We prove that several problems associated to probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 × 47 matrices with nonnegative rational entries is bounded is undecidable.

this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NP-hard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into two halfspaces, and the dynamics in eachhalfspace correspond to a differentlinear system

A set of matrices over the integers is said to be length-k-mortal (with positive integer) if the zero matrix can be expressed as a product of length of matrices in the set. The set is said to be mortal if it is length-k-mortal for some finite k.

We consider the solvability of the equation k Y i=1 A i x i = B and generalizations, where the A i and B are given commuting matrices over an algebraic number field F . In the semigroup membership problem, the variables x i are constrained to be nonnegative integers. While this problem is NP-complete for variable k, we give a polynomial time algorithm if k is fixed. In the group membership problem, the matrices are assumed to be invertible, and the variables x i may take on negative values. In this case we give a polynomial time algorithm for variable k and give an explicit description of the set of all solutions (as an affine lattice). The results generalize recent work of Cai, Lipton, and Zalcstein [CLZ] where the case k = 2 is solved using Jordan Normal Forms (JNF). We achieve greater clarity, simplicity, and generality by eliminating the use of JNF's and referring to elementary concepts of the structure theory of algebras instead (notably, the radical and the local decomposit...

We give a survey of Skolem’s problem for linear recurrence sequences. We cover the known decidable cases for recurrence depths of at most 4, and give detailed proofs for these cases. Moreover, we shall prove that the problem is decidable for linear recurrences of depth 5.

In this paper, we discuss the existence of an algorithm to decide if a given set of 2 2 matrices is mortal: a set F = fA 1 ; : : : ; Am g of 2 2 matrices is said to be mortal if there exist an integer k 1 and some integers i 1 ; i 2 ; : : : ; i k 2 f1; : : : ; mg with A i 1 A i 2 A i k = 0. We survey this problem and propose some new extensions: we prove the problem to be BSS-undecidable for real matrices and Turing-decidable for two rational matrices. We relate the problem for rational matrices to the entry equivalence problem, to the zero in the left upper corner problem and to the reachability problem for piecewise a ne functions. Finally, we state some NP-completeness results.

Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free, 2) contains the identity matrix, 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs to a given finitely generated semigroup. 1

We prove that the Jordan Normal Form of a rational matrix can be computed exactly in polynomial time. We obtain the transformation matrix and its inverse exactly, and we show how to apply the basis transformation to any commuting matrices. 1 Introduction There are two motivations for this work on computing the Jordan Normal Form of a rational matrix exactly. The first is related to the resolution of the complexity of the A B C problem [4], and its application to the complexity problem in finitely generated commutative linear groups and semigroups in general. The second motivation is concerned with the design and analysis of uncheatable benchmarks for numerical algorithms, especially matrix multiplication [3, 1]. Our problem is the following. Given a finite set of commuting matrices over the rational numbers, A; B; : : : ; can we compute, in polynomial time, a basis transformation T , and the matrices under the similarity transformation T \Gamma1 AT ; T \Gamma1 BT ; : : : ; so tha...

Abstract. In this paper, we establish the Čern´y-Pin conjecture for automata with the property that their transition monoid cannot recognize the language {a, b} ∗ ab{a, b} ∗. For the subclass of automata whose transition monoids have the property that each regular J-class is a subsemigroup, we give a tight bound on lengths of reset words for synchronizing automata thereby answering a question of Volkov. 1

this paper, we extend previous results in two directions. First, we allow infinite systems, where even decidability is not clear, and second, we consider semigroups as well. On the other hand, since the general problem is undecidable, some restrictions are needed. In 1980, Kannan and Lipton [15] solved the following orbit problem, the case of a cyclic