1. Which of the following regular expressions over {0, 1} denotes the set of all strings not containing 100 as a sub-string?
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1 Multiple choice type questions. Which of the following regular expressions over {, } denotes the set of all strings not containing as a sub-string? 2. DFA has a) *(*)* b) ** c) ** d) *(+)* a) single final state b) more than one initial states c) unique path to the final state for a set of inputs d) all of these. 3. Which of the following is regular? a) Strings of s whose length is a perfect square. b) Strings of all palindromes made up of s and s c) Strings of s whose length is a prime number d) Strings of odd number of zeroes. 4. The logic of pumping lemma is a good example of a) The pigeon-hole principle b) The divide and conquer technique c) Recursion d) Iteration. 5. If S is the number of state sin NDFA then equivalent DFA can have maximum of a) S states b) S- states c) 2 S states d) 2 S states. 6. Given an arbitrary NDFA with n states, the maximum number of states in an equivalent minimized DFA is at least a) n b) 2 n c) n! d) None of these
2 . 7. a* (a+b)* is equivalent to a) a* + b* b) (ab)* c) a*b* d) None of these. 8. A solution to the equation R=Q+RP is a) R=QP* b) Q=RP* c) P=RQ* d) R=PQ* 9. Which of the following sets is regular? a) {a i i=n2, n>=} b) {a p p is a prime } c) {ww } w is in {a, b}} d) { a 2n n>=}. The value of L (Φ*) is a) Σ b) {ε} c) {} d) None of these. Palindromes can not be recognized by any FSM because a) an FSM can not remember arbitrary, large amount of information b) an FSM cannot deterministically fix the mid point c) even if the mid-point is known, an FSM cannot find whether the second half of the string matches the first half d) all of these 2. The basic limitation of finite state machine is that a) an FSM can not remember arbitrary, large amount of information b) an FSM cannot recognize grammars that are regular c) an FSM sometimes recognizes grammars that are regular d) all of these
3 3. Regular expression (a b)(a b) denotes the set a) {a, b, ab, aa} b) {a, b, ba, bb} c) {a, b} d) {aa, ab, ba, bb} 4. Consider the following regular expression: R = {ab+abb)*bbab Which of the following is not in the set denoted by R? a) ababab b) ababbabbbab c) abbbab d) abbabbbab 5. Which of the following is correct? a) Language can be derived from FA b) Regular expressions can be derived from the FA c) FA can be derived from the language d) Both (a) and (b) 6. A grammar in CNF may contain productions like a) A B b) A BC c) A ab d) A abcd 7. Pumping lemma for CFL is used to show that a) A given language is Regular b) A given language is Context Free c) A given language is Context Sensitive d) A given language is not Context Free
4 Short answer (5 marks). (a) State the pumping lemma for regular language 2 (b) Using pumping lemma prove that L = { n n n>=}is not regular Draw the transition diagram of a finite state automaton that accepts all strings over {, } (a) Having odd number of s. (b) Having even number of s and even number of s 3. State and discuss Myhill and Nerode theorem. 4. Construct a regular grammar G generating the regular set represented by P = a*b (a+b)* 5. (a) State the difference between DFA and NFA 2 (b)design an NFA containing all binary strings containing or as substrings (a) What is regular language? 2 (b) Find regular expressions over Σ={a, b}for the languages defined as follows: i) L = { a m b m m>} ii) L2 = { a 2n b 2m+ n>=, mn>=} iii) L3 = { b m ab n m>, n>} 7. Find the regular expression for the following transition graph: Q Q2 Q
5 8. Convert the following NFA to DFA,, Q Q Q2. The set L={a i b j c k where i, j, k are integer and i, j, k>=}. Is L regular? Justify your answer. Minimize the following machine by determining the set of equivalent states. Present State Next State Input = Next State Input= State Output State Output A E C B C A C B G D G A E F B F E D H F B 2. Prove the following identify: r(s+t) = rs + rt 3. Covert the grammars to Greibach Normal Form (GNF): (i) S asa asb ε (ii) S asb asbs ε 4. Explain context free grammar with example 5. Construct context free grammar for generating the language {a n b m cn m,n>=} 6. Remove useless symbols from the Context Free Grammar: S ab bx A BAd bsx a B asb bbx X SBD abx ad 7. Construct a CFG for the RE: R=O*(+)* 8. Find a grammar generating the language:
6 L = {(x n y m z m ) n>=,m>=} 9. Show that the language L= {(a n b n c n ) n>=} is not context free. 2. Design a CFG for the language L= {(a n b 2n ) n>=} 2. Let G={VN,VT,S,P) be a phase-structure grammar, where VN=(S,B), VT={a,b}, P={S aba, B aba, B b}. Find L(G) 22. a) Show that the following grammar is ambiguous: S a absb aab A bs aaab b) Construct a context free grammar that generates the language L= { wcw R w (a, b)*} 23. Define parse tree. What is zero equivalent state? 24. Find a grammar L = {(a n b n c m ) n>=, m>=} 25. Prove that CFLs are not closed under intersection and complement operation. 26. Convert the following context free grammar into an equivalent grammar in CNF S aabb A abab AA a B bbaa bbb b Long answer type questions. (a) Design a 2-input 2-output Mealy machine, which takes as input a binary stream and generates output only when a sequence of the pattern is foiund in the input stream. Justify the design clearly. 7 (b) From the Mealy machine above find the equivalent Moore machine 4 (c) Check whether the Mealy machine you obtained is a minimal one or not. Give proper justification to your answer (a) Construct a minimum state automaton equivalent to a given automaton M whose transition table is given below: 8 State Input
7 a b Q Q Q3 Q Q2 Q5 Q2 Q3 Q4 Q3 Q Q5 Q4 Q Q6 Q5 Q Q4 Q6 Q Q3 (b) Find the regular expression corresponding to the following figure: 7 Q Q2 Q4 Q3 3. Explain different methods for removing useless productions, ε-productions and unit producions from any context free grammar 4. Convert the grammars to Chomsky Normal form: (i) S abab A bab ε B Baa A ε (ii) S asb ε S AB aa ab aba
8 5. Convert the following gramma in GNF: G=({A,A2,A3},{a,b},P,A) Where P = {A A2A3, A2 A3A, A2 b, A3 AA2, A3 a} 6. The following grammar generats prefix expressions with operands X and Y and binary operators +, - and *. E +EE *EE -EE X Y i) Find leftmost and righmost derivations and a derivation tree for string + * - XYXY ii) Show that the grammar is unanbiguous. 7. a) Explain Chomsky Normal Form and Greibach Normal Form. b) Consider the grammar G=({S,A,B}{a,b}P,S) where P: S ba ab A baa as a B abb bs b Find an equivalent grammar in CNF 8. a) Consider the following context free grammar: G=(V,Σ,P,S) where V,Σ,P,S are in theor usual meaning. P consists of the following production rules: S aabb A aa a B bb b Find a grammar in Chomsky normal form equivalent of the grammar G. b) Write the CFG for the language L = { i j 2 k i=j or j=k} 9. a) What is ambiguous grammar? b) Check whether the following grammar is ambiguous: S icts ictses a C b c) Construct a context free grammar generating following language: L={a n b n n>= {a m b 2m m>=} and also construct PDS for the above derived CFG.. a) For the grammar S ab ba A a as baa
9 B b bs abb Give the left most and right derivation for the string aaabbabbba. b) Design a CFG for the language L(G) = { n m n m}. a) Remove left recursion from given grammar: A Ba b B Bc Ad ε b) Convert the grammar to GNF: S ABb a A aaa B B bab 2. a) Find if the string aaa bbb ccc can be derived from the following productions. S ABSc, S Abc, BA AB, Bb bb, Ab ab, AA aa b) Draw deruvation ntree foir the sentential form baabaab for the productions: S AB, A Aa, A bb, B a, B Sb
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