Improved Normal Form for Grammars with One-Sided Contexts

Abstract

Formal grammars equipped with operators for specifying the form of the context of a substring were recently studied by Barash and Okhotin (“Defining contexts in context-free grammars”, LATA 2012), further extending the author’s (“Conjunctive grammars”, DCAGRS 2000) earlier work on propositional connectives in grammars. These grammars allow two types of context specifications: for a substring w of a string uwv, a left context operator ⊲ D states that u is of the form described by D, while the extended left context operator states that uw is described by E. This paper establishes a normal form for these grammars, in which extended left contexts are never used, while left contexts may be applied only for individual symbols, so that all rules are of the form $A \to B_1 C_1 \mathop{\&} \ldots \mathop{\&} B_n C_n$ or $A \to a \mathop{\&} {\lhd}D$. This eliminates circular dependencies in the grammar and allows simplifying the known parsing algorithm. Some further improvements to the algorithm accelerate it from time O(n ³) to time $O(\frac{n^3}{\log n})$.