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

Æsums of inner probabilitiesŠ,6)¸CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 1476.4.10 SummaryTo summarize, the modified, probabilistic Earley algorithm works by executing the following stepsfor each input position.Apply a single prediction step to all incomplete states in the current state set. All transitive predictions¡are subsumed by consulting the left-corner ©§ matrix .Forward probabilities are computed by multiplying old Æ ’s with rule probabilities. Inner probabilitiesare initialized to their respective rule probabilities.A single scanning step applied to all states with terminals to the right of the dot yield the initial elements¡for the next state set. If the next set remains empty (no scanned states) the parse is aborted.Forward and inner probabilities remain unchanged by scanning.The sum of all forward probabilities of successfully scanned states gives the current prefix probability.Apply iterative completion (highest start index first, breadth-first) to all states, except those correspondingto unit productions. Unit production cycles are subsumed by consulting the © matrix¡.Forward and inner probabilities are updated by multiplying old forward and inner probabilities with theinner probabilities of completed expansions.The probabilities that nonterminals )generate particular substrings of the input can be computed as.—£ 0After processing the entire string in this way, the sentence probability can be read off of either theorŠof the final state.6.5 ExtensionsThis section discusses extensions to the Earley algorithm that go beyond simple parsing and thecomputation of prefix and string probabilities. These extension are all quite straightforward and well-supportedby the original Earley chart structure, which leads us to view them as part of a single, unified algorithm forsolving the tasks mentioned in the introduction.6.5.1 Viterbi parsesDefinition 6.8 A Viterbi parse for a string , in a grammar{, is a left-most derivation that assigns maximal1probability to 1 , among all possible derivations for 1 .Both the definition of Viterbi parse, and its computation are straightforward generalizations of thecorresponding notion for Hidden Markov Models (Rabiner & Juang 1986), where one computes the Viterbi