1. Context-Free Grammars
CSC 4170
Theory of Computation
Context-Free Grammars
Section 2.1

2. A  B A   B  0A1 Terminals: 0,1 Variables: A,B Productions:
What is a CFG
2.1.a
A  B
A  
B  0A1
Terminals: ,1
Variables: A,B
Productions:
Start variable: A
Derivation:
A  B  0A1  0B1  00A11  0011 A B
Parse tree: A B A 

3. Our grammar simplified
2.1.b
A  0A1
A   A
 0A1
 00A11
 000A111
 0000A1111 
What language does this grammar produce?

4. N  men | women | children V  like | hate | respect
A more complex CFG
2.1.c1
S  N’_V_N’
N’  N | N_who_V_N’
N  men | women | children
V  like | hate | respect
SN’_V_N’N’_like_N’N_like_N’N_like_Nwomen_like_Nwomen_like_children S
N’ _ V _ N’
N like N
women children

5. N  men | women | children V  like | hate | respect
A more complex CFG
2.1.c2
S  N’_V_N’
N’  N | N_who_V_N’
N  men | women | children
V  like | hate | respect S
N’ _ V _ N’
N respect N _ who _ V _ N’
children men hate N
women

6. A context-free grammar is a 4-tuple (V,,R,S), where
Formal definitions
2.1.d
A context-free grammar is a 4-tuple (V,,R,S), where
1. V is a finite set called the variables;
2.  is a finite set, disjoint from V, called the terminals;
3. R is a finite set of rules, with each rule being a pair of a variable
and a string of variables and terminals;
4. S is an element of V called the start variable.
If u,v, and w are strings of variables and terminals and A w is a rule,
we say that uAv yields uwv, written uAv  uwv.
x * y means that x=y, or x y, or there are z1,…,zn such that
x z1 … zn y.
The language produced (defined, described) by the grammar is
{w | S * w and w is a string of (only) terminals}.
A context-free language is a language produced by some CFG.

7. Ambiguity: An informal example
the girl touches the boy with the flower
Does this mean
the girl touches (the boy with the flower) or
the girl touches
the boy with the flower
(the girl touches the boy) with the flower ?
with the flower
the girl touches the boy

8. An example of an ambiguous CFG
<EXPR>  <EXPR> + <EXPR> | <EXPR>  <EXPR> | a
a + a  a
<EXPR>
<EXPR>
<EXPR> <EXPR>
<EXPR>  <EXPR> a
<EXPR>  <EXPR>
<EXPR> <EXPR> a a a a a
A grammar is ambiguous iff it has two different parse trees for the
same string

9. An equivalent but unambiguous grammar
<EXPR>  <EXPR> + <TERM> | <TERM>
<TERM>  <TERM>  a | a
<EXPR>
<EXPR> <TERM>
<TERM>
<TERM>  a a a
a + a  a

10. A more complex unambiguous grammar
2.1.h
<EXPR>  <EXPR> + <TERM> | <TERM>
<TERM>  <TERM>  <FACTOR> | <FACTOR>
<FACTOR>  (<EXPR>) | a
<EXPR>
<TERM>
<EXPR>
<TERM>  <FACTOR>
<EXPR> <TERM>
<FACTOR> a
<TERM>
<TERM>  <FACTOR>
( <EXPR> )
<FACTOR>
<FACTOR> a
<EXPR> <TERM>
<TERM>
<FACTOR> a a
<FACTOR> a
a + a  a a
(a + a)  a

11. Designing context-free grammars
Design a CFG that produces all regular expressions over the alphabet {0,1}:
<RE> 
Design a CFG G that produces the union of the languages produced by
two given CFGs G1 and G2.
G1:
A1  w1 …
An  wn
G2:
B1  u1 …
Bm  um

12. Converting a DFA into a CFG
2.1.j
Variables: The states of the DFA 1 Q1 Q2 1
Start variable: The start state of the DFA
Productions:
1. Qi  a Qj, whenever there is an a-arrow from Qi to Qj;
2. Qi  , whenever Qi is an accept state.

13. 011001 Testing in work Q1  0 Q1 Q1  1 Q2 Q2  0 Q2 Q2  1 Q1 Q2  
2.1.j
Q1  0 Q1
Q1  1 Q2
Q2  0 Q2
Q2  1 Q1
Q2   1 Q1 Q2 1 Q1
0Q1
01Q2
011Q1
0110Q1
01100Q1
011001Q2
011001
011001