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

= ¸Let t,t) ¸ .V=i£,t,t,6666yyyy¸ Æ—@_,6)1yt,“ ·“ ·Õ· “Æ—@ è ,>è1tÖ0 Ȩ½£=ï1CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 166to show that (6.5)) ¸@B0 be the probability for LR item ) ¸@(with a dot somewhere in the RHS). We want.—£ ˜M0) ¸position of the last input symbol processed; a reduce action of the parser effectively resets H to the beginningof the reduced nonterminal.The computation of LR item sets begins with the initial item £j9, whichthereby agreeing with (6.5).¸The first operation for constructing item sets is closure, whereby for each item ) ¸has t 6 1 by definition,.—£=¤˜ , all items.—£ ˜M0%6@A?for any item ) ¸ 0å…å…å 10.—£ ˜ , regardless of position H and start index". Note that H is not always equal to theC+-,9£j@corresponding to the available productions=¸@are added to the set. This operation is recursiveand corresponds obviously to Earley’s prediction step. Also, the way in whicht values are propagated followsexactly the way forward probabilities are handled during prediction. (The left-corner relation ©ë could beused to compute closure probabilities exactly, but Wright suggests using a truncated recursion instead.) Sinceclosure and prediction are thus isomorphic, and since the prefix relative to the items does not change, (6.5)also remains valid during this step.Finally, a successor set ² Ù of kernel items is constructed from an existing closed set ² in whatcorresponds to Earley’s scanning or completion. Specifically, ) ¸forI ²each current item , an itemis placed in , reachable by scanning a or reducing (completing) a nonterminal=..—£=§˜terminal= ² Ù¸ .V=i£ )let= (We stand for either terminal or nonterminal to treat both cases jointly.) The new item probability iscomputed as) ¸ .V=i£) ¸(6.6)˜.—£=ë˜M0˜M0&6,> Ȩ½£=ï 0This can be understood as scaling the total probability of items matching= to unity.By substituting (6.5) into (6.6) we get“ ·ž'¼å‘îÇ ^ Õ ”'/å‘KÖ0¿j¿j¿^ˆ‡1 H¾zØM0c6Æ @ ,>“ · C+-,90å…å…å@A?10Æ @ ,) ¸.M£=§˜M0ž'¼å‘î0ž'¼å‘îÆ—@;,> ¸~½£=ïÆ @ ,) ¸.M£=§˜M0Š@ è ,=ë00Š@ èè ,) ¸ .V=i£Æ—@˜M0(6.7)ž'¼å‘î,=ë0Æ—@;,> ¸~½£=ï0Ȩ½=Œ£ ïî@ è ,) ¸ .V=i£Æ˜M0(6.8)@ è ?ž'¼ ‘å+-,9 10C 0å…å…å