The hardest language for grammars with context operators

Abstract

In 1973, Greibach ("The hardest context-free language", SIAM J. Comp., 1973) constructed a context-free language L 0 with the property that every context-free language can be reduced to L 0 by a homomorphism, thus representing it as an inverse homomorphic image h −1 (L 0 ). In this paper, a similar characterization is established for a family of grammars equipped with operators for referring to the left context of any substring, recently defined by Barash and Okhotin ("An extension of context-free grammars with one-sided context specifications", Inform. Comput., 2014). An essential step of the argument is a new normal form for grammars with context operators, in which every nonterminal symbol defines only strings of odd length in left contexts of even length: the even-odd normal form.