Introduction to fa and dfa

THEORY OF COMPUTATION

Lecture One:

Automata Theory

1Er. Deepinder KaurAutomata Theory

Theory of Computation

In theoretical computer science and mathematics, the theory

of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: • automata theory, • computability theory• computational complexity theory.

Er. Deepinder Kaur 2Automata Theory

Automata theory

• The word “Automata“ is the plural of “automaton" which simply means any machine.

• automata theory is the study of abstract machines and problems they are able to solve.

• Automata theory is closely related to formal language theory as the automata are often classified by the class of formal languages they are able to recognize.

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Abstract Machine

• An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system used in Automata theory.

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Applications of Automata• A variety of properties concerning the models, grammars, and

languages will be proven.

• These algorithms form the basis of tools for processing languages, e.g., parsers, compilers, assemblers, etc.

• Other algorithms will form the basis of tools that automatically construct language processors, e.g., yacc, lex, etc.– Note that our perspective will be similar to, yet different from a compiler

class.

• Additionally, some things will be proven to be non-computable, e.g., the enhanced compiler.

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Automaton

• An automaton is an abstract model of a digital computer

• It has a mechanism to read input (string over a given alphabet, e.g. strings of 0’s and 1’s on = {0,1}) written on an input file.

• A finite automaton has a set of states • Its control moves from state to state in response to

external “inputs”

Automaton

• With every automaton, a transition function is associated which gives the next state in terms of the current state

• An automaton can be represented by a graph in which the vertices give the internal states and the edges transitions

• The labels on the edges show what happens (in terms of input and output) during the transitions

Components of an automaton

• Input file : Contains strings of input symbols

• Storage unit: consists of an unlimited number of cells, each capable of holding a single symbol from an alphabet

• Control unit : can be in any one of a finite number of internal states and can change states in defined manner

Some Terms used in automaton theory

• Alphabets-Everything in mathematics is based on symbols. This is also true for automata theory. Alphabets are defined as a finite set of symbols. An example of alphabet is a set of decimal numbers ∑={0,1,2,3,4,5,6,7,8,9}• Strings- A string is a finite sequence of symbols selected from some

alphabet If ∑ {a,b} is an alphabet then abab is string over alphabet ∑. A

string is generally denoted by w. The length of string is denoted by |w|

• Empty string is string with zero occurrence of symbols . This string is represented by є

• The set of strings, including empty, over an alphabet ∑ is denoted by ∑*.

• ∑+ = ∑* -{є}• Languages-A set of strings which are chosen from some ∑*,

where ∑ is a particular alphabet, is called a language . If ∑ is an alphabet, and L subset of ∑*, then L is said to be language over alphabet ∑. For example the language of all strings consisting of n 0’s followed by n 1’s for some n>=0:

{є,01,0011,000111,-------}

• Langauge in set forms- {w|some logical view about w} e.g {anbn|n>=1}

• Kleene closure- Given an alphabet, a language in which any string of letters from ∑ is a word, even the null string, is called closure of the alphabet . It is denoted by writing a star, after the name of alphabet as a superscript ∑*.

Finite Automaton

• One of the powerful models of computation which are restricted model of actual computer is called finite automata. These machines are very similar to CPU of a computer .They are restricted model as they lack memory.

• Finite automation is called finite because number of possible states and number of letter in alphabet are both finite and automation because the change of state is totally governed by the input.

2.2 Deterministic Finite Automata

– graphic model for a DFA

13Automata Theory Er. Deepinder Kaur

Main parts of pictorial representation of Finite machine

• Strings are fed into device by means of an input tape which is divided into square with each symbol in each square.

• Main part of machine is a black box which serve that what symbol is written at any position on input tape by means of a movable reading head

• P0,p1,p2,p3,p4 are the states in finite control system and x and y are input symbols.

• At regular intervals, the automation reads one symbol from input tape and then enters in a new state that depends only on current state and the symbol just read.

Main parts of pictorial representation of Finite machine

• After reading an input symbol, reading head moves one square to the right on input tape so that on next move, it will read the symbol in next tape square. This process is repeated again and again

• Automation then indicates approval or disapproval• If it winds up in one of the final states, the input string is considered

to be accepted. The language accepted by the machine is the set of strings it accepts.

DFA: Deterministic Finite Automaton

• An informal definition (formal version later):– A diagram with a finite number of states represented

by circles– An arrow points to one of the states, the unique start

state– Double circles mark any number of the states as

accepting states– For every state, for every symbol in , there is exactly

one arrow labeled with that symbol going to another state (or back to the same state)

Automata TheoryEr. Deepinder Kaur

Finite Automata FA• Its goal is to act as a recognizer for specific a

language/pattern.• Any problem can be presented in form of

decidable problem that can be answered by Yes/No.

• Hence FA (machine with limited memory) can solve any problem.

Deterministic Finite Automata DFAFA = “a 5-tuple “ (Q, Σ, , q0, F)

1. Q: {q0, q1, q2, …} is set of states.2. Σ: {a, b, …} set of alphabet.3. (delta): represents the set of transitions that FA can

take between its states. : Q x Σ→Q

Q x Σ to Q, this function: Takes a state and input symbol as arguments. Returns a single state. : Q x Σ→Q

4. q0 Q is the start state.5. F Q is the set of final/accepting states.

Automata Theory Er. Deepinder Kaur

Transition function : Q x Σ→QMaps from domain of (states, letters) to range

of states.

(q0, a)(q2, b)(q1, b)

Transition function • : Q x Σ→Q• Maps from domain of (states, letters) to range

of states.

(q0, a)(q2, b)(q1, b)

How does FA work?1. Starts from a start state.2. Loop

Reads a sequence of letters3. Until input string finishes4. If the current state is a final state then

Input string is accepted.5. Else

Input string is NOT accepted.• But how can FA be designed and represented?

Transition SystemFA = “a 5-tuple “ (Q, Σ, , q0, F)

1. Q: {q0, q1, q2, …} is set of states.2. Σ: {a, b, …} set of alphabet.3. (delta): represents the set of transitions that FA can

take between its states. : Q x Σ→Q

Q x Σ to Q, this function: Takes a state and input symbol as arguments. Returns a single state. : Q x Σ→Q

4. q0 Q is the start state.5. F Q is the set of final/accepting states.

Transition System

Transition Diagrams Transition Tables

Transition Diagram Notations

• If any state q in Q is the starting state then it is represented by the circle with arrow as

• Nodes corresponding to accepting states are marked by a double circle

Transition DiagramCan be represented by directed labeled graph/Transition tableVertex is a stateStates= nodes

Starting/Initial state denoted by circle and arrow/-Final state(s) denoted by two concentric circles/+Other states with circle

Transition function =directed arrows connecting states.

Alternative Representation: Transition Table

A A BB A CC C C

Rows = states

Columns =input symbols

Final statesstarred

*Arrow forstart state

Acceptability of a stringA string is accepted by a transition system if

• There exist a path from initial state to final state

• Path traversed is equal to w

Example1.1• Build an FA that accepts only aab

a b+S4

a bS1 S2 ?

S2 S3 ?

S3 ? ?

S4 ? ?Automata Theory Er. Deepinder Kaur

Example withTransition Table

A C BB D AC A DD B C

Check for 110101

Example withTransition Table

Solution:-

(A,110101)= (B,10101) (A,0101)

(C,101)

(D,01)

(B,1)A*Automata Theory Er. Deepinder Kaur

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Properties of transition function1. (q,λ)=q

• It comes back to same state

• It requires an input symbol to change the state of a system.

2. (q,aw)=((q,a),w)

(q,w,a)=((q,w),a)

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Facts in designing FA

First of all we have to analyze set of strings.

Make sure that every state is check for output state and for every input symbol from given set.

No state must have two different outputs for single input symbol

There must be one initial and atleast one final state in FA

Language of a DFA

• Automata of all kinds define languages.• If A is an automaton, L(A) is its language.• For a DFA A, L(A) is the set of strings

labeling paths from the start state to a final state.

• Formally: L(A) = the set of strings w such that δ(q0, w) is in F.

Language• A language is a set of strings. For example, {0, 1}, {all English words}, {0, 0, 0, ...} are

all languages.

Example #8:

• Let Σ = {0, 1}. Give DFAs for {}, {ε}, Σ*, and Σ+.

For {}:

For Σ*: For Σ+:

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Automata Theory 35

Example: Design a FA that accepts set of strings such that every string ends in 00, over the alphabet

{0,1} i,e ∑={0, 1}

Inorder to design any FA, first try to fulfill the minimum condition.

Start 0 0 0

Being DFA, we must check every input symbol for output state from every state. So we have to decide output state at symbol 1 from q0,q1 and q2. Then it will be complete FA

q0 q1 q2

• 1 0 0start 0

q0 q1 q2q2

Ex 2 –

• Construct a DFA that accepts a’s and b’s and ‘aa’ must be substring

Example: String in a Language

Minimal condition : aa

There may be aabbaa.bbbbaa,aa,aab,aabb,….

Ex : (0+1)*00(0+1)*

• Idea: Suppose the string x1x2 ···xn is on the tape. Then we check x1x2, x2x3, ..., xn-1xn in turn.• Step 1. Build a checker

• Step 2. Find all edges by the following consideration:

Consider x1x2.• If x1=1, then we give up x1x2 and continue to

check x2x3. So, we have δ(q0, 1) = q0. • If x1x2 = 01, then we also give up x1x2 and

continue to check x2x3. So, δ(q1, 1) = δ(q0, 1) =q0.• If x1x2 = 00, then x1x2··· xn is accepted for any

x3···xn. So, δ(q2,0)=δ(q2,1)=q2.

(0+1)*00(0+1)*

Ex 1All words that start with “a” over the alphabet {a,b}

a(a+b)*

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Ex3• All words that start with triple letter

(aaa+bbb)(a+b)*

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Ex4• All words with even count of letters having “a” in an

even position from the start, where the first letter is letter number one.(a+b)a((a+b)a)*

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EX5:Construct DFA to accept (0+1)*

Ex 6:Construct DFA to accept 00(0+1)*

(0+1)*(00+01)

Construct a FA that accepts set of strings where the number of 0s in every string is multiple of 3 over alphabet ∑={0,1}

1 1 1 start 0 0

0 As 0 existence of 0 is also multiple of 3, we have to consider starting

state as the final state.

q1 q2q0

Design FA which accepts set of strings containing exactly four 1s in every string over alphabet ∑={0,1}

q2q4q0 q1 q21 1

0 0 00 0

q5 is called the trap state or dead state. Dead states are those states which transit to themselves for all input symbols.

Design a FA that accepts strings containing exactly one 1 over alphabet {0,1}. Also draw the transition table for the FA generated

q2q2q1

q3 is the dead state

Transition table for previous problem

δ/∑ 0 1

q1 q1 q2

*q2 q2 q3

q3 q3 q3

Non final state that transit in self loop for all inputs

Design an FA that accepts the language

L={w ϵ (0,1)*/ second symbol of w is ‘0’ and fourth input is ‘1’}

q0 q3q1 q20 11/0

startq4

Design DFA for the language L={w ϵ (a,b)*/nb(w) mod 3 > 1}

As given in the language, this can be interpreted that number of b mod 3 has to be greater than 1 and there is no restriction on number of a’s. Thus it will accept string with 2 bs,5 bs, 8bs and so on.

q0 q1 q2

Q={q0,q1,q2}F={q2}

Design FA over alphabet ∑= {0,1} which accepts the set of strings either start with 01 or end with 01

Example #4:

• Give a DFA M such that:

L(M) = {x | x is a string of 0’s and 1’s and |x| >= 2}

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q1q0 q20/1

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Example #5:

L(M) = {x | x is a string of (zero or more) a’s and b’s such that x does not contain the substring aa}

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Automata Theory 61

Example #6:

L(M) = {x | x is a string of a’s, b’s and c’s such that x contains the substring aba}

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Automata Theory 62

DFA Practice

• Design a FA which accepts the only input 101 over input set {0,1}

• Strings that end in ab

• Strings that contain aba

• String start with 0 and ends with 1 over {0,1}

• Strings made up of letters in word ‘CHARIOT’ and recognize those strings that contain the word ‘CAT’ as a substring

Er. Deepinder Kaur 63Automata Theory

Introduction to fa and dfa

transitions automata

deepinder kaurautomata

automaton theory alphabets

defined manner automata

word automata

formal language theory

applications of automata

finite automaton

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