The dissertation of Andreas Stolcke is approved: University of ...

for all pairs of states d =¸+= ŠÙ…Ù´ ´Ù+= Ḿ© ´Ù…Ù¸¸ 99 9W9 ,òu óCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 152is non-zero. Then@>£ and6)¸.M£j>Θ in the chart, such that the unit-productionrelation ©,> CE=-0 C©=-0The first summation is carried out once for each state :6) ¸,>applied for each choice of>, but only if ) ¸ .t>ΘRationale. This increments ḾÙ…Ù the equivalent of ©, whereas the second summation isnot a unit production..—£j>͘times, accounting for the infinity of C«=ë0 ,>issurroundings which= in can occur if it can be derived through cyclic productions. Note that the computationis unchanged, sinceŠÙ…Ù already includes an infinity of cyclically generated subtrees for=, whereof´Ùappropriate.6.5.3 Parsing bracketed inputsThe estimation procedure described above (and EM-based estimators in general) are only guaranteedto find locally optimal parameter estimates. Unfortunately, it seems that in the case of unconstrained SCFGestimation local maxima present a very real problem, and make success dependent on chance and initialconditions (Lari & Young 1990). Pereira & Schabes (1992) showed that partially bracketed input samplescan alleviate the problem in certain cases. The bracketing information constrains the parse of the inputs, andtherefore the parameter estimates, steering it clear from some of the suboptimal solutions that could otherwisebe found.A second advantage of bracketed inputs is that they potentially allow more efficient processing,since the space of potential derivations (or equivalently, Earley paths) is reduced. It is therefore interesting tosee how any given parser can incorporate partial bracketing information. This is typically not a big problem,but in the case of Earley’s algorithm there is a particularly simple and elegant solution.Consider again the grammartróthat any candidate parse has to be consistent with, e.g., there cannot be a parse that has a constituent spanningthe first and second , or the third and fourth. The supplied bracketing can be nested, of course, and need notbe complete, i.e., within a bracketing there are still potentially several ways of parsing a substring.The Earley parser can deal efficiently with partial bracketing information as follows. A partiallybracketed input is processed as usual, left-to-right. When a bracketed portion is encountered, the parserinvokes itself recursively on the substring delimited by the pair of parentheses encountered. More precisely:òA partially bracketed input for this grammar would be 07 . The parentheses indicate phrase boundaries¡The recursive parser instance gets to see only the substring as input.