Logically defined subsets of Nk
Theoretical Computer Science
Journal of the ACM (JACM)
Reversal-Bounded Multicounter Machines and Their Decision Problems
Journal of the ACM (JACM)
The Mathematical Theory of Context-Free Languages
The Mathematical Theory of Context-Free Languages
Multitape NFA: weak synchronization of the input heads
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
Weak synchronization and synchronizability of multitape pushdown automata and turing machines
LATA'12 Proceedings of the 6th international conference on Language and Automata Theory and Applications
How to synchronize the heads of a multitape automaton
CIAA'12 Proceedings of the 17th international conference on Implementation and Application of Automata
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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=(x"1,...,x"n) 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 Ginsburg@?s problem is equivalent to deciding for an arbitrary n-tape NFA M accepting L(M)@?a"1^@?x...xa"n^@? (where n=1 and a"1,...,a"n are distinct symbols) whether there exists an equivalent 0-synchronized n-tape NPDA M^'. Ginsburg@?s problem is decidable if 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.