for such problems, we suggest a new coding technique to cover the case of diagonal matrices.