Improved normal form for grammars with one-sided contexts

Formal grammars equipped with operators for specifying the form of the context of a substring were recently studied by Barash and Okhotin ("An extension of context-free grammars with one-sided context specifications", Inform. and Comput., 2014) further extending the author's earlier work on propositional connectives in grammars (A. Okhotin, "Conjunctive grammars", J. Autom. Lang. Comb., 2001). These grammars allow two types of context specifications: for a substring v of a string uvx, a proper left context operator ?D states that u is of the form described by D, and an extended left context operator ?E states that uv is described by E. This paper establishes a normal form for these grammars, in which extended left contexts are never used, whereas left contexts may be applied only to individual symbols, so that all rules are of the form A ? B 1 C 1 & ? & B m C m or A ? a & ? D . This eliminates circular dependencies in the grammar and leads to a new, clearer version of the known cubic-time parsing algorithm. With some further improvements, the algorithm is accelerated from time O ( n 3 ) to time O ( n 3 log ? n ) .