Push Down Automata New

8/3/2019 Push Down Automata New

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Push Down Automata

• Introduction

• Schematic representation of Pushdown automata andits working

•

Formal definition of PDA> variations of Transition functions and their meaning

> Move of a PDA

• Instantaneous Description of PDA

• Language acceptance of PDA

> Language Acceptance from the final state

> Language Acceptance from the empty stack



• Graphical representaions of PDA

• Construction of PDA for CFL

•

Equivalence of PDA’s and CFG’s • Deterministic Push Down Automata



Introduction

• A simple Automata can accept only Regular

languages and they can cannot accept context

free languages. This is because FA have limited

memory and recognition of CFL requires

unbounded memory. In order to have this

ability for FA, extra storage mechanism is

provided and is called as STACK. So, A FA withstack as storage Mechanism is called Push

Automata



Formal Defination

A Non deterministic PDA(NPDA) is defined as a 7-tupleM=(Q,∑,Г,δ,q0,Z,F)

where Q= Finite set of internal states

∑= Input symbols called input Alphabet

Г= stack symbols called as stack Alphabet

δ= A transition function

q0= A start state

Z = A symbol on the stack indicates bottom of the

stack

F= A finite set of final states





PDA consists of 4 components namely, Input buffer, Stack,

control Unit and output

1.Input Buffer : This holds the string to be processed. It is divided

into n number of cells and each cell is capable of holding a

single character. There is a reading mechanism and individual

characters are read from left to right, one character at a time

2.Stack : It is an extra storage mechanism which is divided into n

number cells and each cell is capable of holding a single

character. The character Z is used to indicate the bottom andthe reading and writing is done with usual stack operation

i.e., LIFO and decided by the Transition function



Control Unit : This is used to control the Overall

operation of PDA. Initially it is assumed to be in start

state q0, input pointer is pointing to the first

character ’ a ‘of the input string to be processed andZ is the symbol on top of the stack the next state and

STACK operation is decided by the transition function

δ and we say the PDA has taken a move. When this

happens state is changed or it can stay in the samestate and stack is operated with usual stack

operation



It means that the transition function δ takes 3

arguments and is defined as follows

δ : Q X (∑ U {ε}) X Г = Q X Г*

i.e., It takes 3 arguments namely current state of them/c, current input symbol and the current symbol on

the top of the stack and the result of the transition

function is a set pairs (Q,x) where Q is the next state

and x is string that decides the stack operation and

they are as follows



1. δ(q0,a,Z)=(p, aZ)

It means in state q0 ,on i/p symbol ‘a’ and when top of stack

is Z, the PDA enters into state ‘p’ and the i/p symbol ‘a’ is

pushed on to stack over Z.2. δ(q0,a,bZ)=(p,ε)


is ‘b’, the PDA enters into state ‘p’ and the stack top symbol‘b’ is popped from the stack and Z is now on top.



3. δ(q0,a,bZ)=(p,r)


is ‘b’, the PDA enters into state ‘p’ and the stack top symbol

‘b’ is replaced by ‘r’ and r is now on top of the stack. 4. δ(q0,ε ,ε )=(q0,Z)

It means in state q0 ,on empty i/p symbol and when stack

is empty , the PDA enters into state ‘q0 and the symbol Z ispushed onto the stack .



5. δ(q0, ε ,Z )=(q0, ε)

It means in state q0 ,on empty i/p symbol and

when top of stack is Z , the PDA enters into state

‘q0 and the symbol Z is popped from the stack .

4. OUTPUT : whenever input string is exhausted

and PDA stays in the final state or stack is

empty the string is accepted otherwise it is

rejected



Instantaneous Description (ID) of PDA

While processing the string ‘w’ the current and

successive configurations are described by the term

Instantaneous Description ID which is defined astriplet (q,w,u)

where q -> current state

w -> string to be processedu -> current content of stack

Note: |- is used to indicate the Different ID’s



Example: Let the current configuration of PDA be

(q0,aw,bZ)

if transition function is defined as

δ(q0,a,b) = (p, ab) then next configuration of PDA isdescribed as (q0,aw,bZ) |- (p, w, abZ)


δ(q0,a,b) = (p, ε) then next configuration of PDA isdescribed as (q0,aw,bZ) |- (p, w, Z)




δ(q0,a,b) = (p, c) then next configuration of PDA

is described as (q0,aw,bZ) |- (p, w, cZ)



Language Acceptance of a PDA:

1. Language acceptance by final state:

Here PDA has to process entire string w andfinally it has to stay in a state that belongs to

final state. At this moment of time stack

contents are irrelevant. This is defined as

follows

L(M)={ w ε ∑* : (q0,w,z) |- (p, ε,u) }

where p is in F and u is Г*



2. Language acceptance by empty stack:

Here PDA has to process entire string w and

finally it has to empty the stack. This moment

of time current state is irrelevant. This is

defined as follows

L(M)={ w ε ∑* : (q0,w,z) |- (p, ε, ε ) }

where p is current state which is

irrelevant.



Graphical Representations of PDA

PDA’s are represented by Transition diagram and is as

follows :

1. The nodes corresponds to the state of the PDA

2. An arrow labeled start indicates the start state and

doubly circled states are final states

3. If δ(q,a,x) = (p, α) is a transition function then there isan arrow labeled a,x/α from state q to state p.



• Construction of PDA for CFL

L={anbn | n>=0}

L={a

n

b

2n

| n>=0}L={w | w Є (a+b)* and na(w)=nb(w) }

L={w | w Є (a+b)* and na(w)>nb(w) }

L={w | w Є (a+b)* and na(w)<nb(w) }L={anbm c(n+m) | n>=0}

L={a2nbn | n>=0}



Equivalence of CFG and PDA’s

We Know that the language accepted by

PDA is Context free language and these

languages are represented by context-free-

grammar. From every context-free-grammar

we can construct PDA and vice versa. Let us

discuss the construction of PDA fromContext-free-grammar



• Construction of PDA from Context-free-

grammar

1 By GNF notation of CFG

2 By Empty stack( Without GNF notation)

f



1.By GNF Notation of CFG:

Step1 : Convert the grammar into GNFStep2 : Let q0 be the start state and Z be the initial

symbol on the stack. Without consuming any Input,

push the start symbol onto the stack and change the

state to q1. The transition is as follows:

δ(q0,ε,Z)= (q1,SZ)

Step3 : For each production of the form

a) A-> aα introduce the transition δ(q1,a,A)= (q1,α)

b) A-> a introduce the transition δ(q1,a,A)= (q1,ε)



Step 4 Finally, in state q1, without consuming

the input, change the state to qf which is an

accepting state. The transition as follows

δ(q0,ε,Z)= (qf,Z)

Example : S->aABC

A-> aB | a

B-> bA | b

C-> a



• Its PDA construction is as follows

δ(q0,ε,Z)= (q1,SZ)

S->aABC δ(q1,a, S)= (q1,ABC)

A-> aB | a δ(q1,a,A)= (q1,B)

δ(q1,a,A)= (q1,ε)

B-> bA | b δ(q1,b,B)= (q1,A)

δ(q1,b,B)= (q1,ε)

C-> a δ(q1,a,C)= (q1,ε)

δ(q1,ε,Z)= (qf, Z)



2 By Empty stack

( Without GNF notation)

Let G=(V,T,S,P) be a CFG then PDA M that

accepts L(G) by empty as follows

M =({q),T, VUT, δ, q, S) where transition

function δ is defined as follows

1.For each variable A

δ(q,ε,A) = (q,β) such that A->β is Production in P

2. For each terminal a , δ(q, a, a) = (q,ε)



Deterministic Push Down Automata

Definition : Let M=(Q,∑,Г,δ,q0,Z,F) be a PDA. It is

said to be deterministic if it satisfies the

following conditions

1.δ(q,a,A) is unquily defined for each

qЄQ, aЄ∑ and AЄГ

2.δ(q,Є,A) is defined then δ(q,a,A) is

undefined for all a Є ∑



Properties of Context free

Languages

•Closure Properties

•

Pumping Lemma



Closure properties of CFG’s

We know that the family of regularlanguages are closed under all the set

operations namely, Union, concatenation,Kleen star, intersection and complement.Similarly, The Family of context free languagesare closed under Union, concatenation, Kleenstar only and not closed under Intersection.Let discuss these properties one by one



1 Theorem: If L1 and L2 are context free languages

then L1UL2 is a context free language

Proof: If L1 and L2 are context free languages, then

each of them has a context free grammar

Let G1 and G2 are the grammars for L1 and

L2 respectively. Proof requires that grammars

have no Non terminal common. So, subscript allof G1’s non terminals with 1 and subscript all of

G2’s non terminal with 2



Now we combine the two grammars into one grammar thatwill generate the union of two languages. To do this, weadd one new non terminal S and two new productions andis as follows

S->S1 | S2

Here S is starting non terminal for the new UNIONgrammar and can replaced by either S1 ( Start Symbol of G1) or S2 ( start symbol of G2), there by generating either astring from L1 or from L2. Since the non terminals of thetwo original languages are different in the derivation If wereplace S by S1 we use only the production of grammar G1and if we replace S by S2 we use only the production of grammar G2 . The resulting grammar G is usually a Union of G1 and G2. Hence the argument is proved.



2 Theorem : If L1 and L2 are context free languages , thenL1L2 is a context free languages

Proof: Here also we subscript all of the non terminals of G1 with a 1 and all of the non terminals of G2 with a 2.

Then we add a new non terminal S and one new ruleto the combined grammar

S->S1S2

The resulting grammar G, in the derivationgenerates first strings of L1 by start symbol S1 and thenit generates strings of L2. The resulting strings arenothing but the concatenation of strings of L1 and L2.Hence the argument is proved



3 Theorem : If L is a context free language, then L*is also context free language.

Proof: Subscript all the non terminals of theGrammar for L with a 1. Then add a new start

symbol S and the new rule as followsS->S1 S |Є

Here the rule S->S1 S is used to generate thestrings of several instances and the rule S->|Є isused to terminate the strings generated from S1.The resulting strings are nothing the star closureof L. Hence the argument is proved.



4 Theorem: If L1 and L2 are context free

languages then L1ПL2 is not context free

language.

Proof: The two context free languages are not

closed under Intersection is proved by

considering the examples

Let L1= {an bn cm | n, m>=0} is Context Free

language



Also, Let L2= {am bn cn | n, m>=0} is Context Free

language

Since, both are context free languages there

exits a Context free grammar and is as follows

G1 : S->AB G2: S->AB

A->aAb |Є A->aA |Є

B-> cB |Є B->bBc |Є



The strings in L1 contains the same number of a’s and b’s, while the strings in L2 contain the samenumber b’s and c’s. The intersections of L1 and L2contains the strings of both in L1 and L2. i.e samenumber of a’s, b’s and c’s

Thus L1ПL2= {an bn cn | n >=0}

Using pumping lemma the above language can

proved it is not Context free language. So, theclass of context free languages are not closedunder Intersection.



5 Theorem: If L is a context free language thenComplement of L is not context free language.

Proof: From set Identity. Complement can be

expressed as followsL1ПL2 = Complement

(Complement(L1)UComplent(L2))

we know that Context free languages are notclosed under Intersection. Hence context freelanguages are not closed Complement



Pumping Lemma for context free

languages

Statement : Let L be an infinite context free

language. Then there exists some positive

integer ‘m’ such that any wЄ L with |w|>=m

can be decomposed as

w=uvxyz

with |vxy|<=m and |vy|>=1

Such that uvixyiZ Є L

for all i=1,2,3……



Proof: consider the languages L-{ε} and assume

that we have grammar G for it without unit

productions and ε-productions

Since the length of the RHS of a productions isbounded say K and number of variables are finite

, the length of the derivations of any wЄL must be

at least |w|/k. Therefore since L is infinite , thereexists arbitrarily long derivation and derivation

tree of arbitrary height.



Since the number of variables are finite for G theremust be some variable that repeats in thederivation path and is shown in fig…

so the derivation is as follows

S=>uAz=>uvAyz=>uvxyz

where u,v,x,y and z are all strings of terminals.From the above we see that

A=>vAy and A=>xwe can generate all the strings uvixyiz, i=0,1,2..of the grammar and are in L



In the derivation A=>vAy and A=>x no variable

repeats and no unit production and ε-

production v and y cannot be empty strings

because |vy|>=1. hence completes theargument.



Turing Machine(TM)•Introduction

•Schematic representation of Turing Machineand its working

•Formal definition of Turing Machine

> Transition function Move of a TM

•Instantaneous Description of TM

•Language acceptance of TM

•Representation of TM

•Examples on TM



Turing Machines

(At last!)



Introduction



Turing Machine is modified version of PDA and itis more powerful than PDA. Instead of stack it

uses the Tape to store the input symbols.Turing Machine is going to accept all types of languages ( regular (Type-3), context free (type-2)and context sensitive languages(type-1)). Apart

from these languages it can also accept Type-0languages generated by unrestricted grammar.Hence it is called as generalized finite automata.



Schematic representation of Turing Machine and itsworking

It is generalized finite automata and has three maincomponents, namely Tape, Read-write head andcontrol unit



1. Tape : It is storage device of infinite length

and is divided into n number cells. Each cell is

capable of holding a single character. The

string to be processed is stored and read fromleft to right. Before and after the strings the

blank character are stored



2. Read-write Head

It is a reading / writing mechanism that canread and write a symbol from where it is

pointing. The reading/writing and movementof read-write head is controlled by control unit

3. Control unit: This component drives theworking of Turing Machine. It controlsreading/writing operations and movement of read-write head.



At any given instance of time TM is assumed to be in somestate(Initially start state) and the read/write head ispointing to some symbol in the tape, the next state, read /write operations and movement of read/write head isdecided by the transition function δ and is as follows:

Next Symbol to be

state replaced in tape

↓ ↓

δ(q, a) = (p, b, R/L)

↑ ↑ ↑Current I/P symbol Movement of head (R-right, L-left)

state pointed by R/W head



Formal Definition:The Turing M = (Q,∑,Г,δ,q0,B,F) where

Q = Finite set of internal states

∑ = Finite set of input alphabets Г = Finite set of Tape alphabets

δ = Transition Function (δ :(Q,Г)→Q x Г x (R,L)

q0 = Start state belongs to Q

B = A special symbol indicating Blank CharacterF =Set of final states and is subset of Q



Instantaneous Description (ID) of TM :

While processing the string W the current

and successive configuration of the TM is

described by the term Instantaneous

Description (ID). Here it is defined on the

whole string and (not on the string that is to

processed) the current state of the Machineand formally it is defined as follows



An ID of Turing machine is a string in αqβ, whereq is the current state, αβ is string made from Tapesymbols. The read/write head is pointing to thefirst character of the string β

The initial ID is denoted by q0αβ where q0 is thestart state and the read/write is pointing to thefirst character of α from left

The final ID is denoted as αβqB where qЄF is thefinal state and the read/write head is pointing tothe blank character denoted by B



Example : If the Initial ID of TM is Say

……BBBBa1a2a3a4qa5a6a7a8a9BBBB….

α ↑ β

If the Transition function δ(q,a5)=(p,b5,R) then

Next ID is defined as follows

a1a2a3a4qa5a6a7a8a9 |- a1a2a3a4b5pa6a7a8a9



If the Transition function δ(q,a5)=(p,c5,L) then

Next ID is defined as follows

a1a2a3a4qa5a6a7a8a9 |- a1a2a3pa4 c 5a6a7a8a9



Language acceptance of TM:

After processing the entire string the machinehas to stay in final state and read/write head

should point to blank Character. Formally it isdefined as follows

Let M = (Q,∑,Г,δ,q0,B,F) be the Turing Machine.The language accepted by M is defined as

L(M) = {W |q0 w |- α1pα2, where wЄ∑*, pЄFand α1,α2 Є Г* }

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Push Down Automata New