Example of undecidable problems for 2-generator matrix semigroups
Theoretical Computer Science - Special issue: papers dedicated to the memory of Marcel-Paul Schützenberger
Rainbow Induced Subgraphs in Proper Vertex Colorings
Fundamenta Informaticae
On the Computational Complexity of Matrix Semigroup Problems
Fundamenta Informaticae - Words, Graphs, Automata, and Languages; Special Issue Honoring the 60th Birthday of Professor Tero Harju
| Hi-index | 5.23 |
The following problem looking as a high-school exercise hides an unexpected difficulty. Do the matrices A=(2003)andB=(3505) satisfy any nontrivial equation with the multiplication symbol only? This problem was mentioned as open in Cassaigne et al. [J. Cassaigne, T. Harju, J. Karhumaki, On the undecidability of freeness of matrix semigroups, Internat. J. Algebra Comput. 9 (3-4) (1999) 295-305] and in a book by Blondel et al. [V. Blondel, J. Cassaigne, J. Karhumaki, Problem 10.3: Freeness of multiplicative matrix semigroups, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press, 2004, pp. 309-314] as an intriguing instance of a natural computational problem of deciding whether a given finitely generated semigroup of 2x2 matrices is free. In this paper we present a new partial algorithm for the latter which, in particular, easily finds that the following equation AB^1^0A^2BA^2BA^1^0=B^2A^6B^2A^2BABABA^2B^2A^2BAB^2 holds for the matrices above. Our algorithm turns out quite practical and allows us to settle also other related open questions posed in the mentioned article.