On the existence of regular approximations

We approximate context-free, or more general, languages using finite automata. The degree of approximation is measured, roughly speaking, by counting the number of incorrect answers an automaton gives on inputs of length m and observing how these values behave for large m. More restrictive variants are obtained by requiring that the automaton never accepts words outside the language or that it accepts all words in the language. A further distinction is whether a given (context-free) language has a regular approximation which is optimal under the measure of approximation degree or an approximation which is arbitrarily close to optimal. We study closure and decision properties of the approximation measure.