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

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1 1. Define automata? ANNA UNIVERSITY, CHENNAI B.E/B.Tech DEGREE EXAMINATION APR/MAY 2008 Fifth Semester Computer Science and Engineering CS1303-Theory of Computation Part-A An automaton is an abstract computing device. It is an mathematical model of a system, with discrete inputs, outputs, states and set of transitions from state to state that occurs on input symbol from alphabet. 2. What is the principle of mathematical induction? Let P(n) be a statement about a non-negative integer n. Then the principle of mathematical induction is that P(n) follows from (i) (ii) P(1) and 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 (n-1) is called the induction hypothesis. 3. Construct the DFA for the regular expression aa*/bb*. Given: RE=aa*/bb* Solution: The required DFA is 4. Construct a DFA over =(a,b) which produces not more than 3 a's. Given: =(a,b) Solution: DFA which produces strings with not more than 3 a's: The DFA accepts strings with no a's,one a,two a's and three a's with any number of b's.

2 5. Let S->aB/bA, A->aS/bAA/a, B->bS/aBB/b. Derive the string aaabbabba as left most derivation. Given: A->aB ba A->aS/bAA/a B->bS/aBB/b w=aaabbabbba Solution S ab aabb(b->abb) aaabbb(b->abb) aaabbb(b->b) aaabbb(b->b) 6. What is meant by empty production removal in PDA? The productions with useless symbols are removed, to derive a deterministic,definite set of strings only. 7. State the pumping lemma for CFG. Let L be an infinite context-free language. Then there exists some positive integer m such that any w L with w m can be decomposed as w=uvxyz...1 with vxy m..2 and vy 1..3

3 such that uv'xy"z L.4 for all i=0,1,... This is known as pumping lemma for context free languages. 8. Define turing machine. Turing machine T(M) is defined as M =(,Γ,,q 0,B,F) where is the finite set of states Γ is the finite set of allowable tape symbols B is the symbol of Γ is the blank is the subset of Γ not including B, is the set of input symbols. is the next move function, a kind mapping of function, defined as : Γ-> Γ {L, R} q 0 is the initial state F is a set of final states. 9. What is meant by halting problem? Given any TM M and any input string w, does M halt on w? This problem is undecidable. 10. What is post correspondence problem? An instance of post's correspondence problem (PCP) consist of two lists A=w 1,w 2,w 3...w k and B=x 1,x 2...x k of strings over some alphabets. This instance of PCB has a solution if there is any sequence of integers i 1,i 2,...i m,with m 1,such that w i1,w i2,...w im =x i1,x i2,...x im. The sequence i 1,i 2,...i m is a solution to this instance of PCP. PART-B 11.(a). (i) Prove that for every integer n>=0 the number 4 2n n+2 is a multiple of 13. This type of problems can be solved by using basic and induction steps Basic step Step 1: take n=0,prove L.H.s=R.H.S of the given equation Step2: take n=1,prove L.H.s=R.H.S of the given equation

4 Step 3 n=k, prove L.H.S=R.H.S of the given equation Induction step Take n=k+1,and prove prove L.H.s=R.H.S of the given equation with the help of step 3. (a). (ii) Construct a DFA that will accept strings on {a, b} where the number of b s divisible by 3. Step 1: Write possible number of strings.in which the number of b s divisible by 3. Step 2: If there are n symbols in the string, then there must be n+ 1 state in the DFA Step 3: Take q 0 as initial state and q f be some final state. Every state must have only one transition for each input symbol. Step 4: Write the Transition diagram and table for the given requirement. Step 5: Define the tuples of DFA. 11. (b).construct a finite automaton that accepts the set of all strings in {a,b,c}* such that the last symbol in input string appears earlier in the string. 12. (a).construct a regular expression to the transition diagram.

5 Use Arden s formula to solve this problem. Arden s theorem : 1. Let q1 be the initial state. 2. There are q2,q3,q4..qn number of states.the final state may be some qj where j<=n. 3. Let α ji represents the transition from qj to qi. 4. Calculate qi such that q i = α ji.q j If qi is a start state q i = α ji.q j + 5. Similarly compute the final state which ultimately gives the regular expression r. The required Regular expression is RE=(0+1(1+01)*00)* 12.(b).Construct a NFA for the regular expression (a/b)*abb and draw its equivalent DFA. Step 1: Use the Thompson construction rules to construct the NFA with The required NFA is Step 2: Write the transition table. To construct a DFA from NFA- Step 1: Find the -closure of each state. -closure can be obtained by writing reachable state through input. (Including that state also). Step 2: Give a new name to each set of states derived in step1 Step3: Apply -closure transition function with all input symbols to the states. Step 4: Follow the Step 3, until there is no new states Step 5: Draw the transition diagram for DFA

6 Note: The same problem can also be solved by converting NFA- to NFA and NFA to DFA 13.(a) Construct a CFG accepting L={a m b n /n<m} and construct a PDA accepting L by empty store. Ans: Given L={a m b n /n<m}