The Cocke-Younger-Kasami Algorithm* Chung, Sei Kwang *Alfred Aho, Jeffrey Ullman의 “The Theory of Parsing, Translation, and Compiling”과 인터넷을 참고하여 작성되었습니다.

Contents Preliminaries CYK algorithm Context Free Grammar Chomsky Normal Form Dynamic Programming CYK algorithm Purpose of parsing Premise Constructing the parse table Left parsing from the parse table

Preliminaries(1) Context Free Grammar(1) Grammar Notation ; G = (N, Σ, P, S) N ; a finite set of non-terminal symbols Σ ; a finite set of terminal symbols P ; a finite subset of (N∪Σ)*N(N∪Σ)*×(N∪Σ)* @ Production : (α, β) ∈ P will be written α → β S ; the start symbol in N

Preliminaries(2) Context Free Grammar(2) Chomsky Normal Form CFG G ; if each production in P is of the form A → α , where A is in N and α is in (N∪Σ)* Chomsky Normal Form Production can be 1 of 2 formats A → α A → BC @ e – production ; ex) 00A1 → 001 (∵A → e ∈ P )

Preliminaries(3) Dynamic Programming Optimal substructure Solution of problem = Σ Solution of subproblem Overlapping subproblem X = S1 + S2 S1 = T1 + T2 + T3 S2 = T2 + T3 + T4 T2, T3 overlapped Recording solutions to reduce calculation Reuse the recorded solutions

CYK algorithm(1) Premise Purpose of parsing G = (N, Σ, P, S) ; a Chomsky normal form CFG with no e-production The input string w = a1a2…an Each ai ∈ Σ (1≤i ≤n) The element of the parse table, T ; tij Purpose of parsing To determine whether string w is in L(G) Input string w is in L(G) ⇔ S is in t1n

CYK algorithm(2) Constructing the parse table(1) Input ; w = a1a2…an ∈ Σ+ Output ; The parse table T for w such that tij contains A ⇔ A +⇒ aiai+1…ai+j-1 Method 1st, ti1 = {A|A→ai ∈ P, 1≤i≤n} 2nd, 1≤k<j, tij = {A|for some k, A→BC ∈ P, B is in tik, C is in ti+k, j-k} 3rd, repeat 2nd step until 1≤i≤n, 1≤j≤n-i+1

CYK algorithm(3) Constructing the parse table(2) Example Input string; abaab(n=5) Productions; S→AA|AS|b A→SA|AS|a Parse table → 5 A,S 4 3 S 2 A 1 j i

CYK algorithm(4) Left parsing from the parse table(1) Input ; Output ; A Chomsky normal form CFG G = (N, Σ, P, S) Numbered productions Input string w The parse table Output ; a left parse for w or the signal “error”

CYK algorithm(5) Left parsing from the parse table(2) Method ; A recursive routine gen(i,j,A); generate a left parse corresoding to the derivation A +⇒ aiai+1…ai+j-1 1st, if j = 1, the mth production in P is A→ai then output m 2nd, if j > 1, k(1≤k<j) is the smallest integer, A→BC ∈P then output m

CYK algorithm(6) Left parsing from the parse table(3) Example Input ; w = abaab Numbered productions 1. S → AA 2. S → AS 3. S → b 4. A → SA 5. A → AS 6. A → a Output ; 164356263 1: S → AA 6: A → a 4: A → SA 3: S → b 5: A → AS 2: S → AS 5 A,S 4 3 S 2 A 1 j i

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