Context-Free Grammars - PowerPoint PPT Presentation

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CSC 4170 Theory of Computation. Context-Free Grammars. Section 2.1. 2.1.a. What is a CFG. A  B A   B  0A1. Terminals: 0,1. Variables: A,B. Productions:. Start variable: A. A  B  0A1  0B1  00A11  0011. Derivation:. A. B. Parse tree:. 0 1. A. B.

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Context-Free Grammars

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CSC 4170

Theory of Computation

Section 2.1

2.1.a

What is a CFG

A  B

A  

B  0A1

Terminals: 0,1

Variables: A,B

Productions:

Start variable: A

A  B  0A1  0B1  00A11  0011

Derivation:

A

B

Parse tree:

0 1

A

B

0 1

A

2.1.b

Our grammar simplified

A  0A1

A  

 0A1

 00A11

 000A111

 0000A1111

 00001111

A

What language does this grammar produce?

2.1.c1

A more complex CFG

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

2.1.c2

A more complex CFG

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

2.1.d

Formal definitions

Acontext-free grammaris a 4-tuple (V,,R,S), where

1. V is a finite set called thevariables;

2. is a finite set, disjoint from V, called theterminals;

3. R is a finite set ofrules, with each rule being a pair of a variable

and a string of variables and terminals;

4. S is an element of V called thestart 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.

Thelanguage produced(defined, described) by the grammar is

{w | S * w and w is a string of (only) terminals}.

Acontext-free languageis a language produced by some CFG.

2.1.e

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

2.1.f

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 isambiguous iff it has two different parse trees for the

same string

2.1.g

An equivalent but unambiguous grammar

<EXPR>  <EXPR> + <TERM> | <TERM>

<TERM>  <TERM>  a | a

<EXPR>

<EXPR> + <TERM>

<TERM>

<TERM>  a

a

a

a + a  a

2.1.h

A more complex unambiguous grammar

<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

2.1.i

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

2.1.j

Converting a DFA into a CFG

0

0

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.

2.1.j

0

0

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