Left-Corner Grammar Transform
and
Incremental Parsing
Aria Haghighi
Grammar Transform
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A CFG G
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Nonterminals N
Terminals T
Productions (Rules) P
A grammar Transform
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:G ! G’
Parsing Strategies
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Bottom-Up
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Top-Down
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e.g. CKY
Inefficient and may not terminate
Left-Corner
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Recognize Left-Child predict parent
Left-Corner’s Arch-Nemesis
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Left recursion
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X !* X 
Nontermination
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Recognize X starting at position 1
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Recognize X starting at position 1
Recognize X starting at position 1
Recognize X starting at position 1
…..
Old School
Left-Recursion Removal
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Due to Hopcraft and Ullman
For
A ! 1 | … | m
A ! A 1 | … | A k
(i  A )
(i  )
Add
A ! 1 | 1 A’|…|m | m A’
A’ ! 1 | 1 A’ | …. | k | k A’
Problem with the old school
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It’s a big school!!
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Can have exponential
Blowup in number of productions
PCFG Estimation
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Lots of sparse rules
Select Left-Corner Transform
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Left-corner productions L
Add states D-X
Example Transform
Keeping it Small
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Do we need each D-X ?
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Only if D !L* X 
How do we choose L ?
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Need all left recursive rules
Suffice to have rules where
first child is a non-terminal
Binarizing the Output
For Top-Down Rules: (td)
For Left-Corner Rules: (lc)
Grammar Sizes
Incremental Parsing
Incremental Parsing
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Operate on flattened selective leftcorner algorithm
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Partial Parses include left-chain
with heuristic filtering
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Beam Search