A characterization of exponential-time languages by alternating context-free grammars

Abstract

We show that the class of exponential-time languages or, equivalently, the class of languages accepted by alternating pushdown automata (APDAs), is exactly the class of languages generated by linear-erasing alternating context-free grammars (ACFGs). An ACFG is generalization of an ordinary context-free grammar in which we allow the use of universal nonterminals in much the same way as universal states are used in APDAs. It was recently claimed in [9] that APDAs are equivalent to ACFGs. However, the proofs in both directions have major flaws which do not seem to be correctable. As it turns out, the proof of the claim does not follow from a simple extension of the well-known constructions for the nonalternating case. Our proof is, in fact, for a modified claim: APDAs are equivalent to linear-erasing ACFGs, where linear-erasing means that there is a constant c such that every string of length n in the language generated by the ACFG has a derivation in which all intermediate sentential forms are at most c n long.