Weak Synchronization and Synchronizability of Multitape Pushdown Automata and Turing Machines

Abstract

Given an n-tape automaton M with a one-way read-only head per tape which is delimited by an end marker $ 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 an accepting computation on x such that no pair of input heads, neither of which is on $, are more than k tape cells apart at any time during the computation. When a head reaches the marker, it can no longer move. 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 look at the following problems: (1) Given an n-tape automaton M, is it weakly k-synchronized for a given k (for some k)? and (2) Given an n-tape automaton M, does there exist a weakly k-synchronized automaton for a given k (for some k) M′ such that L(M′) = L(M)? In an earlier paper [1], we studied the case of multitape finite automata (NFAs). Here, we investigate the case of multitape pushdown automata (NPDAs), multitape Turing machines, and other multitape models. The results that we obtain contrast those of the earlier results and involve some rather intricate constructions.