By Jeff Erickson

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Thus, whenever we reason about or design NFAs, we are free to either allow or forbid -transitions, whichever is more convenient for the task at hand. 1. A language L can be described by a regular expression if and only if L is the language accepted by a DFA. We will prove Kleene’s fundamental theorem in four stages: • Every DFA can be transformed into an equivalent NFA. • Every NFA can be transformed into an equivalent DFA. • Every regular expression can be transformed into an NFA. • Every NFA can be transformed into an equivalent regular expression.

However, the following examples show several useful design strategies. 1 Superstrings Perhaps the simplest rule of thumb is to try to construct an algorithm that looks like MultipleOf5: A simple for-loop through the symbols, using a constant number of variables, where each variable (except the loop index) has only a constant number of possible values. Here, “constant” means an actual number that is not a function of the input size n. You should be able to compute the number of possible values for each variable at compile time.

We have just shown that any NFA M = (Σ, Q, s, A, δ) is equivalent to a DFA M = (Σ, Q , s , A , δ ) defined as follows: Q := 2Q s := {s} A := S ⊆ Q S ∩ A = ∅ δ (q , a) := δ(p, a) for all q ⊆ Q and a ∈ Σ. p∈q Similarly, any NFA with -transitions is equivalent to a DFA with the transition function δ (q , a) = δ(r, a) p∈q r∈ -reach(p) for all q ⊆ Q and a ∈ Σ. This conversion from NFA to DFA is often called the subset construction, but that name is somewhat misleading; it’s not a “construction” so much as a change in perspective.