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We show that several problems concerning probabilistic finite automata of a fixed dimension and a fixed number of letters for bounded cut-point and strict cut-point languages are algorithmically undecidable by a reduction of Hilbert's tenth problem. We then consider the set of so called 'F-Problems' (emptiness, infiniteness, containment, disjointness, universe and equivalence) and show that they are also undecidable for bounded (non-)strict cut-point languages on probabilistic finite automata. For a finite set of matrices $\{M_1, M_2, \ldots ,M_k\} \subseteq \mathbb{Q}^{t \times t}$, we then consider the decidability of computing the maximal spectral radius of any matrix in the set $X = \{M^{j_1}_1 M^{j_2}_2 \cdot M^{j_k}_k \vert j_1, j_2,\ldots, j_k \geq 0\}$, which we call a bounded matrix language. Using an encoding of a probabilistic finite automaton shown in the paper, we prove the surprising result that determining if the maximal spectral radius of a bounded matrix language is less than or equal to one is undecidable, but determining whether it is strictly less than one is in fact decidable (which is similar to a result recently shown for quantum automata). [ABSTRACT FROM AUTHOR]

Most computational problems for matrix semigroups and groups are inherently difficult to solve and even undecidable starting from dimension three. The questions about the decidability and complexity of problems for two-dimensional matrix semigroups remain open and are directly linked with other challenging problems in the field. In this paper we study the computational complexity of the problem of determining whether the identity matrix belongs to a matrix semigroup (the Identity Problem) generated by a finite set of 2 × 2 integral unimodular matrices. The Identity Problem for matrix semigroups is a well-known challenging problem, which has remained open in any dimension until recently. It is currently known that the problem is decidable in dimension two and undecidable starting from dimension four. In particular, we show that the Identity Problem for 2 × 2 integral unimodular matrices is NP-hard by a reduction of the Subset Sum Problem and several new encoding techniques. An upper bound for the nontrivial decidability result by C. Choffrut and J. Karhumäki is unknown. However, we derive a lower bound on the minimum length solution to the Identity Problem for a constructible set of instances, which is exponential in the number of matrices of the generator set and the maximal element of the matrices. This shows that the most obvious candidate for an NP algorithm, which is to guess the shortest sequence of matrices which multiply to give the identity matrix, does not work correctly since the certificate would have a length which is exponential in the size of the instance. Both results lead to a number of corollaries confirming the same bounds for vector reachability, scalar reachability and zero in the right upper corner problems. [ABSTRACT FROM AUTHOR]

We consider decidability questions for the emptiness problem of intersections of matrix semigroups. This problem was studied by A. Markov [7] and more recently by V. Halava and T. Harju [5]. We give slightly strengthened results of their theorems by using a different matrix encoding. In particular, we show that given two finitely generated semigroups of non-singular upper triangular 3 × 3 matrices over the natural numbers, checking the emptiness of their intersections is undecidable. We also show that the problem is undecidable even for unimodular matrices over 3 × 3 rational matrices. [ABSTRACT FROM AUTHOR]