The orbit problem is in the GapL hierarchy

Abstract

The Orbit problem is defined as follows: Given a matrix A?? n×n and vectors x,y?? n , does there exist a non-negative integer i such that A i x=y. This problem was shown to be in deterministic polynomial time by Kannan and Lipton (J. ACM 33(4):808---821, 1986). In this paper we place the problem in the logspace counting hierarchy GapLH. We also show that the problem is hard for C=L with respect to logspace many-one reductions.