Chapter 1 Intro to Formal Lang and Automata

8/7/2019 Chapter 1 Intro to Formal Lang and Automata

1/31

Formal Languages and Automata

Theory


2/31

What do we do here? We construct abstract models of

computers and computations

These models possess the importantfeatures that are common to bothhardware and software

These features are essential to many ofthe special and complex constructs weencounter while working with computers


3/31

Models The construction of models is one of the

essentials of any scientific discipline.

The usefulness of a discipline is often

dependent on the existence of simple,

yet powerful, theories and laws.


4/31

Automata To model the hardware of the computer, a

construct called automaton is used.

An automaton is a construct that

possesses all the indispensable features

of a digital computer

Many automaton form an automata


5/31

Automaton It accepts input, produces output, may

have some temporary storage and can

make decisions in transforming the input

into the output


6/31

Formal Language It is an abstraction of the general

characteristics of programminglanguages

It consists of a set of symbols and somerules of formation of sentences

Sentences are formed by grouping thesymbols


7/31

Formal Language A formal language is the set of all

strings permitted by the rules offormation

Study of formal languages is very usefulin learning about the different

programming languages


8/31

What is a language? A system for the expression of certain ideas,

facts, or concepts, including a set of

symbols and rules for their manipulation


9/31

Mathematical definition of a

language This shall require us to understand the

following concepts first

Alphabets

Strings

Concatenation of strings etc.


10/31

Alphabet An ALPHABET is a nonempty set of

symbols

It is denoted by 7

Example:

7!{a,b}where a and b are symbols


11/31

Strings Strings are constructed from the

individual symbols

Strings are finite sequences of symbolsfrom the alphabet

Example : aabba, ababaaa, abbbaaa,etc are the strings formed by t hesymbols of the alphabet


12/31

Assumptions Lower case letters a,b,c, are used for

elements of the alphabet

Lower case letters u,v,w, for stringnames eg w=aabbaba

This indicates that w is a string havingspecific value aabbaba


13/31

Concatenation of the strings Two strings are concatenated by

appending the symbols of one string tothe end of the other string

Example u=aaabbb

v=abbabba

Concatenated string uv=aaabbbabbabba


14/31

Reverse of the string The reverse of a string is obtained by

writing the symbols in reverse order.

Example

w

R

= anan-1an-2a0Where w= a0a1a2an


15/31

Length of the string The length of the string is the number

of symbols in the string

|w| = 5 if w = aabaa

Empty String has no symbols and isdenoted by P

`P` = 0


16/31

7

and 7

7 is an alphabet

7

is the set of all strings obtained byconcatenating zero or more symbolsfrom 7

7always contains Pthen

7 !7 _Pa


17/31

Finiteness 7is always finite

7

and 7

are always infinite


18/31

Language A language L is defined very generally

as a subset of7

Astring in a language L will be called asentence of L


19/31

Problem 7!_a,ba

7

!_Pa,b,aa,ab,ba,bb,aaa,aab,aba,abb, baa,bab,bba,bbb,aaaa, .}

L = {a,aa,aab} is a language on 7as Lis a subset of7 and is finite

L = {anbn:n>0} is also a subset of7

but it is infinite


20/31

Concatenation of two

Languages L1L2 = {xy :x L1 and y L2 }


21/31

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 0s and 1s on 7 = {0,1})

The automaton can produce output ofsome form


22/31

Automaton It has a mechanism for reading input

where input is a string over a givenalphabet written on an input file, whichan automaton can read but not change


23/31

Automaton An automaton is assumed to operate in

discrete time frame

A finite automaton has a set of states

Its control moves from state to state inresponse to external inputs


24/31

Components of an automaton Input file : Contains strings of input

symbols

Storage unit: consists of an unlimitednumber of cells, each capable of holdinga single symbol from an alphabet

Controlunit: can be in any one of a finitenumber of internal states and can change

states in defined manner


25/31

Automaton An automaton operates in discrete time

frame

With every automaton, a transitionfunction is associated which gives thenext state in terms of the current state


26/31

States of a systemLetus have a code

int a,b,x;

a=5;b=0; x=a+b;

printf(%d%d%d\n,a,b,x);

a=a*3

printf(%d%d%d\n,a,b,x);

b=a+x;printf(%d%d%d\n,a,b,x);

States of thissystem arerepresented by

a tuple (a,b,x)

(15,0,5)

(5,0,5)

(15,20,5)


27/31

Transitions from start state to

end state

There are 4 objectsCabbage, Man, Wolf and

Goat on one bank and thereis a boat

All 4 have to cross the riverso that (C,G) or (W,G) arenot left at one place

Man rows the boat andtakes one at a time

Present state

M,C,G,W = J

final state

J= M,C,G,W

transition


28/31

Intermediate states

Class assignment

(M

,C,G,W=J=..= (J=M

,C,G,W)


29/31

Automaton An automaton can be represented by a

graph in which the vertices give theinternal states and the edges transitions

The labels on the edges show whathappens (in terms of input and output)

during the transitions


30/31

Applications of automata Compiler design

Digital design


31/31

Pictorial representation of an

automaton: Transition Graphs

q0

Start State

Average

q1

Work hard withlearningWork hard/work

hard withoutlearning

IntermediateState

Good

Workhard

q2

Work hard withintelligent learning

Final State

Excellent

Chapter 1 Intro to Formal Lang and Automata

Documents

Transcript of Chapter 1 Intro to Formal Lang and Automata