A Note on the Succinctness of Descriptions of Deterministic Languages

The result proved in this paper is that for the elements of some infinite class of deterministic context-free languages the size of deterministic pushdown amomata needed to describe them is not recursively bounded by the size of the smallest unambiguous context-free grammars that generate them. This is a quantitative explanation of the fact that some languages require large descriptions in terms of LR(1) grammars (Knuth, 1965; Aho and Ullman, 1972), or strict deterministic grammars (Harrison and Havel, 1973), even though they can be described very succinctly in terms of general, even unambiguous, context-free grammars. It therefore illustrates one of the tangible advantages of using parsing mechanisms more powerful than a single pushdown stack (e.g., Earley, 1970) even for languages for which that may be sufficient. The most closely related result previously known is that a similar nonrecursive relationship exists between the succinctness of descriptions of regular sets by finite automata and (ambiguous) context-free languages respectively (Meyer and Fischer, 1971). In contrast a fairly precise recursive relationship is known to exist between finite automata and deterministic pushdown automata (Stearns, 1967; Meyer and Fischer, 1971; Valiant, 1975). However, the two further questions of analogously relating finite automata with unambiguous grammars, and unambiguous grammars with ambiguous ones, both remain open.