Abstract
There has been much research into freeness properties of finitely generated matrix semigroups under various constraints. Most freeness problems are undecidable starting from dimension three, even for upper-triangular integer matrices. A recent paper investigated freeness properties of bounded languages of matrices, which are matrices from a set M1⁎M2⁎⋯Mk⁎⊆Fn×n for some semiring F and fixed k∈N>0.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∈Fn are vectors and S⊆Fn×n is a finitely generated matrix semigroup. Ambiguity and freeness problems are defined in terms of the uniqueness of factorizations for each scalar. We show various undecidability results and connections to weighted and probabilistic finite automata.
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