New characterizations of exponential, elementary, and non-elementary time-bounded Turing machines

Abstract

Machine models with multiple types of stores are studied. Deterministic two-way pushdown automata augmented by some number of checking stacks are known to accept exactly the class of elementary languages, which is very general but still has a decidable membership problem. First, we define such a machine to be synchronous if, when a checking stack starts to read from its stack, all other checking stacks can no longer write. For such a synchronous machine, the multiple checking stacks are equivalent to machines with 2 synchronous checking stacks which, in turn, are equivalent to exponential time-bounded deterministic Turing machines. Next, we also show that for any (reasonably defined) one-way deterministic machine model M with a decidable membership problem, the two-way deterministic multi-head variant of M augmented by any number of checking stacks also has a decidable membership problem. We also examine a model, two-way deterministic pushdown automata augmented with some number of non-erasing stacks, where the machine starts reading from the stacks at most a linear number of times. We show that this model accepts non-elementary languages but still has a decidable membership problem, resolving an open problem from the literature.