Keele Research Repository

There has been much research into freeness properties of finitely generated matrix semigroups under various constraints, such as the dimensions of the generator matrices and the semiring over which the matrices are defined. Most freeness problems have been shown to be undecidable starting from dimension three, even for upper-triangular matrices over the natural numbers. There are many open problems still remaining in dimension two. A recent paper has also investigated freeness properties of bounded languages of matrices, which are matrices from a set M∗1 M∗2· · · M∗k ⊆ Fn×n for some semiring F and a fixed value k ∈ N>0, where matrices M1, . . . , Mk are given [1]. We consider a notion of freeness and ambiguity for scalar reachability problems in matrix semigroups and bounded languages of matrices. Scalar reachability concerns the set {ρTMτ |M ∈ S}, where ρ, τ ∈ F n are vectors and S ⊆ F n×n is a finitely generated matrix semigroup. Ambiguity and freeness problems are defined in terms of the uniqueness of factorizations for each scalar. Such problems have also been studied in connection to formal power series. We show various undecidability results and their connections to weighted and probabilistic finite automata.