Theory of Computation — Winter 2022

What are the closure properties of regular languages?

Explain reflexivity, symmetry, and transitivity properties of relations.

Using principle of Mathematical Induction, prove that for every n >= 1, 7 + 13 + 19 + …. + (6n + 1) = n(3n + 4) Using Principle of Mathematical Induction, prove that for every n

D>e=ri v1e, the string “aabbababbaa“ using leftmost derivation for the fo7l l+o w13in +g 1g9ra +m .m . a. r+. (6n + 1) = n(3n +4) S  aA | bC | b A U si nagS P| briBnc iple of Mathematical Induction, prove that for every n B> = 1 a, C | bA | a C7  + 1a3B +| b1S9 + . . . + (6n + 1) = n(3n +4)

Write Regular Expressions for following (i )UsTihneg lParnignuciapglee ooff aMlla stthreinmgast iicna {l 0In,1d}u*c ttihoant ,d por onvoet ethnadt for every n > = w 1i,t h 11. (i7i) + T 1h3e +la n1g9u +a g. e. o. f+ a(l6l ns t+ri n1g) s= c no(n3tnai +ni4n)g both 101 and 010 as substrings.

Consider the NFA-Λ depicted in following table Λ a b c p Φ {p} {q} {r} q {p} {q} {r} Φ *r {q} {r} Φ {p}

Compute the Λ-closure of each state. (ii) Convert the NFA-Λ to a DFA.

Draw FA for following languages:  L1 = {w | 00 is not substring of w}  L2 = {w | w ends in 01} Find FA accepting languages (i)L1 U L2 and (ii)L1 ∩ L2

Define Bounded Minimalization.

Find the CFG for the regular expression : (011+1)∗ (01)∗

Convert the following CFG into CNF. S → bA | aB A → bAA | aS | a B → aBB | bS | b

Define Primitive Recursive Functions.

Prove that following CFG is Ambiguous. S  S + S | S * S | (S) | a

Find CFG for following language: L = { 0i 1j 0k / j > i + k }

Define Pushdown Automata.

Construct a PDA equivalent to the following CFG. S  0BB B  0S | 1S | 0

Design the pushdown automata for language {0n1n|n≥ 0}.

Suppose the PDA M = ({q , q }, {a, b, c}, {a, b, Z }, δ, q , Z , {q }) has the following transition function. 1. δ(q , a, Λ) = (q , a)00 2. δ(q , b, Λ) = (q , b)00 3. δ(q , c, Λ) = (q , Λ)01 4. δ(q , a, a) = (q , Λ)11 5. δ(q , b, b) = (q , Λ)11 Show the acceptance of abbcbba by the above PDA.

Prove that L = {anbncn | n ≥ 1} is not a CFL.

Design deterministic PDA accepting strings with more a’s than b’s.

Enlist limitations of Turing machines.

Write a short note on Halting problem.

Design a Turing machine to reverse the string over alphabet {0, 1}.

Discuss Universal Turing Machine.

Write a short note on Church-Turing Thesis.

Draw a transition diagram for a Turing machine for the language of all palindromes over {a, b}.