THEORY OF COMPUTATION

DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

Third Year Computer Science and Engineering, 6^th Semester

Subject Code & Name : THEORY OF COMPUTATION

UNIT-I CHURCH-TURING THESIS

1. What is aTuring Machine?

Turing machine is a simple mathematical model of a computer. TM has unlimited an 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.

2. What are the difference between finite automata and Turing Machines?

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.

3. Define Turing Machine?

     A turing machine is a 7-tuple (Q, S, Γ, δ, q₀, q_accept, q_reject) where

            Q: finite set of states

            S: input alphabet (cannot include blank symbol, _)

            Γ: tape alphabet, includes S and _

 δ: transition function: Q ´ Γ ®  Q ´ Γ ´ {L, R}

            q₀: start state, q₀ Î Q   

             q_accept: accepting state, q_accept Î Q   

             q_reject: rejecting state, q_reject Î Q   

4. What is configuration?

Turing machine computes, changes occur in the current state, the current tape contents and the current head location.a setting of these three items is called a  configuration.

5. What is a recursively enumerable language?

         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.

6. Define variants of Turing Machine?

        Variants are

                   Non deterministic turing machine.

                   Mutlitape turing machine.

  .                 Enumerators       

7. What is a multitape TM?

    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.

8. Define nondeterministic TM?

•         Arbitrarily chooses move when more than one possibility exists

•         Accepts if there is at least one computation that terminates in an accepting state

9. What is an algorithm?

An algorithm is a collection of simple instruction for carrying out some task. It    is sometimes called procedures or recipes.

10 What is a polynomial?

It is a sum of terms where each term is a product of certain variables and  a    constant called a co-efficient.

                Eg.6 x*x*x*y*z*z=6x³yz²

11. Define Church-Turing thesis?

In 1936 Alonzo church used a notational system called λ-calculus to define   algorithms and Alan Turing did it with his “machines”. These two definitions were shown to be equivalent .This connection between the informal notion and precise definition has come to be called Church-Turing thesis

12. List the types of description.

                 Formal description.

                 Implementation description.

                 High-level description.

13. What is halting problem?

A {M,w | TM M is a TM and M accepts w }

         TM A is undecidable but TM A is Turing – recognizable hence TM A is                  sometimes called the halting problem.

14. Mention the relationship between classes of languages.

                                

15. Define universal Turing machine.

   A universal Turing machine (UTM) is a TM which is capable of simulating any other Turing machine from the description of the machine.

16. What is a Diagonalization language Ld?

        The diagonalization language consists of all strings w such that the TM whose  code is w does not accept when w is given as input.    

17. Why some languages are not decidable or even Turing – recognizable?

            The reason that there are uncountable many languages yet only countably many Turing machines. Because each Turing machine can recognize a single language and there are more languages than Turing machines, some languages are not recognizable by any Turing machine.

18. What do you mean by co-Turing-recognizable?

      A language is called co-Turing-recognizable if it is the complement of a Turing-recognizable language

19. Define Countable and Uncountable?

•         A set is countable if it is finite or has the same size as N (the set of natural numbers {1, 2, 3, …})

–        Examples: N (natural numbers), Z (integers), Q (rational numbers), E (even numbers), etc.

•         A set is uncountable if it has more elements than N .

–        Examples: R (real numbers) É [0,1] É Cantor set

–         

20. What is a correspondence?

        A function that is both one-to-one and onto is called correspond.

BIG QUESTIONS

1.      Explain the various techniques for Turing machine construction.

2.      Briefly explain the different types of Turing machines.

3.      Design a TM to accept the language L={0ⁿ1ⁿ | n>=1}

4.      Explain how a TM can be used to determine the given number is prime or not?

5.      Prove that if L is recognized by a TM with a two way infinite tape if it is recognized by a  TM with a one way infinite tape

6.      Prove that Halting problem is undecidable.

7.      Design a Turing machine that accepts the language L={aⁿ  bⁿ   cⁿ /n>1}

8.      What are the various models of Turing Machine?

9.       With an example explain the universal Turing machine

10.  Show that halting problem of Turing machine is undecidable

11.  Show that a language is decidable if it is Turing-recognizable and co- recognizable.

UNIT-II REDUCIBILITY

           

1.What  is reduction?

               A  reduction is a way of converting one problem into another in such a way that a  solution to the second problem can be used to solve the first problem.

 2.  What is reducibility?

         The primary method of proving some problems are computationally unsolvable.       It is called reducibility.Reducibility always involves two problems which we call A and B. If A reduces to B, we can use a solution to B to solve A.

3.  What is linear bounded automation?

           

           A linear bounded automation is restricted type of Turing machine where in the tape head isn’t permitted to move off the portion of the tape containing the input. It has a limited amount of memory.

4. What is a accepting computation history?

             

           An accepting computation history is defined as , Let M be a Turing machine and w be a input string,  for M on w is a sequence of configuration c1,c2,…cl, Where c1 is the start configuration of M on w . cl is the accepting configuration of M and each ci legally follows from ci-1 according to the rules of M.

5. What is a rejecting computation history?

            Let M be a Turing machine and w be a input string , for M on w is a sequence of configuration c1, c2 , … cl Where cl is a rejecting configuration of M on W .

6. Write down the algorithm that decides ALBA?

           

            The algorithm that decides ALBA is as follows:

1.  L=” on input {M,w} Where M is an LBA and w is a string .

            1.1  Simulate M on w for qng^n steps until it halts

            1.2  If M has halted accept if it has accepted and reject if it has rejected . If it has not halted , reject”.

7. Write about the three conditions of computational  history?

             

             Suppose consider the input B and it has possible input C1,C2……. Cl. If B ants to check whether the Ci satisfy the three conditions of computational history

1.      C1 is a start configuration for M on w

2.      Each Ci+1 legally follows from Cp

3.      Cl is an accepting configuration for M

8. What is called PCP?

           

            The phenomenon of undecidability is not confined to problems concerning automata. An undecidable problem concerning on simple manipulation of strings is called the post correspondence  problem or PCP.

            An instance of the pcp is the collection P of dominos:

            P = {[t1/b1]}, [t2/b2],……….. [tk/bk]}

and a match is a sequence i1,i2,i3….il.. Where ti1,ti2….til = bi1, bi2…..bil

            PCP={<P>/P is an instance of the post correspondence problem with a match}

9. How can we construct P so that a match is an accepting computation history for M on w?

           

            The task is to make a list of the dominos; so that the string we get by reading off the symbols on the top is same as the string of symbols on the bottom.This list is called a match. For example:   [a/ab][b/ca][ca/a][a/ab][abc/c].Reading off the top strings we get abcaaabc , which is same as reading off the bottom.

10. What is MPCP?

            MPCP à Modified Post Correspondence Problem, in that the match is required to start with the first domino in the list.

      MPCP = {<P>/P is an instance of the post correspondence with a match that starts with the first domino}.

11. What is called mapping reducibility?

            Language “A” is mapping reducible to language B, written A<=B; if there is a computational  function f: €*à€* , Where for every w,   w € a↔ f(w) € B.

12. Define computable function?

           A turning machine computes a function by starting with the input to the function on the tape and halting with the output of the function on the tape a function f: €*à€* is a computable function , if some turning machine M, on every input w, halts with just f(w) on its tape.

    Eg: All arithmetic operations on integers.

13. What is recursion theorem?

               Let T be a Turing machine that computes a function t: €* x €*à€*. There is a Turing machine R that computes a function r: €*à€*, Where for every w :

r(w)=T(<R>,w).

             The recursion theorem is a mathematical result that plays an important role in advanced work in the theory of computability.

14. What is self reference?

             The Turing machine that ignores its input and prints out a copy of its own description,   we call this as SELF.

            There is a computable function q: €*à€*, where for any string w, q(w) is the description of a Turing machine Pw that prints out w and then halt.

15. What is computer virus?

              A computer virus is a computer program that is designed to spread itself among   computers.

            Computer virus are inactive when standing alone as  piece of code, but when placed appropriately in a host computer, thereby infecting it and they can become activated and transmit copies of themselves to other accessible machines.Various media can transmit viruses , including internet and transferable disks.

16. Applications of recursion theorem?

1.      ATM is undecidble.

2.      MINTM is not Turing recognisable

3.      Fixed point theorem.

17. What is a fixed point?

              A fixed point of a function is a value that isn’t changed by application of the function.

18. Define formula.

               A formula is a well formed string over this alphabet.

19. What are the Boolean operations and Quantifiers?

   The form of the alphabet of this language :

{^,v,¬,(,),for all, x , there exists,R1,….Rk}.The symbols ^,v, ¬  are called Boolean operations.The symbols of “for all” and ”there exists” are called Quantifiers. The symbol x denotes the variable R1,R2,..Rk ,called the relations.

20. What is atomic formula?

                  A string of  the form Ri (x1,x2,x3,..xj) is an atomic formula. The value ‘j’ is the arity of the relation symbol Ri.

21. What is called Sentence/Statement and Free variable?

                A variable that isn’t bound within the scope of a quantifier is called a Free variable. A formula with no free variable is called a Sentence or Statement.

22.  What is model?

                 A universe together with an assignment of relations to relation symbol is called a model .

               A model M is a tuple (U, P1,P2..Pk), where U is the universe and P1 through Pk are the relations assigned to symbols R1 through Rk.

23. What is language of a model?

               Language of a model is the collection of formulae that use only the relational symbols the model assign and that use each relation symbol with the correct arity.

               If   Ǿ is a sentence in a language  of a model , Ǿ is either true or false in that model.

24. What is decidable theory?

               Let (N,+) be the model, except without x relation. Its theory is Th(N,+).

  For example: the formula[x+x=y] is true  therefore a member of Th(N,+).

25. What is undecidable theory?

               Let (N, + ,X) be a model whose universe is the Natural numbers with the usual +,X relations.

             Church showed that Th(N,+,X) the theory of this model; is undecidable.

26. What is called Turing Reducibility?

               Language A is Turing reducible to language B, written A<=TB, if A is decidable relative to B.

               Turing reducibility is a generalization of mapping reducibility.

27. What is called Oracle Turing machine?

               An Oracle for a language B is an external device that is capable of reporting whether any string w is a member of B.

             An Oracle Turing machine is a modified Turing machine that has the additional capability of quering an Oracle.

28. What is an information?

               Definition of information depends upon the application.

For example: A= 0101010101010101

                                  B= 11101110111000111

             Sequence A contain little information because it is merely a repetition of pattern 01  twenty times. In contrast sequence B appears to contain more information.

29. What is minimal Description?

                Minimal Description of x, written d(x), is the shortest string (M, w), where  TM,M on input w halts with x on its tape.

30. What is Descriptive complexity?

               The Descriptive complexity of  x, written  k(x) is k(x) = |d(x)|

 where k(x) – length of the minimal description of x.

              The definition of k(x) is intended to capture the amount of information in the string x.

31. When an x is a C-compressible?

               Let x be a string, say that x is C-compressible if   k(x) <= |x| - C.

BIG QUESTIONS

1.      Explain how Post correspondence problem is undecidable?

2.      Prove that HALT_TM is undecidable.

3.      Prove that E_TM is undecidable.

4.      Prove that REGULAR _TM is undecidable.

5.      Prove that EQ _TM is undecidable.

6.      Prove that A _LBA is decidable.

7.      Prove that E _LBA is undecidable.

8.      Prove that ALL _CFG is undecidable.

9.      State and prove PCP is undecidable.

10.  Prove that EQ _TM is neither Turing-recognizable nor co-Turing-recognizable.

11.  State and prove recursion therorem

12.  Explain self-reference in detail.

13.  Explain decidability of logical theories?

14.  Write a brief note on a decidable theory?

15.  Write a brief note on an undecidable theory?

16.  Explain Turing reducibility?

UNIT-III TIME COMPLEXITY

1.Define complexity

•         Complexity Theory = study of what is computationally feasible (or tractable) with limited  resources:

–        running time

–        storage space

–        number of random bits

–        degree of parallelism

–        rounds of interaction

                        others…

2. Define time complexity

The time complexity of a TM M is a function f:N → N, where f(n) is the maximum number of steps   M uses on any input of length n.

3. Define Asymptotic Notation

Running time of an algorithm as a function of input size n for large n. Expressed using only the highest-order term in the expression for the exact running time

4. What do you mean by polynomial and exponential bounds?

      Bounds of the form  n^c for c greater than 0 .such a bound are called polynomial bounds.Bounds of the form  2^{(n∂ )}  . s^.uch a bound are called exponential bounds when ∂ is a real  number greater than 0

5. What is Hamiltonian path?

      A Hamiltonian path in a directed graph G is a directed path that goes through each

node exactly once. We consider a special case of this problem where the start node

and target node are fixed.

6. List the types of Asymptotic Notation.

•         Big Oh (O) notation provides an upper bound for the function f

•         Omega (Ω) notation provides a lower-bound

•         Theta (Q) notation is used when an algorithm can be bounded both from above and below by the same function

•         Little oh (o) defines a loose upper bound.

7. What is Hamiltonian path?

     A Hamiltonian path in a directed graph G is a directed path that goes through each

node exactly once. We consider a special case of this problem where the start node

and target node are fixed.

8. Defines brute –force search

         Exponential time algorithms typically arise when we solve by searching through a spsce of  solutions called brute –force search

9. Define running time.

The running time of  N is the function N®N, wheref(n)  is the maximum of steps that N uses on any branch of its computation on any of length n.

10. Define class P

The class of all sets L that can be recognized in polynomial time by deterministic TM. The    class of all decision problems that can be decided in polynomial time.

11. Define class NP.

           Problems that can be solved in  polynomial time by a nondeterministic TM. Includes all problems   in P and some problems possibly outside P.

12. Define verifier.

            A verifier for a language A is an algorithm V, where

                               A = {w | V accepts <w,c> for some string c}.

       We measure the time of a verifier only in terms of the length of w, so a polynomial time   verifier runs in polynomial time in the length of w.

        A language A is polynomially verifiable if it has a polynomial time verifier. The above string c, used as additional information to verify that wÎA, is called a certificate, or proof, of  membership in A. 

13. What do you mean by NP-completeness?

      If a polynomial time algorithm exists for any of the NP-complete problems, all problems in  NP would be polynomial time solvable. These problems are called by NP-completeness

14. Define polynomial time computable functions.

     A function f:£ ®£* is a polynomial time computable functions if some polynomial time   turing machine exists that halts with just f(w) on its tape. when started on any input w.

15. Define polynomial time mapping reducible.

         We say that A is polynomial time mapping reducible, or simply polynomial time reducible, to B,  written A£_PB, if a polynomial time computable function f: S* ® S*  exists s.t. for every string wÎS*,   wÎA   iff   f(w)ÎB. Such a function f is called a polynomial time reduction of A to B.

16. Define literal.

   A literal is a Boolean variable x or a negated Boolean variable x.

17. Define clause.

   A clause s several literals connected with Ús, as in (x Ú y Ú z Ú t).

18 .Define conjunctive normal form.

         A Boolean formula is in conjunctive normal form, called a cnf-formula, if it comprises     several clauses connected with Ùs, as in                     

                  (x Ú y Ú z Ú t)  Ù (x Ú z)  Ù (x Ú yÚ t)

•          A cnf-formula is a 3cnf-formula if all the clauses have 3 literals, as in

             (x Ú y Ú z)  Ù (x Ú z Ú t)  Ù (x Ú y Ú t) Ù (z Ú

                   BIG QUESTIONS

1.      Show that every CFL is a member of class P problem

2.      Show that a language is in NP if it is decided by some non deterministic polynomial time Turing machine.

3.      Explain the cook-Levin theorem

4.      Explain about the vertex cover problem.

5.      Explain about Hamiltonian path problem.

6.      Briefly discuss about the subset sum problem?

7.      With an example explain BIG-O and SMALL-O notation.

8.      Explain about Analyzing algorithms in detail?

9.      Explain CLASS P in briefly with an example?

10.  State and prove PATH € P.

11.  State and prove RELPRIME € P.

12.  Prove that every context-free language is a member of P.

13.  Explain with an example CLASS-NP?

UNIT –V Advanced Topics in Complexity Theory

1. Define Approximation algorithm?

An Approximation algorithm is designed to find approximately optimal solution. An approximation algorithm for a minimization problem is K-Optimal if it always find a solution that is not more than K times optimal.

2. What is MAX-CUT, Cut, and Cut edge?

An Approximation Algorithm for the maximization problem called MAX-CUT . A Cut is an undirected graph is a separation of vertices V in to two disjoint subsets S and T.

3. What is Cut edge and Uncut edge?

A Cut edge is an edge that goes between a node in S and a node in

T. An uncut edge is an edge that is not a cut edge.

4. What is Probabilistic Algorithm?

A Probabilistic Algorithm is an algorithm designed to use the outcome

            of a random process .

5. What is a Probabilistic Turing Machine?

A Probabilistic Turing Machine is a type of non deterministic Turing

Machine .it is defined as

                 Pr[b] = 2^-k

Where K be the number coin flip steps occur on branch B.

6. What is the Class BPP?

BPP is the class of languages that are recognized by probabilistic

Polynomial time Turing Machine with an error probability of 1/3.

7. What is Small Probability of an error?

Error Probability bounds that depends on input length n. if Probability €=2^-n indicates an exponentially small probability of an error.

8. What is a Prime, Composite?

A Prime Number is an integer greater than  1 that is not divisible by

positive numbers other than 1 and itself. A non prime number greater than 1 is

Called Composite.

9. Define Chinese remainder theorem/ Fermat’s little theorem?

The Chinese remainder theorem says that a one-to one correspondence  exists between £_pq and  £p x £q if p and q are relatively prime. Each number r є £_pq  Correspond to the pair (a,b), where a є £p  and b є £q. such that

                    r ≡ a (mod p)

                    r ≡ b (mod q),

10. what is Fermat’s Test?

A type of “test” for primality called a fermat test.

11. What is RP?

RP is the class of languages that are recognized by probabilistic polynomial time Turing machine s where inputs in language are accepted with a probability of atleast ½ and inputs not in a language are rejected with a probability of 1.

12. What is called a Branching Program?

A Branching Program is a model of computation used in complexity

theory and in certain practical areas such as computer aided design. This

 model represents a decision process that queries the values of input variables and basis decisions  about the way to proceed on the answers to those queries.

13. What are query nodes?

A Branching Program is a directed acyclic graph where all nodes are labeled by variables, except for two output nodes labeled nodes 0 0r 1. The nodes that are labeled by variables are called Query nodes.

14. What is a Read-Once branching program?

                  A Read-Once branching program is one that can query each variable at most one time on every directed path from the start node to an output node.

      EQ_{ROBP =} {(B1,B2)| B1 and B2 are equivalent read-once branching program}

15. What is Alternation?

Alternation is a generalization of non determinism that has proven to

be useful in elucidating relationships among complexity classes and in classifying specific problems according to their complexity.

16. What is an Alternating Turing Machine?

An Alternating Turing Machine is a Non deterministic Turing

machine with an additional feature. It states, except for q_accept  and q_reject are divided into universal states and existential states

17. What is an Alternating Time and Space?

An Alternating Time and Space complexity classes are defined as

ATIME (t(n) )   = {L|L is a language decided by an O(t(n)) time

                                                Alternating Turing Machine}

ASPACE (f(n)) = {L|L is a language decided by O(f(n) space   

                                                Alternating Turing Machine}

18. Define Tautology?

A tautology is a Boolean formula that evaluates to 1 on every assignment to its variables. Let TAUT = ((ф) |ф is a tautology)

19. Give the algorithm to show that TAUT is in AP?

                  The following algorithm shows that TAUT is in AP:

“On input (ф):

1.      Universally select all assignments to the variables of ф.

2.      for the particular assignment, evaluate ф

3.      If ф evaluates to 1 , accept, otherwise Reject

20. Define Minimal Formula?

A minimal formula is one that has no shorter equivalent. Let Minimal

            Formula is denoted by:

MIN-FORMULA = {(ф) |ф is a minimal Boolean formula}

21. Give the algorithm to show that MIN-FORMULA is in AP?

                  The following algorithm shows that TAUT is in AP:

“On input (ф):

1.        Universally select all formulas Ψ that are shorter than ф

2.        Existentially select an assignment to the variables of ф

3.        Accept if the formulas evaluate to different values. Reject if they evaluate to the same values

22. What is Σ_i – alternating Turing Machine and π_{i –}Turing Machine?

Σ_i – alternating Turing Machine is an alternating Turing Machine that

            contain at most i runs of universal or existential steps , starting with existential 

            steps

π_{i –}Turing Machine is similar except that it starts with universal steps.

23. What is Interactive Proof Systems?

Interactive Proof Systems provide a way to define a probabilistic

analog of the class NP, much as probabilistic polynomial time algorithm provides a probabilistic analog to P. The development of interactive proof system has profoundly affected complexity theory and has led to important advances in the field of cryptography and approximation algorithm.

24. Define Verifier

We define a verifier to be a function V that computes its next transmission

            .the function v has three inputs:

1.Input String

2.Random Input

3.Partial message History

25. Define Prover

Prover is a party with unlimited computational ability We define it to be

            function P with two inputs:

1.      Input string

2.      Partial Message History.

26. Define Parallel Computation

A Parallel Computer is one that can perform multiple operations simultaneously. Parallel computers may solve certain problems must faster than sequential computers, which can do a single operation at time.

27. What is PRAM?

Parallel Random Access Machine: In the PRAM model idealized processors with a simple instruction set patterned on actual computers interact through the shared memory

28. What are the factors to be consider to achieve a particular parallel time complexity?

We have to consider the simultaneous size and depth of the single circuit family in order to achieve a particular parallel time.

29. When language B is stated to be P-Complete?

1. B Є P and

2. Every A in P is log space reducible to B

30. Why we should maintain a secret key?

            The Secret Key is a piece of information that is used by encrypting and decrypting algorithm. Maintain the secrecy of the key is crucial to the security of the code because any person with access to the key can encrypt and decrypt messages 

31. What is called one time pad?

A key that is as long as the combined message length is called a one time pad. Every bit of a one-time pad key is used to encrypt a bit of the message and then that bit of the key is discarded.

32. Define One Way Permutation

A One Way Permutation is a length –preserving permutation  f with the following two properties

1.      It is computable in polynomial time

2.      For every probabilistic polynomial time Turing Machine M, every K, and sufficiently large n, if we pick a random w of length n and run M on input w,

            Pr_M,w[M(f(w))=w] ≤ n^-k

Here Pr_M,w means that the probability is taken over the random choices made by M and random selection of w

33. Define One Way function

A one way function is a length –preserving function f with the following two properties

1.      It is computable in polynomial time

2.      For every probabilistic polynomial time Turing Machine M, every K, and sufficiently large n, if we pick a random w of length n and run M on input w,

Pr[M(f(w))=y where f(y)=f(w)] ≤ n^-k

34. Define Trapdoor?

A trapdoor function is a length –Preserving indexing function

f: ∑^*×∑^*→∑^*  with an auxiliary probabilistic polynomial time TM G and an auxiliary function h: ∑^*×∑^*→∑^*.