# CT32 COMPUTER NETWORKS DEC 2015

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1 Q.2 a. Using the principle of mathematical induction, prove that (10 (2n-1) +1) is divisible by 11 for all n N (8) Let P(n): (10 (2n-1) +1) is divisible by 11 For n = 1, the given expression becomes (10 (2*1-1) +1) = 11, which is divisible by 11. So, the given statement is true for n=1, i.e. P(1) is true. Let P(k) be true. Then P(k): (10 (2k-1) +1) is divisible by 11 => (10 (2k-1) +1) = 11m, for some natural number m. Now, {(10 (2(k+1)-1) +1) } = (10 (2k+1) +1) = { (2k-1) +1 } = 100 x {10 (2k-1) +1} 99 = (100x 11m)-99 = 11 x (100m - 9), which is divisible by 11 => P(k+1): (10 (2(k+1)-1) +1) is divisible by 11 => P(k+1) is true, whenever P(k) is true. Thus, P(1) is true and P(k+1) is true, whenever P(k) is true. Hence by the principle of mathematical induction, P(n) is true for all n N. b. Discuss the description of Finite Automata. Why we study Automata Theory in computer science? (8) Finite Automata A finite automaton is an abstract model of a digital computer. A finite automaton has a mechanism to read input, which is a string over a given alphabet. This input is actually written on an input file, which can be read by the automaton but cannot change it. Input file is divided into cells, each of which can hold one symbol. The automaton has a temporary storage device, which has unlimited number of cells, the contents of which can be altered by the automaton. Automaton has a control unit, which is said to be in one of a finite number of internal states. IETE 1

2 The automaton can change state in a defined way. Types of Finite Automaton (a) Deterministic Finite Automata (b) Non-deterministic Finite Automata A deterministic automata is one in which each move (transition from one state to another) is unequally determined by the current configuration. If the internal state, input and contents of the storage are known, it is possible to predict the future behaviour of the automaton. This is said to be deterministic finite automata otherwise it is nondeterministic finite automata. Definition of Deterministic Finite Automaton A Deterministic Finite Automata (DFA) is a collection of 5-tuples as: The input mechanism can move only from left to right and reads exactly one symbol on each step. The transition from one internal state to another is governed by the transition function δ. If δ(q0, a) = q1 then if the DFA is in state q0 and the current input symbol is a, the DFA will go into state q1. Definition of Nondeterministic Finite Automaton A Nondeterministic Finite Automata (NFA) is defined by a collection of 5-tuples: A string is accepted by an NFA if there is some sequence of possible moves that will put the machine in the final state at the end of the string. Need of Study and Applications of Finite Automata String Processing Consider finding all occurrences of a short string (pattern string) within a long string (text string). This can be done by processing the text through a DFA: the DFA for all strings that end with the pattern string. Each time accept state is reached; the current position in the text is output. IETE 2

3 Finite-State Machines A finite-state machine is an FA together with actions on the arcs. Statecharts Statecharts model tasks as a set of states and actions. They extend FA diagrams. Lexical Analysis In compiling a program, the first step is lexical analysis. This isolates keywords, identifiers etc., while eliminating irrelevant symbols. A token is a category, for example identifier, relation operator or specific keyword. Q.3 a. Solve the following: (3+2) (i) Construct a DFA that behaves equivalent to the NDFA given by M such that M = ( {q0, q1, q2, q3}, {a, b}, δ, q0, {q3}) where δ is given by STATE Input a Input b Initial state q0 q0, q1 q0 q1 q2 q1 q2 q3 q3 Final State *q3 -- q2 (ii) Find a DFA machine that accepts an even number of either 0 s or 1 s or both 0 s and 1 s over input symbol = {0, 1}. (i) (ii) The transition system (δ) of the DFA machine is shown in the following figure: IETE 3

4 Hence the DFA machine is defined as ({q1, q2, q3, q4}, {0, 1}, δ, q1, {q1}). b. Find a DFA machine for the language L = {0w0 w ϵ {0, 1}*}. (4) The transition graph of the given DFA machine is shown in the following figure. This transition graph consists of 5 tuples as (Q,, δ, q0, F) that is defined as ({q0, q1, q2, q3}, {0, 1}, δ, q0, {q2}). c. Make a minimum state finite automaton that is equivalent to a DFA whose transition table is given as: (7) Definition State a b Initial State q0 q1 q2 q1 q4 q3 q2 q4 q3 Final State *q3 q5 q6 Final State *q4 q7 q6 IETE 4

5 q5 q3 q6 q6 q6 q6 q7 q4 q6 IETE 5

6 Q.4 a. Find the regular expressions corresponding to the following finite automata; consider q 1 as initial state in both automata (s): (8) (i) (i) (ii) (ii) IETE 6

7 (i) b. Obtain a Deterministic Finite Automaton with minimized or reduced states for the following regular expressions (8) (i) (0 + 1)* ( ) (0 + 1)* (ii) 10 + (0 + 11) 0* 1 IETE 7

8 IETE 8

9 Sol (ii): IETE 9

10 IETE 10

11 Q.5 a. Solve the following: (3+3) (i) Show that the language denoted by L = { 0 i 1 i i >= 1} is not a regular language. (ii) Check whether the language represented as L = { a i b j c k k > i + j} is regular or not? Also justify your answer using pumping Lemma. (i) (ii) b. Test the equivalence of two regular languages represented by the regular expressions P = (a + b)* and Q = a*(b a*)* respectively. Is P = Q Explain your answer with proper justification? (4) IETE 11

12 IETE 12

13 (i) L.H.S. (ii) c. Solve the following: (3+3) (i) Consider the two regular Languages represented by the regular expressions P and Q respectively, where P is defined as P = (b + a a* b) and Q is any regular expression then show that P + P Q*Q = a*b Q* (ii) Consider a context free grammar G that consists of the productions S 0B 1A A 0 0S 1AA B 1 1S 0BB For the string , Find the (i) the Leftmost Derivation (ii) the Rightmost Derivation (iii) Parse Tree = PΛ + P Q*Q = P(Λ + Q*Q) = PQ* = (b + aa*b)q* (by definition of P) = (Λb + aa*b)q* = a*bq* = R.H.S. IETE 13

14 Q.6 a. Solve the following: (3+2) (i) Make a deterministic PDA by Final State that accepts the language L = { w ϵ {a, b}* the number of a s in w equal to the no. of b s in w}. (ii) Convert the context free grammar S asb A, A bsa S ϵ to a equivalent PDA by empty stack. (i) (ii) IETE 14

15 b. Construct a Pushdown Automata (PDA) that accepts the language L defined as: L = {a n b m a n m, n 1} by empty stack. Also make the corresponding CFG productions accepting the same set or language. (8) IETE 15

16 c. Define Inherent Ambiguity and Inherently Ambiguous Languages. If G is a context free grammar whose production rules are given as S SbS a. Check whether the given grammar G is ambiguous or inherently ambiguous. (3) Inherent Ambiguity and Inherently Ambiguous Languages The languages generated by a grammar, that have both ambiguous and unambiguous grammars but there exist languages for which no unambiguous grammar can exist. Such types of languages are called inherently ambiguous languages and the property is known as inherent ambiguity. IETE 16

17 IETE 17

18 Q.7 a. Consider a context free grammar G whose production rules are defined as S ASA ba, A B S, B c. Reduce it into Chomsky Normal Form (CNF). (6) IETE 18

19 b. Reduce the given CFG defined as S aabb, A aa a, B bb b into Chomsky Normal form. (6) c. Check whether the language defined as L = {a p p is a prime} is a context free language or not. Justify your answer by using the help of Pumping Lemma. (4) Q.8 a. Design Turing Machines that recognizes the following languages: (5+5) (i) L = {0 n 1 n n 1} and (ii) Set L of all strings over {0, 1} ending with 010 IETE 19

20 Sol (ii): IETE 20

21 b. Define a Turing machine. Construct a Turing Machine that accepts the language given by the expression (0 1* + 1 0*) (6) Definition of Turing Machine A Turing Machine M is a collection of 7-tuples as: As the Turing machine will have to be able to find its input, and to know when it has processed all of that input, we require: IETE 21

22 (a) The tape is initially blank (every symbol is #) except possibly for a finite, contiguous sequence of symbols. (b) If there are initially nonblank symbols on the tape, the tape head is initially positioned on one of them. This emphasises the fact that the input viz., the non-blank symbols on the tape does not contain #. Q.9 a. State the Post correspondence problem (PCP). Find at least three solutions to the PCP defined by the following sets: (8) A = {1, 10, 10111} and B = {111, 0, 10} Post correspondence problem (PCP): An instance of PCP consists of two lines of strings over some alphabet Σ; the two lists must be equal length. We generally refer to the A and B lists, and write A = w 1, w 2,,w k and B = x 1, x 2,,x k, for some integer k. For each i, the pair (w i, x i ) is said to be a corresponding pair. We say this instance of PCP has a solution, if there is a sequence of one or more integers i 1,i 2,, i m that, when interpreted as indexes for strings in the A and B lists, yield the same string. That is, w i1 w i2 w im = x i1 x i2 x im. We say the sequence i 1, i 2,.,i m is a solution to this instance of PCP, if so, the Post correspondence problem is: Given an instance of PCP, tell whether this has a solution. Given : A = (1, 10, 10111) B = (111, 0, 10) From the above we conclude that A1 = 1, A2 = 10, A3 = B1 = 111, B2 = 0, B3 = 10 Then A 3 A 1 A 1 A 2 = B 3 B 1 B 1 B 2 IETE 22

23 Hence the PCP with the given list has a solution. Repeating the sequence 3, 1, 1, 2 we can get more solutions. As a example: A 3 A 1 A 1 A 2 A 3 A 1 A 1 A 2 = B 3 B 1 B 1 B 2 B 3 B 1 B 1 B 2 = Similarly we can get an another solution as having the sequence 3, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2. So the three solutions to the PCP defined by the given sets: (i) 3, 1, 1, 2 (ii) 3, 1, 1, 2, 3, 1, 1, 2 (iii) 3, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2 b. Differentiate between Recursive Languages and Recursively Enumerable Languages. Show that if L1 and L2 are Recursively Enumerable Languages than L1 U L2 is also Recursively Enumerable as well as if L1 and L2 are Recursive Languages than L1 U L2 is also recursive. (8) IETE 23

24 IETE 24

25 IETE 25

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