Parallel Parsing for Unification Grammars

The parsing problem for a rb i t r a ry un i f icat ion grammars is unsolvable We present a class of un i f i ca t ion grammars for whcih the parsing problem is solvable and a para l le l parsing algori thm for th is class of grammars 1. In t roduc t ion Uni f icat ion grammars have the power of a Turing machine, and one can easily prove this by showing that a un i f i ca t ion grammar can simulate any Prolog program It follows that the problem of f inding all possible parses of a sentence in a given un i f icat ion grammar is unsolvable The best we can do is an algori thm that sometimes finds a set of parses and sometimes goes in to an in f in i te loop The t op -down , l e f t t o r i g h t parser used with def in i te clause grammars is of th is k ind if the grammar contains left recurs ion the parser may run forever If we want to wr i te a paral le l parsing algori thm for un i f ica t ion grammar, we f i rs t need to f ind a subset of un i f i ca t ion grammar for which the parsing problem is solvable Indeed th is is the hard par t of the problem We shall see that once we have a parsing algor i thm, f inding the paral lel ism is s t ra igh t fo rward Part 1 reviews the algor i thm of Cocke. Kasami and Younger for parsing c o n t e x t f r e e grammars in Chomsky normal form This a lgor i thm is beaut i fu l ly simple and easily extends to a parsing algori thm for uni f icat ion grammars in Chomsky normal form Therefore the parsing problem is solvable for un i f ica t ion grammars in Chomsky normal form Unfor tunate ly th is subset of un i f i ca t ion grammar is too res t r i c ted to describe human language Part 2 there fore considers an extension of Chomsky Normal Form which allows chain rules rules having one non t e rm ina l symbol on the r igh t side We general ize the CKY algor i thm to handle con tex t f r ee grammars with chain ru les, and extend th is algorithm to un i f ica t ion grammars. The extension works only if one places a res t r i c t i on on the use of chain rules in a un i f i ca t ion grammar, and tha t res t r i c t i on is one main point of the paper Once we have the pars ing algorithm for un i f ica t ion grammars wi th chain rules, we can easily extend it to un i f ica t ion grammars wi th any number of symbols on the r ight side of a ru le Final ly we consider the possibi l i t ies of paral lel ism in the new parsing algor i thm 2. Parsing in Choaaky Normal Form A c o n t e x t f r e e grammar in Chomsky normal form contains two kinds of rules. Terminal rules have a single terminal on the r igh t side, branching rules have exact ly two non t e rm ina l symbols on the r ight side Since no ru le has an empty r igh t side, no symbol can generate the empty s t r ing We use the capi ta l le t ters A.B.C as var iables ranging over n o n t e r m i n a l symbols To describe the substr ings of an input sentence we number the spaces between words 0 is the space before the f i rs t word and n is the space after the n t h word If I < j. i npu t [ i j ] is the s t r ing of words between space i and space j The CKY algori thm bui lds a matr ix M such that M[i j] | A | A »>• i n p u t [ i j ] | Str ic t ly speaking th is is a recognizer not a parser, but it is easily extended to a parser, and the same is t rue for the other algori thms in this paper If SI and S2 are sets of non te rm ina l symbols define the product of SI and S2. Si * S2, by