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

The t?£9 £x¸¸9¸9 x£ òx9òòó u1ó ??1u1t 6 1¥Š"6TtM£¥Š1 factors are a result of the left-corner N u N u 2sum,?1 L u 0 1 N|£££61 : 0¸£ÆÆÆÆ??Æ???1u Š£x9 production.CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 1376.4.5.2 Completion loopsAs in prediction, the completion step in the Earley algorithm may imply an infinite summation, andcould lead to an infinite loop if computed naively. However, only unit productions 12 can give rise to cycliccompletions.The problem is best explained by studying an example. Consider the grammartówhere u 61 LËt . Presented with the input (the only string the grammar generates), after one cycle ofprediction, the Earley chart contains the following states.0 : 090 : 0¸ £96 1¤Š6 11u ¥Š6 u0 : 0x1Æ6¬t0 : 09¸Æ6¬t¸Æ66/t1 .¸is completed, yielding a complete state 0xtocomplete state for9, etc. The non-probabilistic Earley parser can just stop here, but as in prediction, this¸, which allows 9%£ ¸ 09be completed, leading to anotherwould lead to truncated probabilities. The sum of probabilities that needs to be computed to arrive at thecorrect result contains infinitely many terms, one for each possible loop thex¸through Eachsuch loop adds a factor of to the forward and inner probabilities. The summations for all completed statesuturn out as£j9After scanning¸ 09£… , completion without truncation would enter an infinite loop. First 0x¸1 : 0911 : 0xu NÜt u 2N £££6 16 1Šã6/t¸9%£t§NÜt u NËt u 2N|£££ 0&6¬t6/t1u ,1 : 096/t§N t u NËt u 2NÉ£££6 1)Šã6/t§NÜt u NÜt u 2N|£££6 11u Š¸6¬t§Ngt9%£t§NÜt u Ngt u 2N|£££ 0&6 uu ,1u ,xÍ£6¬ttëNgt u NÜt u 2N{£££ 0&6/tThe approach taken here to compute exact probabilities in cyclic completions is mostly analogousto that for left-recursive predictions. The main difference is that unit productions, rather than left-corners,form the underlying transitive relation. Before proceeding we can convince ourselves that this is indeed theonly case we have to worry about.¸612 Some authors refer to unit productions as chain productions.