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Finite Automata andNon Determinism

- http//cis.k.hosei.ac.jp/yukita/

Definition 1.1 Finite Automaton

State Diagram for M1

1

0

q3

0

q1

q2

1

0, 1

Data Representation for M1

Task 01DFA

- Implement M1 with your favorite programming

language. - GUI
- Two buttons for input 0 and 1
- State chart with the current state highlighted

Language of M1

State Diagram for M5

0

q1

1

2, ltRESETgt

0, ltRESETgt

0

1

2

q2

q0

2

1, ltRESETgt

Data Representation for M5

Informal Description of M5

- M5 keeps a running count of the sum of the

numerical symbols it reads, modulo 3. - Every time it receives the ltRESETgt symbol it

resets the count to 0. - M5 accepts if the sum is 0, modulo 3.

Definition 1.7 Regular Language

- A language is called a regular language if some

finite automaton recognizes it.

Example 1.9 A finite automaton E2

- E2 recognizes the regular language of all strings

that contain the string 001 as a substring. - 0010, 1001, 001, and 1111110011110 are all

accepted, - but 11 and 0000 are not.

Find a set of states of E2

- You
- havent just seen any symbols of the pattern,
- have just seen a 0,
- have just seen 00 or,
- have just seen the entire pattern 001.
- Assign the states q,q0,q00, and q001 to these

possibilities.

Draw a State Diagram for E2

0, 1

0

1

q00

q001

q0

0

q

1

0

1

Regular Operations on Languages

Example 1.11

Theorem 1.12 Closedness for Union

Proof of Theorem 1.12

Construction of M

Correctness

- You should check the following.
- For any string recognized by M1 is recognized by

M. - For any string recognized by M2 is recognized by

M. - For any string recognized by M is recognized by

M1 or M2.

Theorem 1.13 Closedness for concatenation

Nondeterminism

- To prove Theorem 1.13, we need nondeterminism.
- Nondeterminism is a generalization of

determinism. So, every deterministic automaton is

automatically a nondeterministic automaton.

Nondetermistic Finite Automata

- A nondeterministic finite automaton can be

different from a deterministic one in that - for any input symbol, nondeterministic one can

transit to more than one states. - epsilon transition
- NFA and DFA stand for nondeterministic finite

automaton and deterministic finite automaton,

respectively.

NFA N1

0,1

0,1

q3

q4

1

q1

q2

1

0,e

Parallel world and NFA

...

...

accept

Example 1.14 NFA N2

0,1

q3

q4

q1

q2

0,1

1

0,1

Let language A consist of all strings over 0,1

containing a 1 in the third position from the

end. N2 recognizes A.

A DFA equivalent to N2

0

0

q010

q110

q000

q100

0

0

1

0

1

0

1

0

0

1

1

q011

q111

q001

q101

1

1

1

Example 1.15 NFA N3

0

0

e

0

e

0

0

Let language A consist of all strings 0k , where

k is a multiple of 2 or 3. N3 recognizes A.

A DFA equivalent to N3

0, 1

q-1

1

1

1

1

1

1

q5

q4

q2

q1

q3

q0

0

0

0

0

0

0

Example 1.16 NFA N4

q1

a

b

e

a

q3

q2

a,b

N4 accepts e, a, baba, and baa. N4 does not

accept b, nor babba.

Definition 1.17 NFA

Example 1.18 NFA N1

In what situation is Non Determinism relevant?

- Von Neumann machines are deterministic.
- However, there are many cases where machine

specification is all we need.

Theorem 1.19

- Every nondeterministic finite automaton has an

equivalent deterministic finite automaton. - Def. The two machines are equivalent is they

recognize the same language.

Proof of Th. 1.19

Incorporate e arrows

Corollary 1.20

- A language is regular if and only if some

nondeterministic finite automaton recognizes it.

Example 1.21 NFA N4 to DFA

1

b

a

e

a

3

2

a,b

Task 02Parallel World

- Write a program that simulates N4.
- GUI
- Three buttons for input 0, 1, and epsion.
- State chart that reflect the branching of the

world.

Start and Accept states

The state diagram of D

2

1,2

a,b

a

b

f

1

a,b

b

a

b

b

a

a

2,3

1,2,3

3

a

1,3

a

b

b

Theorem 1.22 The class of regular languages is

closed under the union operation.

N

N1

e

e

N2

Proof of Th. 1.22

Theorem 1.23 The class of regular languages is

closed under the concatenation operation.

N

e

e

Proof of Th. 1.23

Theorem 1.24 The class of regular languages is

closed under the star operation.

N

e

e

e

Proof of Th. 1.24

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