TOC 2marks Q & A

SKR ENGINEERING COLLEGE

DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

SUBJECT NAME : THEORY OF COMPUTATION(CS2303)

Year: Third Year CSE( Sem:V) Faculty Name: L.Sharmila

2 marks Questions and Answers

UNIT I AUTOMATA

1. Define: (i) Finite Automaton(FA) (ii)Transition diagram April /May 2008

FA consists of a finite set of states and a set of transitions from state to state that occur on input symbols chosen from an alphabet _. Finite Automaton is denoted by a 5- tuple(Q,Σ, δ,q0,F), where Q is the finite set of states , _ is a finite input alphabet, q0 in Q is the initial state, F is the set of final states and _ is the transition mapping function Q * _ to Q.

Transition diagram is a directed graph in which the vertices of the graph correspond to the states of FA. If there is a transition from state q to state p on input a, then there is an arc labeled ‘ a ‘ from q to p in the transition diagram.

2. What is a : (a) String (b) Regular language

A string x is accepted by a Finite Automaton M=(Q, Σ, δ.q0,F) if δ (q0,x)=p, for some p in F.FA accepts a string x if the sequence of transitions corresponding to the symbols of x leads from the start state to accepting state.

The language accepted by M is L(M) is the set {x | _(q0,x) is in F}. A language is regular if it is accepted by some finite automaton.

3. Why are switching circuits called as finite state systems?

A switching circuit consists of a finite number of gates, each of which can be in any one of the two conditions 0 or 1.Although the voltages assume infinite set of values,the electronic circuitry is designed so that the voltages orresponding to 0 or 1 are stable and all others adjust to these value. Thus control unit of a computer is a finite statesystem.

4. What is Deductive proofs?

A deductive proof consists of a sequence of statements which starts from a hypothesis or a given statement to a conclusion. Each step is satisfying some logical principle.

5. Define proof by contrapositive.

It is the other form of if then statement. The contrapositive of the statement “ if H then C “ is If not C then not H “.

6. Define the concatenation of two strings.

Suppose x and y are two strings then the concatenation of x and y is xy.

Ex: if x = 0011 and y = 1100 then xy = 00111100.

7. What are the applications of automata theory?

_ In compiler construction.

_ In switching theory and design of digital circuits.

_ To verify the correctness of a program.

_ Design and analysis of complex software and hardware systems.

_ To design finite state machines such as Moore and mealy machines.

8. What is Moore machine and Mealy machine?

A special case of FA is Moore machine in which the output depends on the state of the machine. An automaton in whch the output depends on the transition and current input is called Mealy machine.

9. Define Language.

A set of strings all of which are chosen from some S^*, where S is a particular alphabet is called a language.

Ex: The language L over {0,1} where set of strings with an equal number 0’s and 1’s.

L = { e, 01, 10, 0011, 0101, 1001, 1100,……}

10. What are the components of Finite automaton model?

The components of FA model are Input tape, Read control and finite control.

(a)The input tape is divided into number of cells. Each cell can hold one i/p symbol

. (b)The read head reads one symbol at a time and moves ahead.

( c)Finite control acts like a CPU. Depending on the current state and input symbol read from the input tape it changes state.

11. Differentiate NFA and DFA

NFA or Non Deterministic Finite Automaton is the one in which there exists many paths for a specific input from current state to next state. NFA can be used in theory of computation because they are more flexible and easier to use than DFA .

Deterministic Finite Automaton is a FA in which there is only one path for a specific input from current state to next state. There is a unique transition on each input symbol.(Write examples with diagrams).

12. What is ε-closure of a state q0?

ε -closure(q0 ) denotes a set of all vertices p such that there is a path from q0 to p labeled ε. Example :closure(q0)={q0,q1}

13. Give the examples/applications designed as finite state system.

Text editors and lexical analyzers are designed as finite state systems. A lexical analyzer scans the symbols of a program to locate strings corresponding to identifiers, constants etc, and it has to remember limited amount of information .

14. Define automaton.

Automaton is a abstact computing device. It is a mathematical model of a system,with discrete inputs, outputs, states and set of transitions from state to state that occurs on input symbols from alphabet Σ.

15. What is the principle of mathematical induction. April/May 2008

Let P(n) be a ststement about a non negative integer n. Then the principle of mathematical induction is that P(n) follows from

(i) P(1) and

(ii) P(n-1) implies P(n) for all n>1.

Condition(i) is called the basis step and condition (ii) is called the inductive step. P(n-1) is called the induction hypothesis.

16. List any four ways of theorem proving

(i) Deductive

(ii) If and only if

(iii) Induction

(iv) Proof by contradiction.

17. Define TOC

TOC describes the basic ideas and models underlying computing. TOC suggests various abstract models of computation, represented mathematically.

18. What are the applications of TOC?

Compiler Design

Robotics

Artificial Intelligence

Knowledge Engineering.

19. Define a Deterministic Finite Automaton.

A Determinstic finite automaton consists of :

a. A finite set of states, often denoted by Q

b. A finite set of input symbols, often denoted by S

c. A transition function that takes as arguments a state and an input symbol and returns a state.

d. A start state, one of the states in Q

e. A set of final or accepting states F.

20. Define a Non Deterministic Finite Automaton

A Non Deterministic Finite Automaton consists of

a. A finite set of states, often denoted by Q

b. A finite set of input symbols, often denoted by S

c. A transition function that takes as arguments a state and an input symbol in S, and returns a subset of Q.

d. A start state, one of the states in Q

e. A set of final or accepting states F.

21. Define Transition Diagram.

Transition Diagram associated with DFA is a directed graph whose vertices correspond to states of DFA, The edges are the transitions from one state to another.

22. What are the properties of Transition Function(δ)

(i) δ(q.ε )=q

(ii) For all strings w and input symbol a

Δ(q,aw)= δ(δ(q.a),w)

Δ(q,wa)= δ(δ(q,w).a)

(iii) The transition function δ can be extended that operates on states and strings.

23. Lists the operations on Strings.

(i) Length of a string

(ii) Empty string

(iii) Concatenation of string

(iv) Reverse of a string

(v) Power of an alphabet

(vi) Kleene closure

(vii) Substring

(viii) Palindrome

24. Lists the operations on Languages.

(i) Product

(ii) Reversal

(iii) Power

(iv) Kleene star

(v) Kleene plus

(vi) Union

(vii) Intersection

25. Define Graphs.

A graph denoted by G=(V,E) consists of a finite set of vertices (or) nodes V and a set E, a pair of vertices called edges.

26. Define the language accepted by a NFA

We define the language of a NFA A = (Q, S, d, q₀, F) by

L(A) = { w/d^{^}(q₀, w) ∩F ≠ j}.

27. Define the language accepted by a DFA

We define the language of a DFA A = (Q, S, d, q₀, F) by

L(A) = { w/d^{^}(q₀, w) is in F}.

28. Define Substring.

A string v appears within another string w(w=uv) is called “substring of w.” IF w=uv, then substrings u & v are said to be prefix and suffix of w respectively.

UNIT II REGULAR EXPRESSIONS AND LANGUAGES

1. What is a regular expression?

A regular expression is a string that describes the whole set of strings according to certain syntax rules. These expressions are used by many text editors and utilities to search bodies of text for certain patterns etc. Definition is: Let _ be an alphabet. The regular expression over _ and the sets they denote are:

i. _ is a r.e and denotes empty set.

ii. _ is a r.e and denotes the set {_}

iii. For each ‘a’ in _ , a+ is a r.e and denotes the set {a}.

iv. If ‘r’ and ‘s’ are r.e denoting the languages R and S respectively then (r+s),

(rs) and (r*) are r.e that denote the sets RUS, RS and R* respectively.

2. Differentiate L* and L+

_

L* denotes Kleene closure and is given by L* =U Li i=0

example : 0* ={_ ,0,00,000,…………………………………}

Language includes empty words also.

_

L+ denotes Positive closure and is given by L+= U Li i=1 q0 q1

3. What is Arden’s Theorem?

Arden’s theorem helps in checking the equivalence of two regular expressions. Let P and Q be the two regular expressions over the input alphabet _. The regular expression R is given as : R=Q+RP Which has a unique solution as R=QP*.

4. Write a r.e to denote a language L which accepts all the strings which begin or end with either 00 or 11.

The r.e consists of two parts:

L1=(00+11) (any no of 0’s and 1’s) =(00+11)(0+1)*

L2=(any no of 0’s and 1’s)(00+11) =(0+1)*(00+11)

Hence r.e R=L1+L2 =[(00+11)(0+1)*] + [(0+1)* (00+11)]

5. Construct a r.e for the language over the set _={a,b} in which total number of a’s are divisible by 3

( b* a b* a b* a b*)*

6. What is: (i) (0+1)* (ii)(01)* (iii)(0+1) (iv)(0+1)+

(0+1)*= { _ , 0 , 1 , 01 , 10 ,001 ,101 ,101001,…………………}

Any combinations of 0’s and 1’s.

(01)*={_ , 01 ,0101 ,010101 ,…………………………………..}

All combinations with the pattern 01.

(0+1)= 0 or 1,No other possibilities.

(0+1)+= {0,1,01,10,1000,0101,………………………………….}

7. Reg exp denoting a language over _ ={1} having (i) even length of string (ii) odd length of a string

(i) Even length of string R=(11)*

(ii) Odd length of the string R=1(11)*

8. Reg exp for: (i) All strings over {0,1} with the substring ‘0101’ (ii) All strings beginning with ’11 ‘ and ending with ‘ab’ (iii) Set of all strings over {a,b}with 3 consecutive b’s. (iv) Set of all strings that end with ‘1’and has no substring ‘00’

(i)(0+1)* 0101(0+1)*

(ii)11(1+a+b)* ab

(iii)(a+b)* bbb (a+b)*

(iv)(1+01)* (10+11)* 1

9. Construct a r.e for the language which accepts all strings with atleast two c’s over the set Σ={c,b}

(b+c)* c (b+c)* c (b+c)*

10. What are the applications of Regular expressions and Finite automata Lexical analyzers and Text editors are two applications.

Lexical analyzers:

The tokens of the programming language can be expressed using regular expressions.

The lexical analyzer scans the input program and separates the tokens.For eg identifier can be expressed as a regular expression

as: (letter)(letter+digit)*

If anything in the source language matches with this reg exp then it is recognized as an identifier.The letter is{A,B,C,………..Z,a,b,c….z} and digit is {0,1,…9}.Thus reg exp identifies token in a language.

Text editors:

These are programs used for processing the text. For example UNIX text editors uses the reg exp for substituting the strings such as: S/bbb*/b/

Gives the substitute a single blank for the first string of two or more blanks in a given line. In UNIX text editors any reg exp is converted to an NFA with Єtransitions, this NFA can be then simulated directly.

11. .Reg exp for the language that accepts all strings in which ‘a’ appears tripled overthe set Σ ={a}

reg exp=(aaa)*

12. .What are the applications of pumping lemma?

Pumping lemma is used to check if a language is regular or not.

(i) Assume that the language(L) is regular.

(ii) Select a constant ‘n’.

(iii) Select a string(z) in L, such that |z|>n.

(iv) Split the word z into u,v and w such that |uv|<=n and |v|>=1.

(v) You achieve a contradiction to pumping lemma that there exists an ‘i’ Such that uvi

w is not in L.Then L is not a regular language.

13. What is the closure property of regular sets?

The regular sets are closed under union, concatenation and Kleene closure.

r1Ur2= r1 +r2

r1.r2= r1r2

( r )*=r*

The class of regular sets are closed under complementation, substitution, homomorphism and inverse homomorphism.

14. .Reg exp for the language such that every string will have atleast one ‘a’ followed by atleast one ‘b’.

R=a⁺b⁺

15. Write the exp for the language starting with and has no consecutive b’s .

reg exp=(a+ab)*

16. Construct a regular expression denoting odd numbers in their binary representation

{0/1}^*1

17. Construct a regular expression denoting even numbers in their binary representation

{0/1}^*0

18. Construct a regular expression denoting the set of all strings over {a,b} such that all starts with a and ends with b

a{a/b}^*b

19. Construct a regular expression denoting the set of all strings over {a,b} such that all starts with a and ends with ab

a{a/b}^*ab

20. Construct a regular expression denoting the set of all strings over {a,b} such that all ends with abb

{a/b}^*abb

21. Construct a regular expression denoting the set of all strings over {a,b} such that all contains three a’s.

b^*ab^*ab^*ab^*

^22. What does the following regular expression denote 0^*1^*2^*

The set of all words over {0,1} such that all starts with 0 number of 0’s or 1 0’s or more number of 0’s followed by similar patterns of 1’s and 2’s.

23. Construct a regular expression for the set of strings that consist alternate 0’s and 1’s

(01)^* + (10)^* + 0(10)^* + 1(01)^*

24. State Pumping lemma? April/May 2008

Let L be a regular language. Then there exists a constant n (which depends on L) such that for every string w in L such that | w | > n we can break w into three strings w = xyz, such that,

(i) v ≠ e

(ii) | xy | < n

(iii) For all k > 0 the string xy^kz is also in L.

25. When do you say two states p and q are equivalent?

We say that two states p and q are equivalent if

for all input strings w, d^{^}(p, w) is an accepting state if and only if d^{^}(q, w) is an accepting state.

26. Lists on the closure properties of Regular sets.

(i) Union

(ii) Concatenation

(iii) Closure

(iv) Complementation

(v) Intersection

(vi) Transpose

(vii) Substitutions

(viii) Homomorphism

27. Let R be any set of regular languages. IsUR regular? Prove it.

Yes. Let P,Q be any two regular languages .As per theorem

L( R )=L(P UQ)

=L(P+Q)

Since ‘+’ is a operator for regular expresstions L( R ) is also regular.

28. Show that (r^*)^*=r^* for a regular expression r,

(r^*)^*=={ε,r,rr,………….}= r^*

29. What are the three methods of conversion of DFA to RE?

(i) Regular Expression equation method

(ii) Arden’s Theorem.

(iii) State elimination technique,

30. What are the algorithms of minimization DFA?

(i) Myhill-Nerode Theorem

(ii) Construction of π_final from π.

UNIT III CONTEXT-FREE GRAMMARS AND LANGUAGES

1. What are the applications of Context free languages

Context free languages are used in :

(i) Defining programming languages.

(ii) Formalizing the notion of parsing.

(iii) Translation of programming languages.

(iV) String processing applications.

2. What are the uses of Context free grammars?

Ø Construction of compilers.

Ø Simplified the definition of programming languages.

Ø Describes the arithmetic expressions with arbitrary nesting of balanced parenthesis { (, ) }.

Ø Describes block structure in programming languages.

Ø Model neural nets.

3. Define a Context Free Grammar

A context free grammar (CFG) is denoted as G=(V,T,P,S) where V and T are finite set of variables and terminals respectively. V and T are disjoint. P is a finite set of productions each is of the form A->_ where A is a variable and _ is a string of symbols from (V U T)*.

4. What is the language generated by CFG or G?

The language generated by G ( L(G) ) is {w | w is in T* and S=>w. That is a G string is in L(G) if:

(1) The string consists solely of terminals.

(2) The string can be derived from S.

5. What is : (a) CFL (b) Sentential form

L is a context free language (CFL) if it is L(G) for some CFG G.

A string of terminals and variables α is called a sentential form if: S => α ,where S is the start symbol of the grammar.

6. What is the language generated by the grammar G=(V,T,P,S) where

P={S->aSb, S->ab}?

S=> aSb=>aaSbb=>…………………………..=>aⁿbⁿ

Thus the language L(G)={ aⁿbⁿ | n>=1}.The language has strings with equal number of a’s and b’s.

7. What is :(a) derivation (b)derivation/parse tree (c) subtree

(a) Let G=(V,T,P,S) be the context free grammar. If A-> β is a production of P and α and γ are any strings in (VUT)* then α A γ => αβγ

(b) A tree is a parse \ derivation tree for G if:

(i) Every vertex has a label which is a symbol of VU TU{_}.

(ii) The label of the root is S.

(iii) If a vertex is interior and has a label A, then A must be in V.

(iv) If n has a label A and vertices n1,n2,….. nk are the sons of the vertex n in order from left with labels X1,X2,………..Xk respectively then A X1X2…..Xk must be in P.

(v) If vertex n has label _ ,then n is a leaf and is the only son of its father.

(c ) A subtree of a derivation tree is a particular vertex of the tree together with all its descendants ,the edges connecting them and their labels.The label of the root may not be the start symbol of the grammar.

8. If S->aSb | aAb , A->bAa , A->ba .Find out the CFL

soln. S->aAb=>abab

S->aSb=>a aAb b =>a a ba b b(sub S->aAb)

S->aSb =>a aSb b =>a a aAb b b=>a a a ba b bb

Thus L={aⁿb^ma^mbⁿ, where n,m>=1}

9. What is a ambiguous grammar?

A grammar is said to be ambiguous if it has more than one derivation trees for a sentence or in other words if it has more than one leftmost derivation or more than one rightmost derivation.

10. Find CFG with no useless symbols equivalent to : S→AB | CA ,

B→BC | AB, A→a , C→aB | b.

S-> AB

S->CA

B->BC

B->AB

A->a

C->aB

C->b are the given productions. A symbol X is useful if S => αXβ => w

The variable B cannot generate terminals as B->BC and B->AB. Hence B is useless symbol and remove B from all productions. Hence useful productions are: S->CA , A->a , C->b

11. Construct CFG without Є production from : S →a | Ab | aBa , A →b | Є , B →b | A.

S->a

S->Ab

S->aBa

A->b

A->Є

B->b

B->A are the given set of production.

A->Є is the only empty production. Remove the empty production

S-> Ab , Put A->Є and hence S-> b.

If B-> A and A->Є then B ->Є

Hence S->aBa becomes S->aa .

Thus S-> a | Ab | b | aBa | aa

A->b

B->b

Finally the productions are: S-> a | Ab | b | aBa | aa A->b

B->b

12. Define a Context Free Grammar

A CFG is a grammar whose productions are of the form

A -> α where A Є V

and α Є (V U T)^*

13. Construct a CFG for the language L(G) = {0ⁿ1ⁿ : n > 1}.

G = {V = {S}, T = {0, 1}, P, S}

where

P ={S -> 0S

S -> 01}

14. Construct a CFG for the language L (G) = {0ⁿ1ⁿ: n > 0}.

G = {V = {S}, T = {0, 1}, P, S}

where

P ={S -> 0S1

S ->e}

15. Find a LM derivation for aaabbabbba with the productions.

P : S à aB | bA, A à a | aS | bAA, B à b | bS | aBB

Solution:

S à aB

S à aaBB

S à aaaBBB

S à aaabBB

S à aaabbB

S à aaabbaBB

S à aaabbabB

S à aaabbabbS

S à aaabbabbbA

S à aaabbabbba

16. Find a L(G) S à aSb, S à ab

Solution: S à aSb

à aaSbb

.

.

.

à aⁱSbⁱ

à aⁱabbⁱ

à aⁿbⁿ

L(G) = {aⁿbⁿ, n > 1}

17. For the grammar S à aCa, C à aCa | b. Find L(G)

Solution:

S à aCa

à aaCaa

.

.

.

à aⁿCaⁿ

S à aⁿbaⁿ

L(G) = {aⁿbaⁿ, n > 0}

18. Construct a CFG for the language over {a,b} which contains palindrome strings.

G = {V = {S}, T = {a, b}, P, S}

where

P ={S -> aSa

S -> bSb

S -> a

S -> b

S ->e}

19. Define the language of a Grammar.

If G = (V, T, P, S) is a CFG, the language of G denoted by L(G), is the set of terminal strings that have derivations from the start symbol i.e. L(G) = {w in T / S -> *w}

20. What are the three ways to simplify a context free grammar?

(i) removing the useless symbols from the set of productions.

(ii) By eliminating the empty productions.

(iii) By eliminating the unit productions.

21. What are the properties of the CFL generated by a CFG?

Each variable and each terminal of G appears in the derivation of some word in L .here are no productions of the form A->B where A and B are variables.

22. Find the grammar for the language L={a ²ⁿ bc ,where n>1 }

let G=( {S,A,B}, {a,b,c} ,P , {S} ) where P:

S->Abc

A->aaA | Є

23. .Find the language generated by :S->0S1 | 0A | 0 |1B | 1 A->0A | 0 , B->1B | 1

The minimum string is S-> 0 | 1

S->0S1=>001

S->0S1=>011

S->0S1=>00S11=>000S111=>0000A111=>00000111 Thus L={ 0 ⁿ 1 ^m | m not equal to n, and n,m >=1}

24. Construct the grammar for the language L={ aⁿ b aⁿ | n>=1}.

The grammar has the production P as:

S->aAa

A->aAa | b

The grammar is thus : G=( {S,A} ,{a,b} ,P,S)

25. . Construct a grammar for the language L which has all the strings which are all palindrome over Σ={a, b}.

G=({S}, {a,b} , P, S )

P:{ S -> aSa ,

S-> b S b,

S-> a,

S->b,

S->Є } which is in palindrome.

26. Differentiate sentences Vs sentential forms

A sentence is a string of terminal symbols.

A sentential form is a string containing a mix of variables and terminal symbols or all variables.This is an intermediate form in doing a derivation.

27. .Define Pushdown Automata.

A pushdown Automata M is a system (Q, Σ, Ґ ,δ ,q₀, Z₀,F) here Q is a finite set of states.

Σ is an alphabet called the input alphabet.

Ґ is an alphabet called stack alphabet.

q₀ in Q is called initial state.

Zo in Ґ is start symbol in stack.

F is the set of final states.

Δ is a mapping from Q X ( Σ U {Є} ) X Ґ to finite subsets of

Q X Ґ*.

28. Specify the two types of moves in PDA.

The move dependent on the input symbol(a) scanned is:

δ(q,a,Z) = { ( p1, γ1 ), ( p2,γ2 ),……..( p^m,γ^m ) }

where q qnd p are states , a is in Σ ,Z is a stack symbol and

γi is in Ґ*. PDA is in state q , with input symbol a and Z the top symbol on state enter state p iReplace symbol Z by string γi

The move independent on input symbol is (Є-move):

δ(q,Є,Z)= { ( p1,γ1 ), ( p2,γ2 ),…………( p^m,γ^m ) }. Is that PDA is in state q , independent of input symbol being scanned and with Z the top symbol on the stack enter a state p i and replace Z by γi.

29. What are the different types of language acceptances by a PDA and define them.

For a PDA M=(Q, Σ ,Ґ ,δ ,q0 ,Z0 ,F ) we define :

(i) Language accepted by final state L(M) as:

* { w | (q0 , w , Z0 ) |-- ( p, Є , γ ) for some p in F and γ in Ґ * }.

(ii) Language accepted by empty / null stack N(M) is:

{ w | (q0,w ,Z0) |----( p, Є, Є ) for some p in Q}.

30. Is it true that the language accepted by a PDA by empty stack and final states are different languages.

No, because the languages accepted by PDA ‘s by final state are exactly the languages accepted by PDA’s by empty stack.

31. Define Deterministic PDA.

A PDA M =( Q, Σ ,Ґ ,δ ,q0 ,Z0 ,F ) is deterministic if:

· For each q in Q and Z in Ґ , whenever δ(q,Є,Z) is nonempty then δ(q,a,Z) is empty for all a in Σ.

· For no q in Q , Z in Ґ , and a in Σ U { Є} does δ(q,a,Z) contains more than one element. (Eg): The PDA accepting {wcw R | w in ( 0+1 ) * }.

32. Define Instantaneous description(ID) in PDA.

ID describe the configuration of a PDA at a given instant.ID is a triple such as (q, w ,γ ) , where q is a state , w is a string of input symbols and

γ is a string of stack symbols.

If M =( Q, Σ ,Ґ ,δ ,q0 ,Z0 ,F ) is a PDA we say that

(q,aw,Zα) |-----( p, , βα) if δ(q,a,Z) contains (p, β ).

M ‘a’ may be Є or an input symbol. Example: (q1, BG) is in δ(q1, 0 , ) ells that (q1, 011, GGR )|---- ( q1, 11,BGGR).

33. What is the significance of PDA?

Finite Automata is used to model regular expression and cannot be used to represent non regular languages. Thus to model a context free language, a Pushdown Automata is used.

34. When is a string accepted by a PDA?

The input string is accepted by the PDA if:

o The final state is reached .

o The stack is empty.

35. Give examples of languages handled by PDA.

(1) L={ a ⁿbⁿ | n>=0 },here n is unbounded , hence counting cannot be done by finite memory. So we require a PDA ,a machine that can count without limit.

(2) L= { wwR | w Є {a,b}* } , to handle this language we need unlimited counting capability .

36. Is NPDA (Nondeterministic PDA) and DPDA (Deterministic PDA)equivalent?

The languages accepted by NPDA and DPDA are not equivalent. For example: wwR is accepted by NPDA and not by any DPDA.

37. Construct a PDA that accepts the language generated by the grammar

S à aSbb

S à aab

The PDA A = ({q}, {a,b}, {S,a,b}, d, q, S}

where d :

i. d(q,z₀,S) = {(q, aSbb), (q, aab)}

ii. d(q, a, a) = {(q, e)}

iii. d(q, b, b) = {(q, e)}

38. Construct a PDA that accepts the language generated by the grammar

S à aABB

A à aB | a

B à bA | b

The PDA is given by

A = ({q}, {a,b}, {S, A, B, Z, a, b}, d, q, S}

where d :

d(q, z, S) = {(q, aABB)}

d(q, z, A) = {(q, aB), (q, a)}

d(q, z, B) = {(q, bA), (q, b)}

d(q, a, a) = {(q, e)}

d(q, b, b) = {(q, e)}

39. What is the main difference between pushdown automata and finite automata?

A pushdown automaton has additional stack facility and it can recognize non – regular languages. i.e. Context free languages.

40. What language does deterministic PDA’s accept?

The DPDA’s accept a class of languages that is between the regular languages and the CFL’s.

41. Is it true that non – deterministic PDA is more powerful than that of deterministic PDA? Justify your answer.

No, NPDA is not more powerful than DPDA. Because, NPDA may produce ambiguous grammar by reaching its final state or by emptying its stack. But DPDA produces only unambiguous grammar.

42. What is the additional feature PDA has when compared with NFA? Is PDA superior over NFA in the sense of language acceptance? Justify your answer.

PDA is superior to NFA by having the following additional features.

i. Stack which is used to store the necessary tape symbols and use the state to remember the conditions.

ii. Two ways of language acceptances, one by reaching its final state and another by emptying its stack.

43. State the equivalence of acceptance by final state and empty stack.

· If L = L(M2) for some PDA M2 , then L = N(M1) for some PDA M1.

· If L = N(M1) for some PDA M1 ,then L = L(M2 ) for some PDA M2

where L(M) = language accepted by PDA by reaching a final state. N(M) = language accepted by PDA by empty stack.

UNIT IV PROPERTIES OF CONTEXT-FREE LANGUAGES

1. What are the two major normal forms for context – free grammar?

The two Normal forms are

· Chomsky Normal Form

· Greibach Normal Form

2. What is a useless symbol? Nov/Dec 2007

A symbol x is useful if there is a derivation.

_{* *}

S ==> a x b ==> w for some a, b, w Î T^*

or else, it is useless.

3. What is e - Production rule?

Any production rule of the form A à e is known as e - production.

4. Define Unit Production.

Any production rule of the form A à B is known as unit production.

5. When do you say a symbol is useful?

We say a symbol is useful either if it derives a string of terminals or it can be used in the middle of a derivation which yields a terminal or a string of terminals.

6. Define Chomsky’s Normal form.

A CFG whose production rules are of the form

A à BC or A à a

where A, B, and C are variables and a is terminal.

7. Write the procedure to eliminate e - productions.

^· For all productions A àe, put A into V¹

· Repeat the following steps until no new variables are added.

a. For all productions

B à A₁ A₂ A₃ … A_n

where A₁ A₂ A₃ … A_n are in V¹

^b. Put B into V^1.

8. Write the procedure to eliminate the unit productions.

· Find all variables B, for each A such that

_*

A ==> B

· The new grammar G’ is obtained by letting into P’ all non – unit productions of P.

_*

· For all A and B satisfying A ==>, add to P’

A à y₁ | y₂ | . . . | y_n

where B à y₁ | y₂ | . . . | y_n is the set of productions in P’

9. Eliminate the useless symbol from the following

S à AB | a

A à b

B is an useless symbol since it doesn’t derive a terminal. Eliminating it we get

S à a

A à b

10. Define Greibach Normal form. Nov/Dec 2009

A CFG whose production rules are of the form A à α where a is a terminal and α is either empty or a string of non – terminals.

11. State pumping lemma for Context free language. April/May 2008

Let L be a CFL. Then there exists a constant n such that if z is any string in L such that |z| < n then we can write z = uvwxy subject to the following conditions.

a) |vwx | < n

b) vx ≠ e

c) for all i > 0 uvⁱwxⁱy is in L.

d)

12. What is the use of pumping lemma for CFG.

It is used to check whether a given language is context free language or not.

13. What operations that preserve CFL’s.

1. Substitution

2. Union

3. Concatenation

4. Closure (star)

5. Reversal

14. What is a formal language?

Language is a set of valid strings from some alphabet. The set may be empty,finite or infinite. L(M) is the language defined by machine M and L( G) is the language defined by Context free grammar. The two notations for specifying formal languages are: Grammar or regular expression Generative approach) Automaton(Recognition approach)

15. What is Backus-Naur Form(BNF)?

Computer scientists describes the programming languages by a notation called Backus- Naur Form. This is a context free grammar notation with minor changes in format and some shorthand.

16. Let G= ( {S,C} ,{a,b},P,S) where P consists of S->aCa , C->aCa |b. FindL(G).

S-> aCa => aba

S->aCa=> a aCa a=>aabaa

S->aCa=> a aCa a=> a a aCa a a =>aaabaaa

Thus L(G)= { aⁿbaⁿ ,where n>=1 }

17. Find L(G) where G= ( {S} ,{0,1}, {S->0S1 ,S->_ },S ) S->_ , _ is in L(G)

S-> 0S1 =>0_1=>01

S->0S1=>0 0S11=>0011

Thus L(G)= { 0n1n | n>=0}

18. What is a parser?

A parser for grammar G is a program that takes as input a string w and produces as output either a parse tree for w ,if w is a sentence of G or an error message indicating that w is not a sentence of G.

19. What are the closure properties of CFL?

CFL are closed under union, concatenation and Kleene closure.

CFL are closed under substitution , homomorphism. CFL are not closed under intersection , complementation. Closure properties of CFL’s are used to prove that certain languages are not context free.

20. State the pumping lemma for CFLs.

Let L be any CFL. Then there is a constant n, depending only on L, such that if z is in L and |z| >=n, then z=uvwxy such that :

(i) |vx| >=1

(ii) |vwx| <=n and

(iii) for all i>=0 uviwxiy is in L.

21. What is the main application of pumping lemma in CFLs?

The pumping lemma can be used to prove a variety of languages are not context free . Some examples are:

L1 ={ aibici | i>=1} is not a CFL.

L2= { aibjcidj | i>=1 and J>=1 } is not a CFL.

22. What is Ogden’s lemma?

Let L be a CFL. Then there is a constant n such that if z is any word in L, and we mark any n or more positions of z “ distinguished” then we can write z=uvwxy suchthat:

(1) v and x together have atleast one distinguished position.

(2) vwx has at most n distinguished positions and

(3) for all i>=0 uviwxiy is in L.

23. Give an example of Deterministic CFL.

The language L={anbn : n>=0} is a deterministic CFL

24. What are the properties of CFL?

Let G=(V,T,P,S) be a CFG

· The fanout of G , _(G) is largest number of symbols on the RHS of any rule in R.

· The height of the parse tree is the length of the longest path from the root to some leaf.

25. What is a turing machine? April/May 2008

Turing machine is a simple mathematical model of a computer. TM has unlimited and unrestricted memory and is a much more accurate model of a general purpose computer. The turing machine is a FA with a R/W Head. It has an infinite tape divided into cells, each cell holding one symbol.

26. What are the special features of TM?

In one move, TM depending upon the symbol scanned by the tape head and state of the finite control:

· Changes state.

· Prints a symbol on the tape cell scanned, replacing what was written there.

· Moves the R/w head left or right one cell.

27. Define Turing machine.

A Turing machine is denoted as M=(Q,Σ,Ґ ,δ ,q0, B,F)

Q s a finite set of states.

Σ is set of i/p symbols ,not including B.

Ґ is the finite set of tape symbols.

q0 in Q is called start state.

B in Ґ is blank symbol.

F is the set of final states.

Δ is a mapping from Q X Ґ to Q X Ґ X {L,R}.

28. Define Instantaneous description of TM.

The ID of a TM M is denoted as α1q α2 . Here q is the current state of M s in Q; α1 α2 is the string in Ґ * that is the contents of the tape up to the rightmost nonblank symbol or the symbol to the left of the head, whichever is the rightmost.

29. What are the applications of TM?

TM can be used as:

· Recognizers of languages.

· Computers of functions on non negative integers.

· Generating devices.

30. What is the basic difference between 2-way FA and TM?

Turing machine can change symbols on its tape , whereas the FA cannot change symbols on tape. Also TM has a tape head that moves both left and right side ,whereas the FA doesn’t have such a tape head.

31. What is (a)total recursive function and (b)partial recursive function

If f(i1,i2,………ik) is defined for all i1,…..ik then we say f is a total recursive function. They are similar to recursive languages as they are computed by TM that always halt.

A function f(i1,…ik) computed by a Turing machine is called a partial recursive function. They are similar to r.e languages as they are computed by TM that may or may not halt on a given input.

32. Define a move in TM.

Let X1 X2…X i-1 q Xi…Xn be an ID.

The left move is: if _ (q, Xi )= (p, Y,L) ,if i>1 then X1 X2…X i-1 q Xi…Xn |---- X1X2… X i-2 p X i-1 Y X i+1…Xn. M The right move is if _ (q, Xi )= (p, Y,R) ,if i>1 then X1 X2…X i-1 q Xi…Xn |---- X1X2… X i-1Y p X i+1…Xn. M

33. What is the language accepted by TM?

The language accepted by M is L(M) , is the set of words in _ * that cause M to enter a final state when placed ,justified at the left on the tape of M, with M at qo and the tape head of M at the leftmost cell. The language accepted by M is: { w | w in _ * and q0w |--- _1 p _2 for some p in F and _1 ,_2 in _ * }.

34. Give examples of total recursive functions.

All common arithmetic functions on integers such as multiplication , n!, [log2n] and 22n are total recursive functions.

UNIT V UNDECIDABALITY

1. What are(a) recursively enumerable languages (b) recursive sets?

The languages that is accepted by TM is said to be recursively enumerable (r. e ) languages. Enumerable means that the strings in the language can be enumerated by the TM. The class of r. e languages include CFL’s.

The recursive sets include languages accepted by at least one TM that halts on all inputs.

2. What are the various representation of TM?

We can describe TM using:

· Instantaneous description.

· Transition table.

· Transition diagram.

3. What is P – type problem?

The problems which can be solved in polynomial time are known as P – type problem.

4. What do you mean by NP type problems?

The problems which can be accepted by a non deterministic TM are known as NP type problems.

5. What are the possibilities of a TM when processing an input string?

· TM can accept the string by entering accepting state.

· It can reject the string by entering non-accepting state.

· It can enter an infinite loop so that it never halts.

6. What are the techniques for Turing machine construction?

· Storage in finite control.

· Multiple tracks.

· Checking off symbols.

· Shifting over

· Subroutines.

7. Define Modified Post’s correspondence problem.

Given lists A and B of K strings each from å^*, say

A = w₁, w₂, … w_k

B = x₁, x₂, … x_k

does there exist a sequence of integers i₁, i₂, … i_r such that

w₁ w_i1 w_i2 … w_ir = x₁ x_i1 x_i2 … x_ir

The sequence of i_1, i_2, … i_m is a solution to this instance of PCP.

8. Define the classes P and NP.

P consists of all those languages or problems accepted by some Turing Machine that runs in some polynomial amount of time, as a function of its input length.

NP is the class of languages or problems that are accepted by nondeterministic TM’s with a polynomial bound on the time taken along any sequence of non – deterministic choices.

9. What is the storage in FC?

The finite control(FC) stores a limited amount of information. The state of the Finite control represents the state and the second element represent a symbol scanned.

10. When is checking off symbols used in TM?

Checking off symbols is useful method when a TM recognizes a language with repeated strings and also to compare the length of substrings.

(eg) : { ww | w _ _ * } or {aⁱbⁱ | i>=1}. This is implemented by using an extra track on the tape with symbols Blank or √.

11. When is shifting over Used ?

A Turing machine can make space on its tape by shifting all nonblank symbols a finite number of cells to the right. The tape head moves to the right , repeatedly storing the symbols in the FC and replacing the symbols read from the cells to the left. The TM can then return to the vacated cells and prints symbols.

12. What is a multihead TM? Nov/Dec 2007

A k-head TM has some k heads. The heads are numbered 1 through k, and move of the TM depends on the state and on the symbol scanned by each head. In one move, the heads may each move independently left or right or remain stationary.

13. What is a 2-way infinite tape TM?

In 2-way infinite tape TM, the tape is infinite in both directions. The leftmost square is not distinguished. Any computation that can be done by 2-way infinite tape can also be done by standard TM.

14. How can a TM used as a transducer?

A TM can be used as a transducer. The most obvious way to do this is to treat the entire nonblank portion of the initial tape as input , and to treat the entire blank portion of the tape when the machine halts as output. Or a TM defines a function y=f(x) for strings x ,y _ _* if: q0X | --- qfY, where qf is the final state.

15. What is a multi-tape Turing machine?

. A multi-tape Turing machine consists of a finite control with k-tape heads and ktapes ; each tape is infinite in both directions. On a single move depending on the state of finite control and symbol scanned by each of tape heads ,the machine can change state print a new symbol on each cells scanned by tape head, move each of its tape head independently one cell to the left or right or remain stationary.

16. What is a multidimensional TM?

The device has a finite control , but the tape consists of a k-dimensional array of cells infinite in all 2k directions, for some fixed k. Depending on the state and symbol scanned , the device changes state , prints a new symbol and moves its tapehead in one of the 2k directions, either positively or negatively ,along one of the k-axes.

13. When a recursively enumerable language is said to be recursive ? Is it true that the language accepted by a non-deterministic Turing machine is different from recursively enumerable language?

A language L is recursively enumerable if there is a TM that accepts L and

recursive if there is a TM that recognizes L. Thus r.e language is Turing acceptable and recursive language is Turing decidable languages. No , the language accepted by non-deterministic Turing machine is same as recursively enumerable language.

17. What is Church’s Hypothesis?

The notion of computable function can be identified with the class of partial recursive functions is known as Church-hypothesis or Church-Turing thesis. The Turing machine is equivalent in computing power to the digital computer.

18. When we say a problem is decidable? Give an example of undecidable

problem?

A problem whose language is recursive is said to be decidable.

Otherwise the problem is said to be undecidable. Decidable problems have an

algorithm that takes as input an instance of the problem and determines whether

the answer to that instance is “yes” or “no”.

(eg) of undecidable problems are

(1)Halting problem of the TM.

15. Give examples of decidable problems.

1. Given a DFSM M and string w, does M accept w?

2. Given a DFSM M is L(M) = _ ?

3. Given two DFSMs M1 and M2 is L(M1)= L(M2) ?

4. Given a regular expression _ and a string w ,does _ generate w?

5. Given a NFSM M and string w ,does M accept w?

16. Give examples of recursive languages?

i. The language L defined as L= { “M” ,”w” : M is a DFSM that

accepts w} is recursive.

ii. L defined as { “M1” U “M2” : DFSMs M1 and M2 and L(M1)

=L(M2) } is recursive.

17. What are UTMs or Universal Turing machines?

Universal TMs are TMs that can be programmed to solve any problem, that can be solved by any Turing machine. A specific Universal Turing machine U is:

Input to U: The encoding “M “ of a Tm M and encoding “w” of a string w.

Behavior : U halts on input “M” “w” if and only if M halts on input w.

18. What is the crucial assumptions for encoding a TM?

There are no transitions from any of the halt states of any given TM .

Apart from the halt state , a given TM is total.

19. What properties of recursive enumerable seta are not decidable?

o Emptiness

o Finiteness

o Regularity

o Context-freedom.

20. Define L .When is a trivial property?

L is defined as the set { | L(M) is in . } is a trivial property if is empty or it consists of all r.e languages.

21. What is a universal language Lu?

The universal language consists of a set of binary strings in the form of

pairs (M,w) where M is TM encoded in binary and w is the binary input string.

Lu = { < M,w> | M accepts w }.

22.What is a Diagonalization language Ld?

The diagonalization language consists of all strings w such that the TM M

whose code is w doesnot accept when w is given as input.

23. What properties of r.e sets are recursively enumerable?

o L ≠Φ

o L contains at least 10 members.

o w is in L for some fixed w.

o L ∩ Lu ≠ Φ

24. What properties of r.e sets are not r.e?

· L = Φ

· L = Σ *.

· L is recursive

· L is not recursive.

· L is singleton.

· L is a regular set.

· L - Lu ≠ Φ

25. What are the conditions for L to be r.e?

L is recursively enumerable iff satisfies the following properties:

i. If L is in and L is a subset of L_ ,then L_ is in (containment property)

ii. If L is an infinite language in ,then there is a finite subset of L in .

iii. The set of finite languages in is enumaerable.

26. What is canonical ordering?

Let Σ * be an input set. The canonical order for Σ *as follows . List words in

order of size, with words of the same size in numerical order. That is let _ ={

x0,x1,…x t-1 } and xi is the digit i in base t.

(e.g) If _ ={ a,b } the canonical order is Σ * , a ,b , aa, ab ,……..

27. How can a TM acts as a generating device?

In a multi-tape TM ,one tape acts as an output tape, on which a symbol, once written can never be changed and whose tape head never moves left. On that output tape , M writes strings over some alphabet _ , separated by a marker symbol # , G(M) ( where G(M) is the set w in Σ * * such that w is finally printed between a pair of #’s on the output device ).

28. What are the different types of grammars/languages?

o Unrestricted or Phase structure grammar.(Type 0 grammar).(for TMs)

o Context sensitive grammar or context dependent grammar (Type1)(for Linear Bounded Automata )

o Context free grammar (Type 2) (for PDA)

o Regular grammar (Type 3) ( for Finite Automata).

This hierarchy is called as Chomsky Hierarchy.

29. Show that AMBIGUITY problem is un-decidable.

Consider the ambiguity problem for CFGs. Use the “yes-no” version of AMB. An algorithm for FIND is used to solve AMB. FIND requires producing a word with two or more parses if one exists and answers “no” otherwise. By the reduction of AMB to FIND we conclude there is no algorithm for FIND and hence no algorithm for AMB.

30.State the halting problem of TMs. April/May 2008

The halting problem for TMs is: Given any TM M and an input string w, does M halt on w? This problem is undecidable as there is no algorithm to solve this problem.

31.Define PCP or Post Correspondence Problem. April/May 2008

An instance of PCP consists of two lists , A = w1,w2,….wk and B = x1,…..xk of strings over some alphabet _ .This instance of PCP has a solution if there is any sequence of integers i1,i2,..im with m >=1 such that wi1, wi2,…wim = xi1,xi2 ,…xim The sequence i1 ,i2 ,…im is a solution to this instance of PCP.

32.Define MPCP or Modified PCP.

The MPCP is : Given lists A and B of K strings from _ * ,say A = w1 ,w2, …wk and B= x1, x2,…..xk does there exists a sequence of integers i1,i2,…ir such that w1wi1wi2…..wir = x1xi1xi2…xir?

33 . What is the difference between PCP and MPCP?

The difference between MPCP and PCP is that in the MPCP ,a solution

is required to start with the first string on each list.

34. What are the concepts used in UTMs?

· Stored program computers.

· Interpretive Implementation of Programming languages.

· Computability.

35. What is NP – Complete problem?

Let L be a language (problem) in NP. We say that L is NP – complete if the following statements are true.

a) L is in NP.

b) For every language L’ in NP there is a polynomial – time reduction of L’ to L.

16 marks Questions

1. Prove that, if L is accepted by an NFA with ε -transitions, then L is accepted by an NFA without ε -transitions.
   Refer page no:26,Theorem 2.2
   
   
2. Prove that for every regular expression there exist an NFA with ε -transitions.
   Refer page no:30,Theorem 2.3
   
   
3. Construct the NFA with ε -transitions from the given regular expression.
   
   
4. Conversion of NFA to DFA
   Draw the NFA’s transition table
   Take the initial state of NFA be the initial state of DFA.
   Transit the initial state for all the input symbols.
   If new state appears transit it again and again to make all state as old state.
   All the new states are the states of the required DFA
   Draw the transition table for DFA
   Draw the DFA from the transition table.

5. Prove the equivalence of NFA and DFA. (8) Nov/Dec 2006

6. Construct a NFA for regular expression (a/b)*abb and draw its equivalent DFA. April/May2008

UNIT-II

1. Consider the following ε- NFA. Compute the ε – closure of each state and find its equivalent DFA. (8) Nov/Dec 2006

2. Prove that a language L is accepted by some DFA iff L is accepted by some NFA. Nov/Dec 2006

3. Convert the following NFA to its equivalent DFA. Nov/ Dec 2006

	0	1
->p	{p,q}	{p}
q	{r}	{r}
r	{s}	{ φ}
*s	{s}	{s}

4. Explain the construction with transition from any regular expression (8) Nov/Dec 2006

^5. Find the regular expression for the set of all strings denoted by R₁₃²

from the deterministic fininte automata given below. (8) Nov/dec 2006

1

1

0

0,1

0

6. Construct the regular expression to the transition diagram. April/may 2008

7. Show that the language p is a prime is not regular. Nov-Dec2007

8. Conversion of DFA into regular expression.
Arden’s theorem is used to find regular expression from the DFA.
using this theorem if the equation is of the form R=Q+RP,we
can write this as R=QP*.
_ Write the equations for all the states.
_ Apply Ardens theorem and eliminate all the states.
_ Find the equation of the final state with only the input symbols.
_ Made the simplifications if possible
_ The equation obtained is the required regular expression.

UNIT-III

1. Construction of reduced grammar.
Elimination of null productions
- In a CFG,productions of the form A->_ can be eliminated, where A is
a variable.
Elimination of unit productions.
- In a CFG,productions of the form A->B can be eliminated, where A
and B are variables.
Elimination of Useless symbols.
- these are the variables in CFG which does not derive any terminal or
not reachable from the start symbols. These can also eliminate.
-
2. Chomsky normal form(CNF)
If the CFG is in CNF if it satisfies the following conditions
- All the production must contain only one terminal or only two
variables in the right hand side.
Example: Consider G with the production of S->aAB , A-> bC , B->b, C->c.
G in CNF is S->EB , E->DA , D-> a , A->FC , F-> b , B->b , C-> c.

3. Conversion of CFL in GNF.
Refer page no: 97,Example 4.10

4. Design a PDA that accepts the language {wwR | w in (0+1)*}.
Refer page no:112, Example 5.2

5. Prove that if L is L(M2) for some PDA M2,then L is N(M1) for some PDA M1.
Refer page no:114,Theorem 5.1

6.If L is a context-free language, then prove that there exists a PDA M such that
L=N(M).
Refer page no: 116,Theorem 5.3

7. Construct a context free grammar for the given language Nov Dec 2006

L={anbn|/n>=1}U{amb2m/m>=1} and hence a PDA accepting L by empty stack

8. Explain Leftmost and rightmost derivations.
If we apply a production only to the leftmost variable at every step to derive the
required string then it is called as leftmost derivation.
If we apply a production only to the rightmost variable at every step to derive the
required string then it is called as rightmost derivation.

Example:
Consider G whose productions are S->aAS|a , A->SbA|SS|ba.For the string
w=aabbaa find the leftmost and rightmost derivation.
LMD: S=>aAS
=>aSbAS
=>aabAS
=>aabbaS
=>aabbaa
RMD: S=>aAS
=>aAa
=>aSbAa
=>aSbbaa
=>aabbaa

9. Prove that for every derivations there exist a derivation tree.
Refer page no: 84,Theorem 4.1

10. Show that L = {an! : n>=0} is not regular (8) Nov 2006

11.Convert the grammar S->ABb|a, A->aaA|B, B->bAb into greibach normal form Nov 2006

12.Construct a context free grammar for the languages L(G1)={aib2i/I>0} and L(G2)={anban/n>0} Nov 2006

UNIT-IV

1. Conversion of PDA into CFL.
Theorem: refer page no:117
Example: refer page no :119

2. State and prove the pumping lemma for CFL
Refer page no: 125,Theorem 6.1

3. Explain the various techniques for Turing machine construction.
storage in finite control
multiple tracks
checking off symbols
shifting over
subroutines.
For explanation refer page no 153-158

4. Briefly explain the different types of Turing machines.
- two way finite tape TM
- multi tape TM
- nondeterministic TM
- multi dimensional TM
- multihead TM
For explanation refer page no 160-165

5.Design a TM to perform proper subtraction.
Refer page no: 151,Example 7.2

6. Design a TM to accept the language L={0n1n | n>=1}
Refer page no:149,Example 7.1

7. Explain how a TM can be used to determine the given number is prime or not?
It takes a binary input greater than 2,written on the first track, and determines
whether it is a prime. The input is surrounded by the symbol $ on the first track.
Theory of Computation 24
To test if the input is a prime, the TM first writes the number 2 in binary on the
second track and copies the first track on to the third. Then the second track is subtracted as many times as possible, from the third track effectively dividing the third track by the second and leaving the remainder.
If the remainder is zero, the number on the first track is not a prime.If the
remainder is non zero,the number on the second track is increased by one.If the second
track equals the first,the number on the first track is the prime.If the second is less than
first,the whole operation is repeated for the new number on the second track.

8. Construct a Turning machine to perform multiplication Nov/Dec 2006

9.Design a Turing Machine to recognize each of the following languages.

Nov/Dec2009
(i) {0n 1n|n = 1} (8)
(ii) {ww^R | w ? (0 + 1)*} (8)

9. (a)Prove that the function fadd (x,y)=x+y is a primitive recursive
(b)Show there exists aTM for which the halting problem is solvable April/May 2008

10.Design Turing Machine M for f(x,y)=x*y where x,y are stored in the tape in the form 0^x10^y1 Nov/Dec 2007

Unit-V

1..State and explain RICE theorem.
Refer page no: 188,Theorem 8.6

2. Define Lu and prove that Lu is recursive enumerable.
Refer page no: 183,Theorem 8.4

3. Define Ld and prove that Ld is undecidable.
Refer page no: 182.

4. Prove that if a language L and its complement are both recursively enumerable, then L is recursive.
Refer page no: 180,Theorem 8.3

5. Prove that the halting problem is undecidable. Nov 2006
Refer page no: 187

6. Prove that there exists an recursively enumerable language whose complement is not recursively enumerable

7.Find whether the following languages are recursive or reclusively enumerable. Nov/Dec 2006

(i) Union of two recursive languages

(ii) Union of two recursively enumerable languages

(iii) Lif L and complement of L are recursively enumerable

_(iv) L_u

8. Consider the turing machine M and w=01, Nov/Dec 2006

where M=({q1,q2,q3},{0,1}, δ ,q1, B, {q3}) and δ is given by

Reduce the above problem to Post correspondence problem and find whether that PCP has a solution or not. (12) Nov/Dec 2007

9.Find the language obtained from the following operations Nov/Dec2007

a)Union of two recursive languages. (6)

b)union of two recursively enumerable language. (6)

c) L if L and complement of L are recursively enumerable(4)