The Real Story on Synchronous Rewriting Systems. Daniel Gildea Computer Science Department University of Rochester

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1 The Real Story on Synchronous Rewriting Systems Daniel Gildea omputer Science Department University of Rochester

2 Overview Factorization Tree Decomposition Factorization of LFRS

3 YK Parsing w 2 : [, x 1, x 2 ] w 1 : [, x 0, x 1 ] w 0 w 1 w 2 : [, x 0, x 2 ]

4 inarization of FG E D D E X D X Y E Y X Y X D Y E O(n r+1 ) O(n 3 ) (homsky Normal Form)

5 Rule factorization for bilexicalized parsing H H O(n 5 ) O(n 4 ) (Eisner and Satta, 1999)

6 Synchronous ontext-free Grammar (SFG) (1) (2) D (3) E (4), (2) E (4) (1) D (3) E D X X D Y Y E

7 Grammar Factorization Factor rule into intermediate rules Lowers parsing complexity onsiders rules independently

8 Grammar Factorization FG: O(n r+1 ) O(n 3 ) ilex: O(n 5 ) O(n 4 ) SFG: O(n 2r+2 ) O(n cr ) Goal: find best factorization for large rules: Machine translation (Galley et al., 2004) Non-projective parsing (Kuhlman and Nivre, 2006; Gildea, 2010)

9 Tree Decomposition DE DEF EFG FGH GHI HIJ IJK E G I K M D F H J L N O P Q R S D GHN JKL HNO KLM NOP OPQ PQR QRS every edge in some cluster clusters containing any vertex are connected

10 Tree Decomposition for 3ST DE DEF EFG FGH GHI HIJ IJK E G I K M D F H J L N O P Q R S D GHN JKL HNO KLM NOP OPQ PQR QRS variable in 3ST node in graph appear in same clause in 3ST edge in graph

11 Tree Decomposition for Dynamic Programming DE DEF EFG FGH GHI HIJ IJK E G I K M D F H J L N O P Q R S D GHN JKL HNO KLM NOP OPQ PQR QRS every edge in some cluster can apply constraint in cluster clusters containing any vertex are connected variable has consistent value in solution

12 D E Factor Graph for FG w 4 : E c w 3 : D w 2 : w 1 : w 1 w 2 w 3 w 4 x 0 x 1 x 2 x 3 x 4 c : (Gildea, omputational Linguistics, 2010)

13 FG inarization Deduction Rule: Factor Graph: w 4 : E w 3 : D w 2 : w 1 : c : c w 1 w 2 w 3 w 4 x 0 x 1 x 2 x 3 x 4 Dependency Graph: Tree Decomposition: x 0 x 1 x 2 x 3 x 4 x 0 x 1 x 2 x 0 x 2 x 3 x 0 x 3 x 4

14 ilexicalized Parsing Deduction Rule: Factor Graph: w l : c w 2 : w 1 : w 1 w l w 2 c : x 0 h x 1 m x 2 Dependency Graph: Tree Decomposition: h m x 0 x 1 x 2 hx 1 mx 2 x 0 hx 1 x 2

15 SFG Parsing Deduction Rule: Factor Graph: c w 4 : E w 3 : D w 2 : w 1 : c : y 0 y 1 y 2 y 3 y 4 w 1 w 2 w 3 w 4 x 0 x 1 x 2 x 3 x 4 Dependency Graph: y 0 y 1 y 2 y 3 y 4 Tree Decomposition: x 0 x 3 x 4 y 0 y 2 y 3 y 4 x 0 x 1 x 2 x 3 x 4 x 0 x 2 x 3 y 0 y 1 y 2 y 3 y 4 x 0 x 1 x 2 y 1 y 2 y 3 y 4

16 Treewidth Width of tree decomposition is size of largest cluster Treewidth tw(g) is minimum width of any tree decomposition of G Optimal parsing complexity = treewidth

17 omplexity Treewidth for general graphs is NP-complete (rnborg et al., 1987) Simple algorithm is O(l k+2 ) forlvertices and treewidth k Parsing with original grammar is O(n l ) Treewidth is O(l) for fixed k (odlaender, 1996) Efficient algorithms for LFRS?

18 Linear ontext-free Rewriting Systems LFRS generalizes FG, TG, G, SFG, STG. Productions p P take the form: p : g( 1, 2,..., r ) where, 1,... r V N, and g is a linear, non-erasing function g( x 1,1,...,x ),..., x 1,ϕ(1 1,1,...,x 1,ϕ(r ) ) = t 1,...,t ϕ() (Vijay-Shankar et al. L 1987)

19 Linear ontext-free Rewriting Systems FG: YES TG: YES SFG: YES E D ilex: NO

20 Treewidth for LFRS lg. for treewidth of LFRS rule would imply 4 (G)-approximation algorithm for treewidth for general graphs. Proof strategy: Take arbitrary graph G From G, construct LFRS production P with dependency graph G Show relation between tw(g) and tw(g )

21 Edge Doubling G ex : G ex : E F D G D H

22 Tree decompositions of G ex and G ex G ex : G ex : D G E H F D E DF D G D H

23 G E F D H Eulerian tour:,,, D, F,, E,, G,, H,, P ex : X g ex (,,, D, E, F, G, H) g ex ( s,1, s,2, s,3, s,1, s,2, s,1, s,2, s,3, s D,1, s E,1, s F,1, s G,1, s H,1 ) = s,1 s,1 s,1 s D,1 s F,1 s,2 s E,1 s,2 s G,1 s,2 s H,1 s,3 s,3

24 Dependency Graph # E E F DF # D G H G H

25 pproximation lg. Map tree decomp. of G to tree decomp. G of treewidth SOL. tw(g) SOL 4 (G)tw(G) Implies 4 (G) approximation alg. for treewidth est known approximation algs: O(log k) (mir, 2001) O( log k) (Feige at al., 2005)

26 Dependency Treebank Experiments nmod sbj root vc pp nmod np tmp hearing is scheduled on the issue today nmod g 1 g 1 = sbj g 2 (nmod,pp) g 2 ( x 1,1, x 2,1 ) = x 1,1 hearing, x 2,1 root g 3 (sbj,vc) g 3 ( x 1,1, x 1,2, x 2,1, x 2,2 ) = x 1,1 is x 2,1 x 1,2 x 2,2 vc g 4 (tmp) g 4 ( x 1,1 ) = scheduled, x 1,1 pp g 5 (np) g 5 ( x 1,1 ) = on x 1,1 nmod g 6 g 6 = the np g 7 (nmod) g 7 ( x 1,1 ) = x 1,1 issue tmp g 8 g 8 = today

27 Dependency Treebank Experiments Kuhlmann and Nivre (L 2006) define mildly non-projective dependency structures. Gomez-Rodriguez et al. (L 2009) define mildly ill-nested dependency structures parsed in O(n 3k+4 ).

28 Treebank Parsing omplexity complexity arabic czech danish dutch german port swedish (Gildea, NL, 2010)

29 onclusion Grammar Factorization = Tree Decomposition Parsing omplexity = Treewidth Optimal factorization is NP-complete Optimal factorization is possible when parsing is possible Optimal factorization of LFRS implies 4 (G) approx alg. for treewidth