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

2. 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

3. 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

4. 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 )

5. 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

6. 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

7. 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

8. 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

9. 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”

10. 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

11. 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 ;
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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