Automaticity III: Polynomial automaticity and context-free languages

If L is a formal language, we define A(L)(n) to be the number of states in the smallest deterministic finite automaton that accepts a language which a grees with L on all inputs of length no more than n. This measure is called automaticity. In this paper, we first study the closure properties of the class DPA of languages of deterministic polynomial automaticity, i.e., thos e languages L for which there exists k such that A(L)(n) = O(n(k)). Next, w e discuss similar results for a nondeterministic analogue of automaticity, introducing the classes NPA (languages of nondeterministic polynomial autom aticity) and NPLA (languages of nondeterministic poly-log automaticity). We conclude by showing how to construct a context-free language of automatici ty arbitrarily close to the maximum possible.