Orbits of linear maps and regular languages

TL;DR

This paper explores the complex relationship between linear map orbits hitting polyhedral sets and the intersection of regular languages with permutation filters, highlighting their undecidability and relation to longstanding problems.

Contribution

It establishes the equivalence between hitting polyhedral sets by linear map orbits and permutation filter realizability, connecting these to well-known open problems.

Findings

The permutation filter realizability problem's borderline decidability status.

Connections to the Skolem problem and linear recurrent sequences.

Identification of related decidable and undecidable problems.

Abstract

We settle the equivalence between the problem of hitting a polyhedral set by the orbit of a linear map and the intersection of a regular language and a language of permutations of binary words (the permutation filter realizability problem). The decidability of the both problems is presently unknown and the first one is a straightforward generalization of the famous Skolem problem and the nonnegativity problem in the theory of linear recurrent sequences. To show a `borderline' status of the permutation filter realizability problem with respect to computability we present some decidable and undecidable problems closely related to it.