Linear-Time Computation by Nondeterministic Multidimensional Iterative Arrays

It is shown by simulation that every language accepted within time $n^d$ by a nondeter-ministic one-dimensional Turing machine is accepted in linear time by a nondeterministic d-dimensional iterative array. Conversely, every language accepted in linear time by such an iterative array is accepted within time $n^{d+1}$ by a nondeterministic one-dimensional Turing machine. It follows that the class of languages accepted in linear time by nondeterministic multidimensional iterative arrays is precisely Karp’s class NP, that nondeterministic $(d + 2)$-dimensional iterative arrays are more powerful than nondeterministic d-dimensional iterative arrays, and that nondeterministic two-dimensional iterative arrays are more powerful than the entire class of nondeterministic multidimensional Turing machines. Related deterministic results are surveyed and summarized for comparison.