Proving Correctness of DFAs and Lower Bounds

Induction is a proof principle that is often used to establish a statement of the form “for all natural numbers i, some property P (i) holds”, i.e., ∀i ∈ N. P (i). In this class, there will be many occassions where we will need to prove that some property holds for all strings, especially when proving the correctness of a DFA design, i.e., ∀w ∈ Σ∗. S(w). We will often prove such statements “by induction on the length of w”. What that means is “We will prove ∀w. S(w) by proving ∀i ∈ N.∀w ∈ Σ. S(w)”. That is, we will take ith statement to be proved by induction to be ∀w ∈ Σ. S(w). Before giving examples of such proofs, we will begin by establishing some basic properties of DFAs that will be useful.