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By Jeff Erickson

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6 Decision Algorithms It’s unclear how much we can say here, since we haven’t yet talked about graph algorithms, or even really about graphs. Perhaps this discussion should simply be moved to the graphtraversal notes. • Is w ∈ L(M)? Follow the unique path from q0 with label w. By definition, w ∈ L(M ) if and only if this path leads to an accepting state. • Is L(M) empty? The language L(M ) is empty if and only if no accepting state is reachable from q0 . This condition can be checked in O(n) time via whatever-first search, where n is the number of states.

Let L ⊆ Σ∗ be an arbitrary regular language. Prove that the following languages are regular. ] 14 Models of Computation Lecture 4: Nondeterministic Automata [Fa’14] (a) repeat(L) := {w ∈ Σ∗ | wn ∈ L for some n ≥ 0} (b) allreps(L) := {w ∈ Σ∗ | wn ∈ L for every n ≥ 0} (c) manyreps(L) := {w ∈ Σ∗ | wn ∈ L for infinitely many n ≥ 0} (d) fewreps(L) := {w ∈ Σ∗ | wn ∈ L for finitely many n ≥ 0} n (e) powers(L) := w ∈ Σ∗ w2 ∈ L for some n ≥ 0 (f) whattheN (L) := {w ∈ Σ∗ | wn ∈ L for some n ∈ N }, where N is an arbitrary fixed set of non-negative integers.

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ 0 1 2 3 4 5 ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ 6 7 8 In the first iteration of the main loop, the algorithm discovers several distinguishable pairs of states. For example, the algorithm sets Dist[0, 2] ← True because Dist[δ(0, 1), δ(2, 1)] = Dist[2, 9] = True. —so the algorithm terminates. 16 Models of Computation Lecture 3: Finite-State Machines [Fa’14] The final table implies that the states of our trimmed DFA fall into exactly three equivalence classes: {0, 1, 3, 5}, {2, 4}, and {6, 7, 8, 9}.

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