On the open problem of Ginsburg concerning semilinear sets and related problems

In his 1966 book, “The Mathematical Theory of Context-Free Languages”, S. Ginsburg posed the following open problem: Find a decision procedure for determining if an arbitrary semilinear set is a finite union of stratified linear sets. It turns out that this problem (which remains open) is equivalent to the problem of synchronizability of multitape machines. Given an n-tape automaton M of a given type (e.g., nondeterministic pushdown automaton (NPDA), nondeterministic finite automaton (NFA), etc.) with a one-way read-only head per tape and a right end marker on each tape, we say that M is 0-synchronized (or aligned) if for every n  -tuple x=(x1,…,xn)x=(x1,…,xn) that is accepted, there is an accepting computation on x such that at any time during the computation, all heads except those that have reached the end marker are on the same position (i.e., aligned). One of our main contributions is to show that Ginsburgʼs problem is equivalent to deciding for an arbitrary n-tape NFA M   accepting L(M)⊆a1⁎×⋯×an⁎ (where n⩾1n⩾1 and a1,…,ana1,…,an are distinct symbols) whether there exists an equivalent 0-synchronized n  -tape NPDA M′M′. Ginsburgʼs problem is decidable if M′M′ is required to be an NFA, and we will generalize this decidability over bounded inputs.It is known that if the inputs are unrestricted, the problem is undecidable. We also show several other related decidability and undecidability results.It may appear, as one of the referees suggested in an earlier version of this paper, that our main result can be written in first order logic. We explain why this is not the case in the Introduction.