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©) 6#=©,©,©,,ÆNNö,²NL++and>+?1£Ù ŠÙCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 136c) The ) relationthere is nonterminal> a such ) ¸ that >defined as the reflexive, transitive closure of ) ¸ ›=,=.i.e., ) iff CÀ›=C/ =ë0 is defined as a seriesCÀ›= isd) The probabilistic reflexive, transitive left-corner relation ©§6|©,{§0 is a matrix of probability sumsor C . Each © CÀœ=ë0 )) +-, ) C=ë0c6)Ð6¢=-0+-,>1 ¸¡) ¸ =90+9,N yvž1) ¸ Ÿ>10=ë0+-,>1 ¸¡+-,>2 ¸›+-,ž1¢ž2+-,y) ¸ Ÿ>10=ë0N/£££Alternatively, © is defined by the recurrence relation ) C=ë0'6+-,) ¸ )g=ë020and zero otherwise).( denotes the Kronecker delta, defined as unity if 6#=, )öThe ©¥ recurrence for can be conveniently written in matrix notation>!0_© C ,>=ë0from which the closed-form solution is derived:© 6 © An existence proof for ©¥ is given in Section 6.4.8. Appendix 6.6.3.1 shows how to speed up the computationof © by inverting only a reduced version of the matrix ² L .© 6 0,;²The significance of the matrix © for the Earley algorithm is that its elements are the sums ofleading from a state6)¸ predicted state £j@, via any number of intermediate states.the probabilities of the potentially infinitely many prediction paths.M£j>Í˜ to a¸ can be computed once for each grammar, and used for table-lookup in the following, modified©prediction step.Prediction (probabilistic, transitive),> for all productions=¸@such that ©H :6) ¸.—£>Î˜Æ)Šóò6QC:¸ @=C¬£=ã0 is non-zero. Then£j@ò…ÆóH@E0 (6.3)+-,= ¸Ù += Æ © C ,>=90@E0 (6.4)Note the © newof all path linking> to=.For>Ð6n= probabilities this covers the case of a single step of prediction;ŠÙ 6¸ factor in the updated forward probability, which accounts for the sum CTF=ë0,>+-,= 1 always, since ©¥ is defined as a reflexive closure. CÀŸ=-0Œ‹,=
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.
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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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4@@@@-@b@6@˜--@@@0@@@@@CHAPTER 2.
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6tt,u ·¥¸¹u ,10ºtu ,2 10Yt ¸
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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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It, u¦¸¹u Ù 0w6¬tt,_, u Ù 0
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, uu!¸¹u Ù 0c6,u ,ÔÔ0 ö1 ö1
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CHAPTER 3. HIDDEN MARKOV MODELS 32T
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CHAPTER 3. HIDDEN MARKOV MODELS 34R
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4ÿ= ê•4TÃE0&Ò¢¡? •ç1 Lht
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6666ò U1ò +9,9. 4 20+-,¡ . 4 10C
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2. For each candidate"I!computeLet"
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6\“ç%&ät\“ç tè ä, u¦¸¹u
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, u1 ¸¼u Ù 0 and , u3 ¸¹u Ù 0
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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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¸= ¸= ¸.1.2¸¸) 1_) 20&6#=,,,,
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CHAPTER 4. STOCHASTIC CONTEXT-FREE
Page 98 and 99: CHAPTER 4. STOCHASTIC CONTEXT-FREE
Page 104 and 105: ==Ì==ÌCHAPTER 4. STOCHASTIC CONTE
Page 106 and 107: ,= ===I¸theybÜ„thiscg„\\ ¸¸
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 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 156 and 157: 9 itself²²NN++9 ¸0ÌLL++??1£,=C
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
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BIBLIOGRAPHY 186DAGAN, IDO, FERNAND
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BIBLIOGRAPHY 188——, & ——. 1
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BIBLIOGRAPHY 190QUINLAN, J. ROSS, &
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BIBLIOGRAPHY 192WALLACE, C. S., & P
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