Orbits of Linear Maps and Regular Languages

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 (PB-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 PB-realizability problem with respect to computability we present some decidable and undecidable problems closely related to it. This paper is an extended version of the journal publication [16] and contains some additional results.