On LR(k)-parsers of polynomial size

Usually, a parser for an $LR(k)$-grammar $G$ is a deterministic pushdown transducer which produces backwards the unique rightmost derivation for a given input string $x \in L(G)$. The best known upper bound for the size of such a parser is $O(2^{{|G||\Sigma|k+k\log} |\Sigma| + \log |G|})$ where $|G|$ and $|\Sigma|$ are the sizes of the grammar $G$ and the terminal alphabet $\Sigma$, respectively. If we add to a parser the possibility to manipulate a directed graph of size $O(|G|n)$ where $n$ is the length of the input then we obtain an extended parser. The graph is used for an efficient parallel simulation of all potential leftmost derivations of the current right sentential form such that the unique rightmost derivation of the input can be computed. Given an arbitrary $LR(k)$-grammar $G$, we show how to construct an extended parser of $O(|G| + #LA |N|2^k k \log k)$ size where $|N|$ is the number of nonterminal symbols and $#LA$ is the number of relevant lookaheads with respect to the grammar $G$. As the usual parser, this extended parser uses only tables as data structure. Using some ingenious data structures and increasing the parsing time by a small constant factor, the size of the extended parser can be reduced to $O(|G| + #LA|N|k^2)$. The parsing time is $O(ld(input) + k|G|n)$ where $ld(input)$ is the length of the derivation of the input. Moreover, we have constructed a one pass parser.