1. Present a (context-free) grammar that generates the language { ambn : m≥n≥0 }.
2. Present a (context-free) grammar that generates the language { w ∈ {a,b}* : na(w) ≥ nb(w) }.
3. Present a dfa that accepts the language { w ∈ {0,1}* : 00101 is a substring of w }.
4. Present a dfa that accepts the language { w ∈ {0,1}* : 00101 is a substring of w but 111 is not }.
5. Present a dfa that accepts the language { w ∈ {a,b}* : na(w) mod 2 = 0 ∧ nb(w) mod 3 ≠ 0 }.
6. Present a dfa that accepts the language { w ∈ {a,b}* : (na(w) + 2nb(w)) mod 3 ≠ 1 }.
7. Present a dfa that accepts the language { w ∈ {0,1}+ : #(w) mod 6 ∈ {1,4} }
where #(w) denotes the numeric value of w when it is interpreted as a binary numeral. For example, #(010110) = 22.
The following facts about # and mod may be useful to you.
| #(0) | = | 0 |
| #(1) | = | 1 |
| #(w0) | = | 2⋅#(w) |
| #(w1) | = | 2⋅#(w) + 1 |
| (2k + m) mod n = (2(k mod n) + m) mod n |
8. For string wa, where w ∈ Σ* and a ∈ Σ, let truncate(wa) = w. That is, truncate "chops off" the last symbol of the string to which it is applied. (Meanwhile, truncate(λ) is undefined.)
Present a coherent argument that if L is a language accepted by some dfa, then so is
Such an argument should be of this form: Assume that you are given a dfa M that accepts L (i.e., L(M) = L); explain how you would modify M to obtain a dfa M' that accepts truncate(L) (i.e., L(M') = truncate(L)).