Context-Free Grammar CSCI-GA.2590 – Lecture 3 - PowerPoint Presentation

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Context-Free GrammarCSCI-GA.2590 – Lecture 3

Ralph Grishman

NYU

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A Grammar FormalismWe have informally described the basic constructs of English grammarNow we want to introduce a formalism for representing these constructsa formalism that we can use as input to a parsing procedure1/16/14NYU2

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Context-Free GrammarA context-free grammar consists ofa set of non-terminal symbols A, B, C, … ε Na set of terminal symbols a, b, c, … ε Ta start symbol S ε Na set of productions P of the form N (N U T)*1/16/14NYU3

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A Simple Context-Free GrammarA simple CFG:S  NP VPNP  catsNP  the catsNP  the old catsNP  miceVP  sleepVP  chase NP

1/16/14NYU4

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Derivation and LanguageIf A  β is a production of the grammar, we can rewrite α A γ  α β γ A derivation is a sequence of rewrite operations  …  …

 NP VP  cats VP  cats chase NPThe language generated by a CFG is the set of strings (sequences of terminals) which can be derived from the start symbol

S  …  …

 T* S NP VP

cats

VP



cats chase NP



cats chase mice

1/16/14

NYU

5

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PreterminalsIt is convenient to include a set of symbols called preterminals (corresponding to the parts of speech) which can be directly rewritten as terminals (words)This allows us to separate the productions into a set which generates sequences of preterminals (the “grammar”) and those which rewrite the preterminals as terminals (the “dictionary”)1/16/14NYU6

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A Grammar with Preterminalsgrammar:S  NP VPNP  NNP  ART NNP  ART ADJ NVP  VVP  V NPdictionary:

N  catsN  miceADJ  oldDET

 theV

 sleepV 

chase

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NYU

7

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Grouping AlternatesTo make the grammar more compact, we group productions with the same left-hand side:S  NP VPNP  N | ART N | ART ADJ NVP  V | V NP1/16/14NYU8

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A grammar can be used togeneraterecognizeparseWhy parse?parsing assigns the sentence a structure that may be helpful in determining its meaning1/16/14NYU9

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vs Finite State LanguageCFGs are more powerful than finite-state grammars (regular expressions)FSG cannot generate center embeddingsS  ( S ) | xeven if FSG can capture the language, it may be unable to assign the nested structures we want

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A slightly bigger CFGsentence  np vpnp  ngroup | ngroup ppngroup  n | art n | art adj nvp  v | v

np | v vp | v np pp (auxilliary)pp  p np

(pp = prepositional phrase)

1/16/14

11

NYU

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AmbiguityMost sentences will have more than one parseGenerally different parses will reflect different meanings …“I saw the man with a telescope.”Can attach pp (“with a telescope”) under np or vp1/16/14NYU12

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A CFG with just 2 nonterminalsS  NP V | NP V NPNP  N | ART NOUN | ART ADJ Nuse this for tracing our parsers1/16/1413NYU

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Top-down parserrepeatexpand leftmost non-terminal using first production (save any alternative productions on backtrack stack)if we have matched entire sentence, quit (success)if we have generated a terminal which doesn't match sentence, pop choice point from stack (if stack is empty, quit (failure))1/16/14NYU14

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Top-down parser1/16/14NYU150: Sthe cat chases mice

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Top-down parser1/16/14NYU160: S1: NPthe cat chases mice2: V

backtrack table

0: S 

NP V NP

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Top-down parser1/16/14NYU170: S1: NP3: Nthe cat chases mice

2: V

backtrack table

0: S



NP V NP

1:NP



ART ADJ N

1: NP



ART N

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Top-down parser1/16/14NYU180: S1: NP3: ARTthe cat chases mice

4: N2: V

backtrack table

0: S



NP V NP

1:NP



ART ADJ N

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Top-down parser1/16/14NYU190: S1: NP3: ARTthe cat chases mice

4: ADJ2: V5: N

backtrack table

0: S



NP V NP

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Top-down parser1/16/14NYU200: S1: NPthe cat chases mice2: V

3: NP

backtrack table

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Top-down parser1/16/14NYU210: S1: NP3: Nthe cat chases mice

2: V3: NP

backtrack table

1:NP



ART ADJ N

1: NP



ART N

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Top-down parser1/16/14NYU220: S1: NP4: ARTthe cat chases mice

2: V3: NP5: N

backtrack table

1:NP



ART ADJ N

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Top-down parser1/16/14NYU230: S1: NP4: ARTthe cat chases mice

2: V3: NP5: N

6: N

parse!

backtrack table

1:NP



ART ADJ N

3: NP



ART ADJ N

3: NP



ART N

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Bottom-up parserBuilds a table where each row represents a parse tree node spanning the words from start up to end1/16/14NYU24symbolstart

endconstituentsN01-

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Bottom-up parserWe initialize the table with the parts-of –speech of each word …1/16/14NYU25symbolstartendconstituents

ART01-N12

-

V23

-

N

3

4

-

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Bottom-up parserWe initialize the table with the parts-of –speech of each word …remembering that many English words have several parts of speech1/16/14NYU26symbolstart

endconstituentsART01-N

1

2-V

2

3

-

N

2

3

-

N

3

4

-

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Bottom-up parserThen if these is a production AB C and we have entries for B and C with endB = startC, we add an entry for A with start = startB and end = endC

[see lecture notes for handling general productions]

1/16/14

NYU

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node #

symbol

start

end

constituents

0

ART

0

1

-

1

N

1

2

-

2

V

2

3

-

3

N

2

3

-

4

N

3

4

-

5

NP

0

2

[0, 1]

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Bottom-up parser1/16/14NYU28node #symbolstart

endconstituents0ART01-

1N

12

-

2

V

2

3

-

3

N

2

3

-

4

N

3

4

-

5

NP

0

2

[0, 1]

6

NP

1

2

[1]

7

NP

2

3

[3]

8

NP

3

4

[4]

9

S

0

4

[5, 2, 8]



10

S

1

4

[6, 2, 8]

several more S’s

parse!