Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
Lectures on Linear Sequential Machines
Lectures on Linear Sequential Machines
The size of numbers in the analysis of certain algorithms
The size of numbers in the analysis of certain algorithms
Graph Theory With Applications
Graph Theory With Applications
Multiplicative equations over commuting matrices
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Hypergeometric dispersion and the orbit problem
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Handbook of automated reasoning
On the membership of invertible diagonal and scalar matrices
Theoretical Computer Science
Reachability in Linear Dynamical Systems
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
The Orbit Problem Is in the GapL Hierarchy
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Polynomial ring automorphisms, rational (w,σ)-canonical forms, and the assignment problem
Journal of Symbolic Computation
The continuous Skolem-Pisot problem
Theoretical Computer Science
Equivalence of set- and bag-valued orbits
Journal of Automata, Languages and Combinatorics
The orbit problem is in the GapL hierarchy
Journal of Combinatorial Optimization
On the membership of invertible diagonal matrices
DLT'05 Proceedings of the 9th international conference on Developments in Language Theory
From post systems to the reachability problems for matrix semigroups and multicounter automata
DLT'04 Proceedings of the 8th international conference on Developments in Language Theory
The orbit problem in higher dimensions
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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The accessibility problem for linear sequential machines [12] is the problem of deciding whether there is an input x such that on x the machine starting in a given state q1 goes to a given state q2. Harrison shows that this problem is reducible to the following simply stated linear algebra problem, which we call the "orbit problem":Given (n, A, x, y), where n is a natural number and A, x, and y are nxn, nx1, and nx1 matrices of rationals, respectively, decide whether there is a natural number I such that Aix=y.He conjectured that the orbit problem is decidable. No progress was made on the conjecture for ten years until Shank [22] showed that if n is fixed at 2, then the problem is decidable. This paper shows that the orbit problem for general n is decidable and indeed decidable in polynomial time. The orbit problem arises in several contexts; two of these, linear recurrences and the discrete logarithm problem for polynomials, are discussed, and we apply our algorithm for the orbit problem in these contexts.