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Synchronous Context-Free Grammars
and Optimal Linear Parsing Strategies
Daniel Gildea Giorgio SattaUniversity of Rochester Università di
Padova
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Synchronous CFG
Context-free Grammar:
X → A B
Synchronous Context-free Grammar (SCFG)
X →A1
B2
C3
D4 , C 3 A 1 D 4 B 2
C →Powell, 鲍威尔
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Synchronous CFG
• Synchronous parsing: find tree from two strings
– used to learn grammar from parallel text
• This talk: parsing strategies for long rules
• Results also apply to translation with n-gram
language model
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Context-Free Grammar
A → B C
B
C
A
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Binary SCFG
A → B1
C2 , C 2 B 1
B
C
A
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SCFG with 4 nonterminals
A → B1
C2
D3
E4 , C 2 E 4 B 1 D 3
E
D
C
B
A
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Fan-Out
Number of spans in nonterminal.
CFG: fan-out 1 BC
A
SCFG: fan-out 2 ED
C
B
A
ϕ(G) = maxN∈G
ϕ(N) (Rambow & Satta, 1999)
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Rank
Number of nonterminals on righthand side of rule.
CFG: rank 2 BC
A
SCFG: rank r ED
C
B
A
ρ(G) = maxP∈G
ρ(P)
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Parsing Strategies
Reduce rankE
D
C
B
A
A → B C D E
C
B
X
D
X
Y
E
Y
A
X → B C Y → X D A → Y E
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Parsing Strategies
Reduce rank, may increase fan-out
E
D
C
B
A
C
B
X
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Rule Length in Synchronous CFG
• Binary grammar (ITG): parsing is O(n6) (Wu, 1997)
– Works in real MT (Zhang et al. 2006)
• Many rules cannot be binarized without
increasing fan-out (Aho and Ullman, 1972)
• Fan-out affects space and time complexity
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Parsing Complexity
Space complexity: O(n2ϕ(A))
Time complexity: O(nϕ(A)+ϕ(B)+ϕ(C))
B
C
A
B
C
A
O(n2) space O(n4) space
O(n3) time O(n6) time
(Seki et al. 1991)
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SCFG Parsing Strategies
E
D
C
B
A
C
B
X
naïve strategy: O(n2r+2) time
best strategy: Ω(ncr ) for some c
(Gildea and Štefanković 2007)
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This Talk
• Finding optimal space complexity is
NP-complete
• Finding optimal time complexity ⇒ better algs
for treewidth
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Example Rule
B8
B7
B6
B5
B4
B3
B2
B1
A
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Optimal Parsing Strategy
n7
n5
B1
n3
B2
n1
B3
B4
n6
B5
n4
B6
n2
B7
B8
B4
B3
n1
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Carving Width
2 3 4
1
G
1 2 3 4
tree layout of G
Carving width: max number edges of G routed
through tree layout
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Cyclic Permutation Multigraph
B1
B2
B3
B4
B5
B6
B7
B8A
A → B1B
2B
3B
4B
5B
6B
7B
8 ,
B5B
7B
3B
1B
8B
6B
2B
4
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Carving Width = Space Complexity
A
n7
n5
n3
n1
n6
n4
n2
B1
B2
B3
B4
B5
B6
B7
B8
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Our Reduction
• Carving width instance: (G, k )
• Construct permutation multigraph G′, integer k ′
• Carving width of G ⇔ Carving width of G′ ⇔
optimal parsing for SCFG
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Our Construction
2 3 4
1
G
1 2 3 4
tree layout of G
X1
G1
X2
G2
X3
G3
X4
G4
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G1
X1
G2
X2
G3
X3
G4
X4
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Space Complexity
Theorem 1: Finding the parsing strategy with optimal
space complexity for an SCFG rule is NP-complete
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Treewidth
A C E G I K M
B D F H J L
N
P
R
O
Q
S
CDE DEF EFG FGH GHI HIJ IJK
BCD GHN JK L
ABC HNO K LM
NOP
OPQ PQR QRS
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Dependency Graph
x0 x1 x2 x3 x4
y0 y1 y2 y3 y4
x0 x1 x2 x3 x4
A → B C D E S → A1B
2C
3D
4 , B 2 D 4 A 1 C 3
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Treewidth = Time Complexity
x0 x1 x2 x3 x4
x0x1x2 x0x2x3 x0x3x4
A → B C D E
C
B
X
D
X
Y
E
Y
A
X → B C Y → X D A → Y E
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Our Reduction
• Treewidth instance: (G, k )
• Construct dependency graph G′, integer k ′
• Approx of treewidth of G ⇔ Treewidth of G′ ⇔
optimal time complexity for SCFG
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Dependency Graph Construction
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Approximation Algorithm for Treewidth
SOL < 8∆(G)(OPT + 1) .
SOL: solution using SCFG parsing strategy
OPT : optimal treewidth of input graph G
∆(G) = degree (max num edges touching one vertex)
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Time Complexity
Theorem 2: Finding the parsing strategy with optimal
time complexity for an SCFG rule implies a
∆(G)-factor approximation algorithm for treewidth.
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Time Complexity
Theorem 3: If finding the parsing strategy with
optimal time complexity for an SCFG rule is
NP-complete, then treewidth for graphs of degree 6 is
NP-complete.
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Conclusion
• Finding parsing strategy with best space
complexity is NP-hard.
• P-time alg for finding parsing strategy with best
time complexity implies better approximation
algs for treewidth
• NP-hardness for time complexity implies
NP-hardness for treewidth of graphs of degree
six
