On LL(k) linear conjunctive grammars

TL;DR

This paper studies the LL(k) subclass of linear conjunctive grammars, showing hierarchy collapse to LL(1) and providing a linear-time, logarithmic-space parser, thus placing these languages within the class L.

Contribution

It proves the hierarchy collapses to LL(1) and constructs an efficient parser, advancing understanding of the computational complexity of linear conjunctive grammars.

Findings

Hierarchy of LL(k) collapses to LL(1).

Parser operates in linear time and logarithmic space.

LL(k) languages are contained in class L.

Abstract

Linear conjunctive grammars are a family of formal grammars with an explicit conjunction operation allowed in the rules, which is notable for its computational equivalence fo one-way real-time cellular automata, also known as trellis automata. This paper investigates the LL($k$) subclass of linear conjunctive grammars, defined by analogy with the classical LL($k$) grammars: these are grammars that admit top-down linear-time parsing with $k$-symbol lookahead. Two results are presented. First, every LL($k$) linear conjunctive grammar can be transformed to an LL(1) linear conjunctive grammar, and, accordingly, the hierarchy with respect to $k$ collapses. Secondly, a parser for these grammars that works in linear time and uses logarithmic space is constructed, showing that the family of LL($k$) linear conjunctive languages is contained in the complexity class $L$.