Multitape NFA: Weak Synchronization of the Input Heads

Abstract

Given an n-tape nondeterministic finite automaton (NFA) M with a one-way read-only head per tape and a right end marker $ on each tape, and a nonnegative integer k, we say that M is weakly k-synchronized if for every n-tuple x = (x ₁, …, x _n ) that is accepted, there is a computation on x such that at any time during the computation, no pair of input heads, neither of which is on $, are more than k cells apart. As usual, an n-tuple x = (x ₁, …, x _n ) is accepted if M eventually reaches the configuration where all n heads are on $ in an accepting state. We show decidable and undecidable results concerning questions such as: (1) Given M, is it weakly k-synchronized for some k (resp., for a specified k) and (2) Given M, is there a weakly k-synchronized M ′ for some k (resp., for a specified k) such that L(M ′) = L(M)? Most of our results are the strongest possible in the sense that slight restrictions on the models make the undecidable problems decidable. A few questions remain open.