The Orbit Problem Is in the GapL Hierarchy

Abstract

The Orbit problem is defined as follows: Given a matrix A εℚn×n and vectors x,y ∈ ℚn, 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 in [7]. In this paper we put the problem in the logspace counting hierarchy GapLH. We also show that the problem is hard for C=L.