When is a pair of matrices mortal?
Information Processing Letters
Example of undecidable problems for 2-generator matrix semigroups
Theoretical Computer Science - Special issue: papers dedicated to the memory of Marcel-Paul Schützenberger
Closed-form analytic maps in one and two dimensions can simulate universal Turing machines
Theoretical Computer Science - Special issue on real numbers and computers
Widening the Boundary between Decidable and Undecidable Hybrid Systems
CONCUR '02 Proceedings of the 13th International Conference on Concurrency Theory
Decision problems for semi-Thue systems with a few rules
Theoretical Computer Science - Insightful theory
Computation in one-dimensional piecewise maps and planar pseudo-billiard systems
UC'05 Proceedings of the 4th international conference on Unconventional Computation
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
Periodic and infinite traces in matrix semigroups
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
Orbits of linear maps and regular languages
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
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In this paper we consider several reachability problems such as vector reachability, membership in matrix semigroups and reachability problems in piecewise linear maps. Since all of these questions are undecidable in general, we work on lowering the bounds for undecidability. In particular, we show an elementary proof of undecidability of the reachability problem for a set of 7 two-dimensional affine transformations. Then, using a modified version of a standard technique, we also prove the vector reachability problem is undecidable for two (rational) matrices in dimension 16. The above result can be used to show that the system of piecewise linear functions of dimension 17 with only two intervals has an undecidable set-to-point reachability problem. We also show that the “zero in the upper right corner” problem is undecidable for two integral matrices of dimension 18 lowering the bound from 23.