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Spring 2016 CIS 262
Automata, Computability and Complexity
Jean Gallier
Review Session
Released February 17, 2016
Problem 1. (a) Give an NFA accepting L = {aa, bb}∗ (with a single -transition).
(b) Convert the NFA of question 1(a) to a DFA.
Problem 2. (a) Given a DFA
D = (
Q,Σ,δ ,q 0, F
), construct an equivalent DFAD = (Q,Σ, δ , q 0, F ) (i.e. such that L(D) = L(D)) and such that there are no incoming
transitions into the start state q 0
of D.
(b) Prove of disprove (by giving a counter-example) the following statement:
A DFA D accepts a finite language iff its underlying directed graph has no cycle.
Problem 3. Given any alphabet Σ = {a1, . . . , ak}, prove by induction on n ≥ 0 that thefunction L maps R(Σ)n into R(Σ)n, that is, L(R(Σ)n) ⊆ R(Σ)n. Conclude that L(R(Σ)) ⊆R(Σ).
Problem 4. Given an alphabet Σ, for any language L over Σ, if LR
= {wR
| w ∈ L} is thereversal language of L, prove that
(L1 ∪ L2)R = LR
1 ∪ LR
2 ;
(L1L2)R = LR
2LR1 ;
(L∗)R = (LR)∗.
Problem 5. (a) Let Σ = {a, b}. Give a DFA accepting
L1 = {w ∈ Σ∗ | w contains an even number of a’s}
(There is one with two states and, by the way, 0 = zero is even.)Give a DFA accepting
L2 = {w ∈ Σ∗ | w contains a number of b’s divisible by 3}
(There is one with three states and the number of b’s is 0, 3, 6, 9, . . ..)
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(c) Give a DFA accepting L3 = L1 ∩ L2.
Problem 6. Describe a method for testing whether L(D1) = L(D2), for any two DFA’s, D1
and D2 over the same alphabet.
Hint : Recall that for any two sets, A and B , we have A = B iff A ⊆ B and B ⊆ A and thatA ⊆ B iff A−B = ∅. Also use the cross-product construction.
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