  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 $L$.
