CYK Algorithm for Context Free Grammar (original) (raw)

Last Updated : 15 Jul, 2025

Prerequisite - Converting Context Free Grammar to Chomsky Normal Form CYK algorithm is a parsing algorithm for context free grammar. In order to apply CYK algorithm to a grammar, it must be in Chomsky Normal Form. It uses a dynamic programming algorithm to tell whether a string is in the language of a grammar.**Algorithm :**Let w be the n length string to be parsed. And G represent the set of rules in our grammar with start state S.

  1. Construct a table DP for size n × n.
  2. If w = e (empty string) and S -> e is a rule in G then we accept the string else we reject.
  3. For i = 1 to n:
    For each variable A:
    We check if A -> b is a rule and b = wi for some i:
    If so, we place A in cell (i, i) of our table.
  4. For l = 2 to n:
    For i = 1 to n-l+1:
    j = i+l-1
    For k = i to j-1:
    For each rule A -> BC:
    We check if (i, k) cell contains B and (k + 1, j) cell contains C:
    If so, we put A in cell (i, j) of our table.
  5. We check if S is in (1, n):
    If so, we accept the string
    Else, we reject.

**Example -**Let our grammar G be:

S -> AB | BC A -> BA | a B -> CC | b C -> AB | a

We check if baaba is in L(G):

  1. We first insert single length rules into our table.
  2. We then fill the remaining cells of our table.
  3. We observe that S is in the cell (1, 5), Hence, the string baaba belongs to L(G).

Time and Space Complexity :