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.
Let w be the n length string to be parsed. And G represent the set of rules in our grammar with start state S.
- Construct a table DP for size n × n.
- If w = e (empty string) and S -> e is a rule in G then we accept the string else we reject.
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.
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.
We check if S is in (1, n): If so, we accept the string Else, we reject.
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):
- We first insert single length rules into our table.
- We then fill the remaining cells of our table.
- We observe that S is in the cell (1, 5), Hence, the string baaba belongs to L(G).
Time and Space Complexity :
- Time Complexity –
Where |G| is the number of rules in the given grammar.
- Space Complexity –
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