On the Undecidability of Freeness of Matrix Semigroups

Abstract

We slightly improve the result of Klarner, Birget and Satterfield, showing that the freeness of finitely presented multiplicative semigroups of 3×3 matrices over ℕ is undecidable even for triangular matrices. This is achieved by proving a new variant of Post correspondence problem. We also consider the freeness problem for 2×2 matrices. On the one hand, we show that it cannot be proved undecidable using the above methods which work in higher dimensions, and, on the other hand, we give some evidence that its decidability, if so, is also a challenging problem.

Scalar Ambiguity and Freeness in Matrix Semigroups over Bounded Languages

“…We prove the result by encoding an instance of the MMPCP problem. The basic idea is inspired by [7]. We start by showing the undecidability of the scalar freeness problem.…”

“…It was proven by Klarner et al that the freeness problem is undecidable over N 3×3 in [12] and this result was improved by Cassaigne et al to hold even for upper-triangular matrices over N 3×3 in [6].…”

Some decision problems on integer matrices

“…The main reason is that a direct product of two free monoids has a faithful representation in the multiplicative semigroup N 3×3 (which extends naturally that of a free monoid in the multiplicative semigroup of N 2×2 ). This allows us to encode Post Correspondence Problem and therefore to establish the undecidabibility of certain problems, see [16], also [11], [3] or [8]. E., g., the freeness of the subsemigroup of a finite number of matrices can be shown to be undecidable when one observes that the "uniquely decipherability property" in A * × B * is undecidable, [5].…”

Reachability Problems in Quaternion Matrix and Rotation Semigroups

“…We use an almost identical proof to that in [7] to show undecidability, but we obtain the result for matrices over H(Q) 2×2 rather than (Z + ) 3×3 :…”

“…Thus the problem of freeness for 2×2 rational quaternion matrix semigroups is undecidable. See [7] for fuller details of the proof method.…”