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Minimizing a DFA

Two steps:

Removing the states that are inaccessiblefrom the start state.

Combining states that are equivalent.

Two states p and q are equivalent iffFor any string , if p r and q s, r is

final iff s is final.

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Example of FA Minimization

aStart

e

fb

c g

d h

1 1

1

1

11

1

1

0 0

0

0

0

00 0

a b c d e f g

h

g

f

e

d

c

b

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Example of FA Minimization

We finally get: Start

1

1

1 1

1

0

0

0

00

[c] [d,f]

[b,h]

[a,e]

[g]

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Class Discussion

Minimize the following DFA:

0

00

1

11

a

bc

d e f g01 0

10

0

1

1

Start

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Context Free Languages (CFL)

Many languages are not regular, e.g.,

balanced parentheses, begin-end matches,

{0n1n| n>0}, {1n2| n>0}

In this second part, we will study another

class of languages higher in the hierarchythat can describe a larger set of languages.

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Context Free Grammar (CFG)

CFG is invented originally for describing naturallanguages:

boy

little

, , , and are called non-terminalsor variables. boy

and little are called terminals. The rules are called

productions.

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Example of CFG

+

*

()

id

Variables: Terminals: +, *, (, ), id

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Formal Definition of CFG

A CFG is denoted by a 4-tuple:

(V, T, P, S)

where V is a set of variables (non-terminals)T is a set of terminals

P is a set of productions (rules) of the form:

A where A is a variable and (VT)*

S is a special variable called start symbol

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Production Rules

A set of productions:

A 1

A 2

..

A k

The above can be written as A 1|2|.. |k|

The previous example for can be

written as E E+E | E*E | (E) | id

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DerivationLets look the CFG G: E E+E | E*E | (E) | id

E * E

(E) * E

(E) * id

( E + E ) * id

( E + id ) * id

( id + id ) * id

We say if can be obtained from by applying aproduction once. We say if can be obtained from

by applying the productions zero or more times. We use

to specify clearly which grammar we are using.

E

Derivation

*

G

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Context Free Languages (CFL)

The language generated by a CFG G, denoted by

L(G), is :

L(G) = {| T* and S }Therefore,

every string consists solely of terminals, and

every string can be derived from S

It is called a context free language (CFL).

*G

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CFL

Notice that:

We must start with the start symbol

We can use any production any number oftimes.

The final string can only contain terminals.

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Examples of CFG

Consider the following CFG G1:

S aSb

S ab

How to generate aabb?S aSb (use Rule 1)

aabb (use Rule 2)

How to generate aaabbb?

S aSb (use Rule 1)

aaSbb (use Rule 1)

aaabbb (use Rule 2)

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Examples of CFG

Consider the following CFG G2:

S SS

S (S)

S

In this example, variables are {S}, terminals are

{(,)}, start symbol is S and productions are the

above three rules.

What language does this grammar generate?

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Examples of CFG

How to generate () with G2?S (S) (use Rule 2)

() (use Rule 3)

How to generate (()()) with G2?S (S) (use Rule 2)

(SS) (use Rule 1)

((S)S) (use Rule 2)

((S)(S)) (use Rule 2)(()(S)) (use Rule 3)

(()()) (use Rule 3)

G2:

S SSS (S)

S

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Examples of CFG

How can we represent all the arithmetic expressionswith plus, minus, parentheses and variables x,

y and z, i.e., {x, y, z, x+y, x-y, y+x, x+(y-z), .. }?

Consider the following CFG G3:E E + E E (E) V x V z

E E - E E V V y

In this example, variables are {E, V}, terminals are{+, -, (, ), x, y, z}, start symbol is S and productions

are the above seven rules.

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Examples of CFG

How to generate x+(y-z)?

E E + E (use Rule 1)

V + E (use Rule 4)

x + E (use Rule 5)x + (E) (use Rule 3)

x + (E - E) (use Rule 2)

x + (V - E) (use Rule 4)

x + (y - E) (use Rule 6)x + (y - V) (use Rule 4)

x + (y - z) (use Rule 7)

G3:E E + E | E - E | (E) | V

Vx | y | z

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Class Discussion

Write a CFG for each of the following

languages:

The set of all palindromes over the alphabet={a,b}

{anbncmdm| n 1, m 1}

{anbmcmdn| n 1, m 1}

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Class Discussion

Consider the grammar:

S aB | bA

A a | aS | bAAB b | bS | aBB

Generate 4 strings from this grammar.

Do you know what language does this CFGrepresent?