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

ÆÆb) The probabilistic left-corner relation 11 +ÆƸ¸ 99 0¸ £9£ òu óò£…06 t§N u t§N u 2t§N|£££6¬t) ¸ =.+ø,??iff?CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 1356.4.5.1 Prediction loopsAs an example, consider the following simple left-recursive SCFG.tówhere u 6 1 L§t . Non-probabilistically, the prediction loop at position 0 would stop after producing the states9%09 ¸09 ¸This would leave the forward probabilities at£9% £,0£…06 t,0£j9%06 u corresponding to just two out of an infinity of possible paths. The correct forward probabilities are obtainedas a sum of infinitely many terms, accounting for all possible paths of length 1.¸¸ 09 09,0¸091 L u 016 1,0u N u 2N u 3N|£££6 u ,1 L u 0£j9%0616¬t¸ 1uIn these eacht sums corresponds to a choice of the first production, each to a choice of the second production.u09If we didn’t care about finite computation the resulting geometric series could be computed by letting theprediction loop (and hence the summation) continue indefinitely.Fortunately, all repeated prediction steps, including those due to left-recursion in the productions,can be collapsed into a single, modified prediction step, and the corresponding sums computed in closed form.For this purpose we need a probabilistic version of the well-known parsing concept of a left corner, which isalso at the heart of the prefix probability algorithm of Jelinek & Lafferty (1991).Definition 6.5 The following definitions are relative to a given SCFG{.a) Two nonterminals ) and= are said to be in a left-corner relation ) ¸ =for )that has a RHS starting with=,there exists a productiondefined as the total probability of choosing a production for )that has= as a left corner:’6£is the matrix of probabilities +-, ) ¸ L=ë0 ,,{§0+-,+9,) ¸ y ”'’‘/“ =ë0&6) ¸ =“.B0£11 If a probabilistic relation”is replaced by its version”œ set-theoretic , i.e.,used here reduce to their traditional discrete counterparts; hence the choice of terminology.§¨+•‡Q–—”œthen the closure operationsiff”io§¨˜•‡š0,