On the Recursion-Theoretic Complexity of Relative Succinctness of Representations of Languages

In Hartmanis (1980) a simple proof is given of the fact (originally proved in Valiant (1976)) that the relative succinctness of representing deterministic context-free languages by deterministic vs. nondeterministic pushdown automata is not recursively bounded, in the following sense: there is no recursive function which, for deterministic context-free languages L, can bound the size of the minimal deterministic pushdown automaton accepting L as a function of the size of a nondeterministic pushdown automaton accepting L. It is then stated that "...even if we would know (or be given) which pushdown automata describe deterministic languages, we still could not effectively write down the corresponding deterministic pushdown automata because of their enormous size which grows nonrecursively in the size of the nondeterministic pushdown automata." This does not, however, rule out a priori the possibility that a partial recursive bound might exist, as a function of the description of the nondeterministic pushdown automaton rather than of its size; indeed, the proof in Hartmanis (1980) uses the fact that the bounding function is total. It is the purpose of this note to make a case for using partial bounding functions in questions of relative succinctness. It will be shown that for the examples considered in Hartmanis (1980) (deterministic context-free languages, unambiguous languages, regular languages), the best possible partial bound as a function of the description of the unrestricted automata, while still nonrecursive, has lower recursiontheoretic complexity than the best possible bound as a functions of the size of the unrestricted automata. An example wi l l be given of a class of languages for which there exists a partial recursive bound on the size of the restricted automata, again as a function of the description of the unrestricted automata, while no (total) recursive bound exists as a function of the size of the unrestricted automata.