Chapter 2: Context-Free Languages
For Your Enjoyment
Chapter 2.1, Context-Free Grammars
Problem 2.3 on page 155
Answer each part for the following context-free grammar G.
R → XRX | S
S → aTb | bTa
T → XTX | X | ε
X → a | b
- What are the variables of G?
- What are the terminals of G?
- What is the start variable of G?
- Give three strings in L(G).
- Give three strings not in L(G).
- True or False. T ⇒ aba.
- True or False. T ⇒ * aba.
- True or False. T ⇒ T.
- True or False. T ⇒ * T.
- True or False. XXX ⇒ * aba.
- True or False. X ⇒ * aba.
- True or False. T ⇒ * XX.
- True or False. T ⇒ * XXX.
- True or False. S ⇒ * ε.
- Give a description in English of L(G).
Context-Free Grammar Definition
A context-free grammar is a 4-tuple (V, Σ, R, S) where
- V is a finite set called the variables,
- Σ is a finite set, disjoint from V, called the terminals,
- R is a finite set of rules, with each rule being a variable and
a string of variables and terminals, and
- S ∈ V is the start variable.
Yields Definition
uAv yields uwv, written uAv ⇒ uwv, if A → w is a rule of
the grammar.
Derives Definition
u derives v, written u ⇒* v, (1) if u = v or (2) if
a sequence u1, u2, ... uk exists
for k ≥ 0 where u ⇒ u1 ⇒ u2 ...
uk ⇒ v.
Language of the Grammar Definition
L(G) = {w ∈ Σ* | S ⇒ * w}
Active Learning Problem
- Produce a DFA that recognizes A = {w | w contains either zero or one 0}
over the binary alphabet.
- Construct a context-free grammar for language A, using the
states of the DFA as the variables.
Ambiguous Definition
A string w is derived ambiguously in context-free grammar G if
it has two or more different leftmost derivations. Grammar G
is ambiguous if it generates some string ambiguously.
Active Learning Problem
Show that the following context-free grammar is ambiguous for
the string aabbcc.
S → WX | YZ
W → ε | aW
X → bXc | ε
Y → aYb | ε
Z → ε | cZ
Chomsky Normal Form
On your own, understand how to convert a context-free grammar to
Chomsky Normal Form. The procedure is described on pages 109-111.
