CSCI 1010 Models of Computa3on. Lecture 17 Parsing Context-Free Languages

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1 CSCI 1010 Models of Computa3on Lecture 17 Parsing Context-Free Languages

2 Overview BoCom-up parsing of CFLs. BoCom-up parsing via the CKY algorithm An O(n 3 ) algorithm John E. Savage CSCI 1010 Lect 17 2

3 Parse Trees for CFLs Example: G 3 = (N 3,T 3,R 3,S) A deriva3on of caacaabcbc and its parse tree. s cmnc camanc ca 2 Ma 2 Nc ca 2 ca 2 Nc ca 2 ca 2 bnbc ca 2 ca 2 bcbc John E. Savage CSCI 1010 Lect 17 3

4 Chomsky Normal Form A CFG G = (N, T, R, S) is in Chomsky normal form if every rule is of the form A BC or A b, b T, except if ε L(G) in which case S ε is also a rule. John E. Savage CSCI 1010 Lect 17 4

5 Top-Down Parsing Parse string such as w=w 1 w 2 w 3 w 4 w 5 with rules of the form A BC or A b, b T for language L. The first rule is S BC where B w 1 w k and C w k+1 w 5 splits w. Same for subsequences. This is top-down parsing. We parse bo2om-up. John E. Savage CSCI 1010 Lect 17 5

6 BoCom-Up Parsing For i = 0, 1,, 5, let N i,i+1 be non-terminals A such that A w i T Let N i,i+2 be non-terminals A such that A BC and B in N i,i+1 and C in N i+1,i+2. Repeat for N i,i+3, N i,i+4,, N 1,6. If S in N 1,6, w in L. John E. Savage CSCI 1010 Lect 17 6

7 Example Convert grammar to Chomsky normal form. John E. Savage CSCI 1010 Lect 17 7

8 BoCom-Up Parsing Parse w=w 1 w 2 w 3 w 4 w 5 =a*b+a. Let w 1 =a, w 2 =*,w 3 =b, w 4 =+,w 5 =a. Let w i,j+1 = w i w i+1 w j. Thus, w i,i+1 = w i w i,i+1 from some rule A a where A ϵ N i,i+1 ={N N w i } N 1,2 ={E,T,F},N 2,3 ={*}, N 3,4 ={E,T,F}, N 4,5 ={+}, N 5,6 ={E,T,F}. All other deriva3ons are of form A BC. John E. Savage CSCI 1010 Lect 17 8

9 BoCom-Up Parsing Parse w=w 1 w 2 w 3 w 4 w 5 =a*b+a where w 1 =a, w 2 =*, w 3 =b, w 4 =+, w 5 =a Other deriva3ons are of form A BC. For i =1,, 5, w i,i+2 = w i w i+1 is derived from A ε N i,i+2 where N i,i+2 ={N N MP for M in N i,i+1 and P in N i+1,i+2 } w i,i+3 = w i w i+1 w i+2 derived from A ε N i,i+3 where N i,i+3 ={N N MP, M in N i,i+2, P in N i+2,i+3 } {N N MP, M in N i,i+1, P in N i+1,i+3 } Con9nue un3l N 1,6. If S in N 1,6, w is in language and its parse is obtained. John E. Savage CSCI 1010 Lect 17 9

10 Quick Sketch of Parsing w = w 1,6 = w 1 w 2 w 3 w 4 w 5 = a*b+a N 1,2 ={E,T,F}, N 2,3 ={*}, N 3,4 ={E,T,F}, N 4,5 ={+}, N 5,6 ={E,T,F} N 1,3 =, N 2,4 ={B, D}, N 3,5 =, N 4,6 ={A} N 1,4 ={E,T}, N 2,5 =, N 3,6 = {E} N 1,5 = N 2,6 =, N 1,6 ={E}, E is the start symbol! w is in L! John E. Savage CSCI 1010 Lect 17 10

11 Matrix Approach to BoCom-Up Parsing Parse w = a*b+a = w 1 w 2 w 3 w 4 w 5, n = w = 5. w 1 =a, w 2 =*, w 3 =b, w 4 =+, w 5 =a N i,i+1 ={N N w i } generates w i, 1 i n-1. Trees of depth one. All other rules are A BC. N i,j E,T,F 2 * 3 E,T,F 4 + N i,j = N i,i+1 N i,i+2 N i,n+1 represented with diagonal nxn matrix. Build it incrementally. 5 E,T,F John E. Savage CSCI 1010 Lect 17 11

12 Matrix Approach to BoCom-Up Parsing N in N i,i+2 ={N N MP, M in N i,i+1, P in N i+1,i+2 } generates w i w i+1, 1 i n-1. Deriva3on depth =2. N 1,3 =, N 2,4 ={B, D}, N 3,5 =, N 4,6 ={A} Add link from B to * and F and from D to * and F N i,j E,T,F 2 * B,D 3 E,T,F 4 + A 5 E,T,F John E. Savage CSCI 1010 Lect 17 12

13 Matrix Approach to BoCom-Up Parsing Compute deriva3ons of w i w i+1 w i+2 for 1 i n-2 N i,i+3 ={N N MP, M in N i,i+1, P in N i+1,i+3 } {N N MP, M in N i,i+2, P in N i+2,i+3 } Deriva3on depth = 3. Add links for deriva3ons. N i,j E,T,F E,T 2 * B,D 3 E,T,F E 4 + A 5 E,T,F John E. Savage CSCI 1010 Lect 17 13

14 Matrix Approach to BoCom-Up Parsing We compute w i w i+1 w i+2 w i+3 for i = 1 to n-3. N i,i+4 ={N N MP, M in N i,i+1, P in N i+1,i+4 } {N N MP, M in N i,i+2, P in N i+2,i+4 } {N N MP, M in N i,i+3, P in N i+3,i+4 } N i,j E,T,F E,T 2 * B,D 3 E,T,F E 4 + A 5 E,T,F John E. Savage CSCI 1010 Lect 17 14

15 Matrix Approach to BoCom-Up Parsing We compute w i w i+1 w i+2 w i+3 w i+4 for i = 1 to n-4 N i,i+5 ={N N MP, M in N i,i+1, P in N i+1,i+5 } {N N MP, M in N i,i+2, P in N i+2,i+5 } {N N MP, M in N i,i+3, P in N i+3,i+5 } {N N MP, M in N i,i+4, P in N i+4,i+5 } N 1,6 ={E} derived as N MP, M in N 1,4, P in N 4,6. w = a*b+a is derived from E EA, A +T, T a, E TB, T a, B *F, F b John E. Savage CSCI 1010 Lect 17 15

16 Parse Tree E EA, A +T, T a, E TB, T a, B *F, F b As matrix assembled, links corresponding to rules and inserted, making it easy to find the deriva3ons and parse! N i,j E,T,F E,T E T B + T 2 * B,D 3 E,T,F E a * F + a 4 + A 5 E,T,F * b John E. Savage CSCI 1010 Lect E A

17 Cocke, Kasami, Younger (CKY) Parser The above parsing algorithm was developed by Cocke, Kasami, and Younger (CKY). We now restate it using set matrices. This allows for fast matrix-matrix mul3plica3on algorithms to speed up the parsing. John E. Savage CSCI 1010 Lect 17 17

18 Set Matrices A set matrix A ={a i,j } is a matrix whose a i,j entry is a subset of N, the non-terminals of a grammar G = (N,T, R,S). Here plays the role of zero. Opera3on is defined on sets S 1, S 2 of non-terminals S 1 S 2 = {A A BC ε R where B ε S 1, C ε S 2 } Thus, N i,i+2 = N i,i+1 N i+1,i+2 Let + denote set union. Then, N i,i+3 = N i,i+1 N i+1,i+3 + N i,i+2 N i+2,i+3 John E. Savage CSCI 1010 Lect 17 18

19 Set Matrices Let D ={d i,j }, E ={e i,j }. Then, C = D E ={c i,j } is m m mul3plica3on of set matrices where (+) denote mul3plica3on (addi3on). c i,j defined as c i,j = 1 k m d i,k e k,j John E. Savage CSCI 1010 Lect 17 19

20 Parsing of Context-Free Languages Let B (1) be (n+1) (n+1) matrix, b i,j = N i,i+1 when j=i+1 and otherwise. Example B (1) = E,T,F * E,T,F + E,T,F John E. Savage CSCI 1010 Lect 17 20

21 Parsing of Context-Free Languages Let B (2) = B (1) B (1), B (3) = B (1) B (2) + B (2) B (1) B (2) = B,D A E,T B (3) = E John E. Savage CSCI 1010 Lect 17 21

22 Parsing of Context-Free Languages Define B (2) = B (1) B (1), B (3) = B (1) B (2) + B (2) B (1) and B (s) = U 1 r s B (r) B (s-r). Note that B (s) has non-empty entries only on diagonal (i,i+s). b (s) i,j = 1 r s-1 b(r) i,r b(s-r) r,j O(s) set matrix ops needed, each using O(n) steps. Repeat for 2 s n or O(n 3 ) basic steps. The CKY algorithm requires O(n 3 ) set products. Fast matrix mul3plica3on can make it faster. John E. Savage CSCI 1010 Lect 17 22

23 Summary BoCom-up parsing via the CKY algorithm This is an O(n 3 ) algorithm. In 1975 Leslie Valiant demonstrated that this problem can be solved in 3me propor3onal to the 3me to mul3ply to n x n matrices. Fast such algorithms exist that run O(n α ) for α ~ 2.4. John E. Savage CSCI 1010 Lect 17 23