Third Year Computer Science and Engineering, 6 th Semester

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1 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 a Turing 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. 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. 4. 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 1

2 language can be enumerated by the TM. The class of r. e languages include CFL s. 5. Define Turing Machine? 6. Define variants of Turing Machine? Variants are 1. Non deterministic turing machine. 2. Mutlitape turing machine.. 3. 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 2

3 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. 11. Define Church-Turing thesis? In 1936 Alonzo church used a notational system called 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. 1. Formal description. 2. Implementation description. 3. 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. 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. 3

4 15. Mention the relationship between classes of languages. 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 Turingrecognizable 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, }) 4

5 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) [0,1] 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={0 n 1 n 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={a n b n c n /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 corecognizable. ************************* END ********************** 5

6 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

7 6. Write down the algorithm that decides ALBA? The algorithm that decides ALBA is as follows: o L= on input {M,w} Where M is an LBA and w is a string. o Simulate M on w for qng^n steps until it halts o 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. 7

8 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 -> 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: *-> *, 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: *-> * 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 *-> *. There is a Turing machine R that computes a function r: *-> *, 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. 8

9 There is a computable function q: *-> *, 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. 9

10 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 is a sentence in a language of a model,?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. 10

11 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= B= 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. 11

12 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-turingrecognizable. 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? ************************* END ********************** 12

13 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 nc for c greater than 0.such a bound are called polynomial bounds. Bounds of the form. Such 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 13

14 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 ( )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 deterministictm. The class of all decision problems that can be decided in polynomial time. 11. Define class NP. 14

15 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 w A, 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 P B, if a polynomial time computable function f: * * exists s.t. for every string w *, w A iff f(w) B. 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. 15

16 A clause s several literals connected with s, as in (x y z 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 s, as in (x y z t) (x z) (x y t). A cnf-formula is a 3cnf-formula if all the clauses have 3 literals, as in (x y z) (x z t) (x y t) (z 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? 16

17 UNIT-IV SPACE COMPLEXITY 1. DEFINE SPACE COMPLEXITY 1. DEFINE SPACE AND NSPACE 2. DEFINE PSPACE 4. DEFINE PSPACE COMPLETE AND PSPACE-HARD 17

18 5. DEFINE QUANTIFIERS 6. DEFINE GENERALIZED GEOGRAPHY In generalized geography we take an arbitrary directed graph with a designated start node instead of the graph associated with the actual cities. For example, the following graph is an example of a generalized geography game. 18

19 7. DEFINE L AND NL 8. DEFINE LOG SPACE TRANSDUCER AND LOG SPACE COMPUTABLE FUNCTION 9. DEFINE NL-COMPLETE 19

20 10. WHAT IS MEANT BY INTRACTABLE Certain computational problems are solvable in principle, but the solutions require so much time or space that they can't be used in practice. Such problems are called intractable. 11. WHAT IS MEANT BY HIERARCHY THEOREMS The hierarchy theorems prove that this intuition is correct, subject to certain conditions described below. We use the term hierarchy theorem because these theorems prove that the time and space complexity classes aren't all the same-they form a hierarchy whereby the classes with larger bounds contain more languages than do the classes with smaller bounds. 12. WHAT IS MEANT BY SPACE CONSTRUCTIBLE 12. WHAT IS MEANT BY TIME CONSTRUCTIBLE 14. WHAT IS MEANT BY EXSPACE-COMPLETE 20

21 15. DEFINE ORACLE 16. DEFINE BOOLEAN CIRCUIT A Boolean circuit is a collection of gates and inputs connected by wires. Cycles aren't permitted. Gates take three forms: AND gates, OR gates, and NOT gates, as shown schematically in the following figure. 21

22 17. DEFINE CIRCUIT FAMILY 18. DEFINE CIRCUIT SIZE COMPLEXITY PART-B 1. EXPLAIN IN DETAIL ABOUT SPACE CONPLEXITY 2. EXPLAIN IN DETAIL ABOUT SAVITCH S THEOREM 3. EXPLAIN IN DETAIL ABOUT THE CLASS PSPACE 4. EXPLAIN IN DETAIL ABOUT PSPACE-COMPLETENESS 5. EXPLAIN IN DETAIL ABOUT THE TQBF PROBLEM 6. EXPLAIN IN DETAIL ABOUT WINNING STRATEGIES FOR GAMES 7. EXPLAIN IN DETAIL ABOUT GENERALIZED GEOGRAPHY 8. EXPLAIN IN DETAIL ABOUT THE CLASSES L AND NL 9. EXPLAIN IN DETAIL ABOUT NP-COMPLETENESS 10. EXPLAIN IN DETAIL ABOUT SEARCHING IN GRAPHS 11. EXPLAIN IN DETAIL ABOUT NL EQUALS CONL 12. EXPLAIN IN DETAIL ABOUT HIERARCHY THEOREMS 22

23 13. EXPLAIN IN DETAIL ABOUT EXPONENTIAL SPACE COMPLETENESS 14. EXPLAIN IN DETAIL ABOUT RELATIVATIO 15. EXPLAIN IN DETAIL ABOUT LIMITS OF DIAGONALIZATION METHOD 16. EXPLAIN IN DETAIL ABOUT CIRCUIT COMPLEXITY 17. PROOF: TQBF IS PSPACE-COMPLETE 23

24 UNIT-V 3. What is meant by Optimization problem? We seek the best solution among a collection of possible solutions called optimization problems. Examples: To find a largest clique in a graph, A smallest vertex cover, or a shortest connecting two nodes. 2. What is meant by cut, cut edge, uncut edge and MAX-cut? A cut in an undirected graph is a separation of the vertices V Into two disjoint subsets S and T. 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. The size of a cut is the number of cut edges. The MAX-CUT problem asks for a largest cut in a graph G. 3. What is meant by PROBABILISTIC ALGORITHMS? A probabilistic algorithm is an algorithm designed to use the outcome of a random process. Typically, such an algorithm would contain an instruction to "flip a coin" and the result of that coin flip would influence the algorithm's subsequent execution and output. Certain types of problems seem to be more easily solvable by probabilistic algorithms than by deterministic algorithms. 5. Define Probabilistic Turing Machine? 24

25 6. Define BPP BPP is the class of languages that are recognized by probabilistic polynomial time Turing machines with an error probability of 1/ What is meant by Prime number and composite? A prime number is an integer greater than 1 that is not divisible by positive integers other than 1 and itself. A nonprime number greater than 1 is called composite. 8. Define Factors One way to determine whether a number is prime is to try all possible integers less than that number and see whether any are divisors, also called factors. 9. What is meant by RP? RP is the class of languages that are recognized by probabilistic polynomial time Turing machines where inputs in the language are accepted 25

26 with a probability of at least 1/2 and inputs not in the language are rejected with a probability of What is meant by branching program? A branching program is a directed acyclic graph where all nodes are labeled by variables, except for two output nodes labeled 0 or 1. The nodes that are labeled by variables are called query nodes. Every query node has two outgoing edges, one labeled 0 and the other labeled 1. Both output nodes have no outgoing edges. One of the nodes in a branching program is designated the start node. 11. Define 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. 12. Define alternating Turing machine An alternating Turing machine is a nondeterministic Turing machine with an additional feature. Its states, except for q accept and q reject are divided into universal states and existential states. When we run an alternating Turing machine on an input string, we label each node of its nondeterministic computation tree with A or V, depending on whether the corresponding configuration contains a universal or existential state. We determine acceptance by designating a node to be accepting if it is labeled with A and all of its children are accepting or if it is labeled with V and any of its children are accepting. 13. Define ATIME and ASPACE? 14. What is meant by input string? 26

27 The objective is to determine whether this string is a member of some language. In the NONISO example, the input string encoded the two graphs. 15. What is meant by Random input? For convenience in making the definition, we provide the Verifier with a randomly chosen input string instead of the equivalent capability to make probabilistic moves during its computation. 16. What is meant by Partial message history? A function has no memory of the dialog that has been sent so far, so we provide the memory externally via a string representing the exchange of messages up to the present point. We use the notation ml#m2# #mi to represent the exchange of messages ml through mi. 17. What is meant by parallel computer? A parallel computer is one that can perform multiple operations simultaneously. Parallel computers may solve certain problems much faster than sequential computers, which can only do a single operation at a time. 18. What is meant by P-complete? A language B is P-complete if 1. B E P, and 2. Every A in P is log space reducible to B. 19. Definition of Trap door Function 27

28 20. Definition of one-way permutation 28

29 PART-B 1. EXPLAIN IN DETAIL ABOUT APPROXIMATION ALGORITHM 2. EXPLAIN IN DETAIL ABOUT PROBABILISTIC ALGORITHMS 3. EXPLAIN IN DETAIL ABOUT THE CLASS BPP 4. EXPLAIN IN DETAIL ABOUT PRIMALITY 5. EXPLAIN IN DEATIL ABOUT READ-ONCE BRANCHING PROGRAMS 6. EXPLAIN IN DETAIL ABOUT ALTERNATION 7. EXPLAIN IN DETAIL ABOUT ALTERNATING TIME AND SPACE 8. EXPLAIN IN DETAIL ABOUT THE POLYNOMIAL TIME HIERARCHY 9. EXPLAIN IN DETAIL ABOUT INTERACTIVE PROOF SYSTEMS 10. EXPLAIN IN DETAIL ABOUT GRAPH NONISOMORPHISM 11. EXPLAIN IN DEATAIL ABOUT PARALLEL COMPUTATION 12. EXPLAIN IN DEATAIL ABOUT UNIFORM BOOLEAN CIRCUITS 13. EXPLAIN IN DEATIL ABOUT THE CLASS NC 14. EXPLAIN IN DEATAIL ABOUT CRYPTOGRAPHY 15. EXPLAIN IN DETAIL ABOUT PUBLIC-KEY CRYPTOGRAPHY 16. EXPLAIN IN DETAIL ABOUT ONE-WAY FUNCTIONS 17. EXPLAIN IN DETAIL ABOUT TRAPDOOR FUNCTIONS 29

30 18. PROOF: FOR EVERY D>=0, A DEGREE-D POLYNOMIAL P ON A SINGLE VARIABLE X EITHER HAS AT MOST D ROOTS, OR IS EVERYWHERE EQUAL TO PROOF: EQ ROBP IS IN BPP 20. PROOF: B IS APOLYNIMAIL TIME, 2-OPTIMA APPROXIMATION ALGORITHM FOR MAX-CUT. 30