Orbits of Linear Maps and Regular Languages

Abstract

The chamber hitting problem (CHP) for linear maps consists in checking whether an orbit of a linear map specified by a rational matrix hits a given rational polyhedral set. The CHP generalizes some well-known open computability problems about linear recurrent sequences (e.g., the Skolem problem, the nonnegativity problem). It is recently shown that the CHP is Turing equivalent to checking whether an intersection of a regular language and the special language of permutations of binary words (the permutation filter) is nonempty ($P_{\mathbb{B}}$E-realizability problem).

In this paper we present some decidable and undecidable problems closely related to $P_{\mathbb{B}}$-realizability problem thus demonstrating its ‘borderline’ status with respect to computability.