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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

Our grammar simplified

A 0A1

A

0A1

00A11

000A111

0000A1111

00001111

A

What language does this grammar produce?

A more complex CFG

S N’_V_N’

N’ N |N_who_V_N’

N men | women | children

V like | hate | respect

SN’_V_N’N’_like_N’N_like_N’N_like_Nwomen_like_Nwomen_like_children

S

N’ _ V _ N’

N like N

women children

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

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.

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

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

An equivalent but unambiguous grammar

<EXPR> <EXPR> + <TERM> | <TERM>

<TERM> <TERM> a | a

<EXPR>

<EXPR> + <TERM>

<TERM>

<TERM> a

a

a

a + a a

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

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

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.

Testing in work

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

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