# 19.5 On unambiguous grammars

The universality problem for context-free grammars is undecidable. For deterministic pushdown automata it is PTIME-complete. For unambiguous context-free grammars, which constitute a model of intermediate expressive power, the problem is still decidable. Using the existential theory of the reals (the existential fragment of Tarski's algebra), it can be shown to be in PSPACE. However, no lower-bound better than PTIME is known. Improving either the PSPACE upper or the PTIME lower-bound would be a major advance on this problem!

A related problem is universality of the union of a deterministic pushdown automaton and an unambiguous finite automaton.

While this problem reduces to the universality problem of unambiguous context-free grammars, a reduction in the other direction seems unlikely. The known complexity bounds are the same as for the more general problem.