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CSE 211: Deterministic Finite Automaton

Md. Shaifur Rahman

Dept. of CSEBangladesh University of Engineering & Technology

Class 2-3

Shaifur (CSE, BUET) CSE211: DFA Class 2-3 1 / 20

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Outline

1 What is Finite Automaton(FA)


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Outline


2 State Diagram of FA


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Outline



3 Formal Definition of FA

4 Examples of FA


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Outline




4 Examples of FA

5 Computation by FA


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Outline




4 Examples of FA

5 Computation by FA

6 What is Regular Language (RL)


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What is Finite Automaton (FA)

Finite Automaton

has a very small amount of memory

Example: the controller for an automatic door


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State diagram of a Finite Automaton

Three states: q1, q2

andq3

Start state: q1

Accept state: q2

Transition: arrows

For Input: 11011 Start in state q12 Read 1, follow transition from q1 to q23 Read 1, follow transition from q2 to q24 Read 0, follow transition from q2 to q35 Read 1, follow transition from q3 to q26 Accept because M1 is in an accept state q2


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State diagram of a Finite Automaton (Contd.)

For input1111, the states are: q1q2q2q2q2 the finalstate: q2, soAccept

For input0110, the states are: q1q1q2q2q3 the final

state: q3, soRejectFor input101010, the states are:q1q2q3q2q3q2q3 the final state: q3, soReject


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State diagram of a Finite Automaton (Contd.)

What types of inputs are accepted by M1?

Accepts:01, 001, 0001, 00001, . . . , 0101, 0100, 0101, . . . , 01101, 011101, . . .

Rejects: all other strings

Observation: M1 accepts any string that ends with a 1 ANDhas evennumber of0s after the last 1


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Formal Definition of Finite Automaton

A Finite Automaton (FA) is a 5-tuple (Q, , , q 0, F) where

Q is the set ofall states

is the set ofall input symbols(also called alphabet): Q Q is thetransition function

q0Q is thestart-state

FQ is the set ofaccept-states


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Example of Formal Definition of FA

Q= {q1, q2, q3} ={0, 1}

is:0 1

q1 q1 q2q2 q3 q2q3 q2 q2

start-state,q0=q1

accept-states, F ={q2}


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FA and its Language

A: is the set of all strings that the FA M accepts

A is thelanguageofM, written as L(M) =A

M recognizeslanguage A

ifM recognizes no strings, L(M) =A= an Mmay accept more than one strings, but it recognizesonly onelanguage

In our previous example,

A={w|w contains at least one 1 and even number of0s following that 1}


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Examples of FA

M2= ({q1, q2}, {0, 1}, , q 1, {q2}) where is:0 1

q1 q1 q2q2 q1 q2

M2 accepts string 1101 as follows: q1q2q2q1q2

M2 rejects string 1010 as follows: q1q2q1q2q1

M2 accepts the language of strings that ends with 1

L(M2) ={w|w ends with 1}


( )

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Examples of FA(Contd.)

M3= ({q1, q2}, {0, 1}, , q 1, {q1}) where is:

0 1

q1 q1 q2q2 q1 q2

M3 accepts string 1101 as follows: q1q2q2q1q2

M3 rejects string 1010 as follows: q1q2q1q2q1

M3 accepts the empty string or the string that ends with 0L(M3) ={w|w is or ends with 0}


E l f FA(C d )

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M4= ({s, q1, q2, r1, r2}, {a, b},,s, {q1, r1}), two accepting states!

M4 starts in State s

ifa, goes to State q1, accepts if the last symbol is a

ifb, goes to State r1, accepts if the last symbol is b

M4 accepts a, aa, aaa, aba, . . . , b, bb, bab, bbb, . . .

L(M4) ={w|w starts and ends with the same symbol}


E l f FA(C d )

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M5= ({q0, q1, q2}, {RESET, 0, 1, 2}, , q 0, {q0})

M5 keeps a running count of the sum of the numerical input symbolsit reads, modulo 3

Accepts when the sum is a multiple of3

RESET sets the sum to zero


E l f FA(C d )

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State Diagram for FA is cool but not always possible!

For example:

Generalized ModuloMachine

For each i1 let Ai ={all strings where the sum of the numbers isa multiple of i}, also the sum is zero when the input is RESET

FA Bi recognizes language Ai

Bi = (Qi, , i, q0, {q0})

Here, Qi ={q0, q1, q2, . . . , q i1}

For each state qj , ifBi is in qj , i is:i(qj , 0) =qji(qj , 1) =qk where k=j+ 1 modulo ii(qj , 2) =qk where k=j+ 2 modulo ii(qj , RESET) =q0


T Y lf

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Try Yourself

Question

Show how M1 and M2 accept/reject the strings aabb, ababandaaabbb.

Formally describe FA M1 and M2.

What do the machines do? Can you comment on their languages?


T Y lf

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Try Yourself

QuestionDraw the state diagram for the following FA:M6= ({q1, q2, q3, q4, q5}, {u, d}, , q 3, {q3}) and is:

u d

q1 q1 q2q2 q1 q3q3 q2 q4q4 q3 q5

q5 q4 q5

What does M6 do? Comment on the language it recognizes.


Formal Definition of Computation

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Formal Definition of Computation

Let M= (Q, , , q 0, F) be an FAw=w1w2. . . wn be a string where wi M accepts w if there arestates r0, r1, . . . , rnQ and

1 r0=q0

2 (ri, wi+1) =ri+1 fori= 0, 1, . . . , n 1 and3 rn F

M recognizeslanguage L(M) =A ifA= {w| M accepts w}

Regular LanguageA is a Regular Language (RL) if it is recognized by some FiniteAutomaton M.


Testing for RL

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Testing for RL

Is the following an RL?

A= {w| the sum of the symbols in w is 0 modulo 3, except thatRESET resets the count to 0}


Testing for RL

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Testing for RL

Is the following an RL?

A= {w| the sum of the symbols in w is 0 modulo 3, except thatRESET resets the count to 0}

YES, Find an FA that recognizes A:

M5

Ifw= 10RESET22RESET012,

q0q1q1q0q2q1q0q0q1q0Shaifur (CSE, BUET) CSE211: DFA Class 2-3 18 / 20

Next Class

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Next Class . . .

Can we design an FA for any arbitrary language?


Next Class

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Next Class . . .

Can we design an FA for any arbitrary language?

Can we combine existing languages to derive/define new languages?


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Question?


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