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

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9 itself²²NN++9 ¸0ÌLL++??1£,=CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 144c){ has no useless nonterminals iff all nonterminals )appear in at least one derivation of some string0.It is useful to translate consistency into ‘process-oriented’ terms.We can view an SCFG as astochastic string-rewritingprocess, in which each step consists of simultaneously replacing all nonterminals ina sentential form with the right-hand sides of productions, randomly drawn according to the rule probabilities.vBooth & Thompson (1973) show that the grammar is consistent if and only if the probabilitythat stochastic.rewriting of the symbol9 start leaves nonterminals remaining ¢ after steps, goes to 0 ¢ asloosely speaking, rewriting9 has to terminate after a finite number of steps with probability 1, or else the¸±°grammar is inconsistent.MoreWe observe that the same property holds not only for9, but for all nonterminals, if the grammarhas no useless terminals. If any )nonterminal admitted infinite derivations with non-zero probability, thenwould have such derivations, since by ) assumption is from9 reachable with non-zero probability.series converge:To prove the existence of © and © , it is sufficient to show that the corresponding geometricwith non-zero probability, i.e., +-,9 CE.>)"˜3Í1 IC1+ 21© 6ãN,2² N|£££6
,,?9 ¸9W9u ó,,,,,CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 145is inconsistent for any choice of u ‹1for all u Ï 1, namely , 1 L u 021 6Ét?, but the left-corner relation (a single number in this case) is well-defined1 . Informally speaking, in these cases the left fringe of a derivation isguaranteed to result in a terminal after finitely many steps, but in the remainder of the derivation nonterminalsare generated at a faster rate than are they are replaced by terminals.SCFG consistency will again play an important role in Chapter 7 where the computation of variousglobal quantities such ¢ as -gram expectation, derivation entropy, etc., for a given SCFG will be discussed.ò6.4.9 Complexity issuesThe probabilistic extension of Earley’s parser preserves the original control structure in most aspects,the major exception being the collapsing of cyclic predictions and unit completions, which can only makethese steps more efficient.We can therefore apply the complexity analysis from Earley (1970) essentiallyunchanged. Below we repeat the highlights, together with proof outlines. We also analyze the dependenceon the size of the grammar, and compare the result with the other known algorithms for SCFGs.The key factor in upper-bounding both time and space complexity in Earley’s algorithm is themaximal number of states created in each state set, for each input position. An Earley state combines aproduction, a dot position, and a start index. Productions and dot positions combine to give a number thatequals the sum of the lengths of all productions, which is roughly the total ‘size’ of the grammar. For fullyparameterized CNF grammars the number of dotted rules ð is(A fully parameterized CNF grammar is one in which each triple of )g=Œ|> nonterminals forms a production¸ )of the input.=>of30 , where ¢ is the number of nonterminals.¢non-zero probability.) In both cases the start index contributes a factor of at most G , the length6.4.9.1 Scaling with input lengthTo determine the complexity in terms of G we note that during prediction and scanning, each stateis processed exactly once, performing operations that depend only on the size of the grammar, not G ; boththerefore ð GY0 take . During completion, the worst case is obtained if each of ð GY0 the states is the result ofcompleting predecessors (with dot positions further left) from all possible previous positions. The total timetaken here is thus ðG3 0 total runtime over the entire string.G2 0 . The completion step thus dominates the computation time, and gives ðEarley (1970) identifies several important classes of context-free grammars on which the algorithmruns faster without special modifications. Grammars with no or bounded ambiguity result in completions thathave to combine at most a fixed number of previous states (combining completions correspond to coalescingmultiple parses for substrings). Such a completion step therefore takes time ðG2 0 .CFGs that can be processed deterministically, i.e., where the correct choice of rule can be determined usingGY0 , to give a total of ðonly the history of the parser and a bounded lookahead into the input, such as LR(") grammars, result in an
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The dissertation of Andreas Stolcke
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Bayesian Learning of Probabilistic
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iAcknowledgmentsLife and work in Be
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iiiContentsList of FiguresList of T
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CONTENTSv4.5.4 Summary and Discussi
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CHAPTER 1. INTRODUCTION 2Instance-b
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CHAPTER 1. INTRODUCTION 4A.0.830.33
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CHAPTER 1. INTRODUCTION 6the ¨ 0 l
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..1 £££1; 450,1 £££1; 450CHAP
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VU=@U@@=U===UCHAPTER 2. FOUNDATIONS
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,,vv,v,v,,directly. However, note t
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CHAPTER 2. FOUNDATIONS 18As more da
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CHAPTER 2. FOUNDATIONS 20Global mod
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CHAPTER 2. FOUNDATIONS 22¡An expli
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ÊS==66@N,ÆÆ=NÆ00ÆÊ=S=N0Æ=#@0
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666CHAPTER 2. FOUNDATIONS 262.5.7 P
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CHAPTER 3. HIDDEN MARKOV MODELS 32T
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CHAPTER 3. HIDDEN MARKOV MODELS 34R
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2. For each candidate"I!computeLet"
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CHAPTER 3. HIDDEN MARKOV MODELS 46l
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CHAPTER 3. HIDDEN MARKOV MODELS 48c
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CHAPTER 3. HIDDEN MARKOV MODELS 50d
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CHAPTER 3. HIDDEN MARKOV MODELS 520
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correlation between initial and fin
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,CHAPTER 3. HIDDEN MARKOV MODELS 56
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,CHAPTER 3. HIDDEN MARKOV MODELS 60
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CHAPTER 3. HIDDEN MARKOV MODELS 62b
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CHAPTER 3. HIDDEN MARKOV MODELS 64t
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CHAPTER 3. HIDDEN MARKOV MODELS 66t
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CHAPTER 3. HIDDEN MARKOV MODELS 68s
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CHAPTER 3. HIDDEN MARKOV MODELS 74b
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domain. 3 In short, we will leave o
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,,,,,£CHAPTER 4. STOCHASTIC CONTEX
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9 ¸)Ô ¸ 9 ¸Ô 1 2 £££;,ÔÔC
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CHAPTER 4. STOCHASTIC CONTEXT-FREE
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==Ì==ÌCHAPTER 4. STOCHASTIC CONTE
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Page 108 and 109: CHAPTER 4. STOCHASTIC CONTEXT-FREE
Page 116 and 117: 104Chapter 5Probabilistic Attribute
Page 118 and 119: ,1makingandCHAPTER 5. PROBABILISTIC
Page 120 and 121: CHAPTER 5. PROBABILISTIC ATTRIBUTE
Page 134 and 135: 122Chapter 6Efficient parsing with
Page 136 and 137: 1z1CHAPTER 6. EFFICIENT PARSING WIT
Page 138 and 139: and each state in set ( -ÏH ):6)
Page 140 and 141: NPDetVTVI P CHAPTER 6. EFFICIENT PA
Page 142 and 143: CHAPTER 6. EFFICIENT PARSING WITH S
Page 144 and 145: ) In particular, the string probabi
Page 146 and 147: H : d=:6) ¸ÆÆÆ:6) ¸ .V=i£Ù )
Page 148 and 149: ©) 6#=©,©,©,,ÆNNö,²NL++and>+
Page 150 and 151: ) The probabilistic unit-production
Page 152 and 153: ¸0 ¸ 29¸¸ 99W9 [;t£ ? 1 ?1u 6
Page 154 and 155: The forward and inner probabilities
Page 158 and 159: ,,by nonterminals. Multiplying this
Page 160 and 161: 6CHAPTER 6. EFFICIENT PARSING WITH
Page 162 and 163: description. Again, we ignore this
Page 164 and 165: for all pairs of states d =¸+= Š
Page 166 and 167: ,9 ¸0 : 0¸ £j9¸ A )z9£CHAPTER
Page 170 and 171: are then summed over all nontermina
Page 172 and 173: +CHAPTER 6. EFFICIENT PARSING WITH
Page 174 and 175: 1CHAPTER 6. EFFICIENT PARSING WITH
Page 178 and 179: = ¸Let t,t) ¸ .V=i£,t,t,6666yyyy
Page 180 and 181: 168Chapter 7-grams from Stochastic
Page 182 and 183: CHAPTER 7. -GRAMS FROM STOCHASTIC
Page 184 and 185: )ÅÆÅÅÅÆÅÅÅÆÅÅÅÆÅÅÅ
Page 186 and 187: -grams CCHAPTER 7. -GRAMS FROM ST
Page 188 and 189: ,?Ó,tÌ?L A0,I 1N I N A A 2 A 3 N
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Page 192 and 193: Consider the following problem: sta
Page 194 and 195: CHAPTER 8. FUTURE DIRECTIONS 1828.2
Page 196 and 197: 184BibliographyAHO, ALFRED V., RAVI
Page 198 and 199: BIBLIOGRAPHY 186DAGAN, IDO, FERNAND
Page 200 and 201: BIBLIOGRAPHY 188——, & ——. 1
Page 202 and 203: BIBLIOGRAPHY 190QUINLAN, J. ROSS, &
Page 204: BIBLIOGRAPHY 192WALLACE, C. S., & P
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