The hardest language for grammars with context operators

TL;DR

This paper characterizes the computational power of grammars with context operators, showing their language family is as expressive as the class of all context-free languages and is closed under inverse homomorphisms.

Contribution

It establishes a new normal form for grammars with context operators and proves their language family is closed under inverse homomorphisms and injective finite transductions.

Findings

Introduces the even-odd normal form for grammars with context operators.

Shows the language family is closed under inverse homomorphisms.

Demonstrates the expressive power matches all context-free languages.

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. The characterization is completed by showing that the language family defined by grammars with context operators is closed under inverse homomorphisms; actually, it is closed under injective nondeterministic finite transductions.