CYK Algorithm for Context Free Grammar

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 :

  • Time Complexity –
    O(n3.|G|) 

    Where |G| is the number of rules in the given grammar.

  • Space Complexity –
    O(n2)  

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