COS 423 Theory of Algorithms Spring 2005
Precept 4
1. Problem 6.5.
2. A fundamental problem in compilers, natural language processing, and theoretical computer science
is determining whether or not a string is in a cont ex t-free gra mmar. A context-free grammar is a
compact notation for expressing a set of strings. It consists of a set of terminals (lowercase letters), a
set of nonterminals (uppercase letters) and a set of productions (replacement rules for nonterminals).
A grammar in Chomsky Normal Form always has production rules with one nonterminal on the left
hand side and either (i) one nonterminal on the right hand side or (ii) two nonterminals on the right
hand side.
terminals a, b
nonterminals S, T, A, B , X
start S
Rules
Sโ๎
SโAB
SโAX
XโTB
TโAB
TโAX
Aโa
Bโb
For example, aabb is a string in the above context free grammar. To see why, we can apply the following
sequence of transformations SโAX โAT B โAABB โaABB โaaBB โaabB โaabb.In
fact, the grammar generates precisely those strings with kaโs followed by kbโs.
Given a context-free grammar in Chomsky Normal Form, design an algorithm to determine whether a
string is in the grammar. Your algorithm should run in O(n3) time where nis the number of characters
in the string. You may assume the number of production rules is a small constant.
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