Computational parallels between the regular and context-free languages

• Authors: H. B. Hunt, III;D. J. Rosenkrantz
• Venue: STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
• Year: 1974

This paper presents a complexity theory of formal languages. The main technique used is that of embedding “&equil;{0,1}*”, “&equil;0*”, and “&equil;&fgr;” into other linguistic predicates. In Section 2, the undecidability of “&equil;{0,1}*” for cfl's is exploited to provide sufficient conditions for the undecidability of predicates on the cfl's. In Section 3, the same techniques are applied to regular sets. Predicates satisfying conditions similar to those of Section 2 are shown to be hard, where how hard depends on the descriptors used to enumerate the regular sets. Section 4 concentrates on the equivalence and containment problems for cfl's. For cfl's, regular sets, and linear cfl's, the complexity of determining equivalence to a fixed language is linked to whether the fixed language is finite, infinite but bounded, or unbounded. In Section 5, the ability of cfg's to generate finite languages whose strings are exponential in the size of the grammar is used to obtain exponential lower bounds on several decidable problems for cfg's generating finite sets. In Section 6, all nontrivial predicates for certain specific classes of languages are shown to be hard. In Section 7, we show that a dpda can always be converted in polynomial time into an equivalent dpda that always halts. Therefore the predicate “&equil;{0,1}*” is in P for dpda's, and embedding this problem into other predicates on the dpda's will not yield nonpolynomial lower bounds. In Section 8, some of the preceding results are generalized to other families of languages.