A new format for FAs

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Theory of Automata

(CS402)

Theory of Automata

Lecture N0. 37

Reading Material

Introduction to Computer Theory

Chapter 14

Summary

New format for FAs, input TAPE, START, ACCEPT , REJECT, READ states Examples of New Format of FA,

PUSH Down STACK , PUSH and POP, Example of PDA

A new format for FAs

A class of machines (FAs) has been discussed accepting the regular language i.e. corresponding to a regular

language there is a machine in this class, accepting that language and corresponding to a machine of this class

there is a regular language accepted by this machine. It has also been discussed that there is a CFG

corresponding to regular language and CFGs also define some nonregular languages, as well

There is a question whether there is a class of machines accepting the CFLs? The answer is yes. The new

machines which are to be defined are more powerful and can be constructed with the help of FAs with new

format.

To define the new format of an FA, the following are to be defined

Input TAPE

The part of an FA, where the input string is placed before it is run, is called the input TAPE.

The input TAPE is supposed to accommodate all possible strings. The input TAPE is partitioned with cells, so

that each letter of the input string can be placed in each cell. The input string abbaa is shown in the following

input TAPE.

Cell i Cell ii Cell iii

The character indicates a blank in the TAPE. The input string is read from the TAPE starting from the cell (i).

It is assumed that when first is read, the rest of the TAPE is supposed to be blank.

The START state

This state is like initial state of an FA and is represented by

START

An ACCEPT state

This state is like a final state of an FA and is expressed by

A REJECT state

This state is like dead-end non final state and is expressed by

REJECT

Note: It may be noted that the ACCEPT and REJECT states are called the halt states.

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Theory of Automata

(CS402)

A READ state

This state is to read an input letter and lead to some other state. The READ state is expressed by

READ

Example

Before some other states are defined consider the following example of an FA along with its new format

Obviously the above FA accepts the language of strings, expressed by (a+b)*a. Following is the new format of

the above FA

START

READ

REJECT

Note

The edge should not be confused with Y-labeled edge. The -edges start only from READ boxes and lead to

halt states.

a,b

Example

The above FA accepts the language expressed by (a+b)*bb(a+b)*

START

a,b

READ

REJECT

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Theory of Automata

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PUSHDOWN STACK or PUSHDOWN STORE

PUSHDOWN STACK is a place where the input letters can be placed until these letters are refered again. It can

store as many letters as one can in a long column.

Initially the STACK is supposed to be empty i.e. each of its storage location contains a blank.

PUSH

A PUSH operator adds a new letter at the top of STACK, for e.g. if the letters a, b, c and d are pushed to the

STACK that was initially blank, the STACK can be shown as

STACK

The PUSH state is expressed by

PUSH a

When a letter is pushed, it replaces the existing letter and pushes it one position below.

POP and STACK

POP is an operation that takes out a letter from the top of the STACK. The rest of the letters are moved one

location up. POP state is expressed as

POP

Note

It may be noted that popping an empty STACK is like reading an empty TAPE, i.e. popping a blank character .

It may also be noted that when the new format of an FA contains PUSH and POP states, it is called

PUSHDOWN Automata or PDAs. It may be observed that if the PUSHDOWN STACK (the memory structure)

is added to an FA then its language recognizing capabilities are increased considerably. Following is an example

of PDA

Example

Consider the following PDA

START

PUSH a

POP2

POP1

a,b

REJECT

STACK

The string aaabbb is to be run on this machine. Before the string is processed,

D the

string is supposed to be placed on the TAPE and the STACK is supposed to

empty as shown below

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Theory of Automata

(CS402)

∫

Reading first a from the TAPE we move from READ1State to PUSH a state, it causes the letter a deleted from

the TAPE and added to the top of the STACK, as shown below

TAPE a a a b b b D D ∫

STACK a

Reading next two a's successively, will delete further two a's from

STACKthe

TAPE and

add these letters to the top of the STACK, as shown below

TAPE a a a b b b D D ∫

///

Then reading the next letter which is b from the TAPE will lead to the POP1 state. The top letter at the STACK

STACK STACK is shown

is a, which is popped out and READ2 state is entered. Situation of TAPE and

below

TAPE a a a b b b D D ∫

////

Reading the next two b's successively will delete two b's from the TAPE, will lead to the

STACK

POP1 state and these b's will be removed from the STACK as shown

below

∫

TAPE /

Now there is only blank character left, to be read from the TAPE, which leads to POP2 state. While the only

blank characters is left in the STACK to be popped out and the ACCEPT state is entered, which shows that the

string aaabbb is accepted by this PDA. It may be observed that the above PDA accepts the language

{anbn: n = 0,1,2, ... }.

Since the null string is like a blank character, so to determine how the null string is accepted, it can be placed in

the TAPE as shown below

TAPE D D D ∫

Reading at state READ1 leads to POP2 state and POP2 state contains only , hence it leads to ACCEPT state

and the null string is accepted.

Note: The process of running the string aaabbb can also be expressed in the table given in the next lecture.

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