Chapter 1: Regular Languages

For Your Enjoyment

• Algorithms can be nondeterministic. Check out the nondeterministic algorithm for primality testing in this Wikipedia article.

Chapter 1.2, Nondeterminism

Lecture Problem

• Design a nondeterministic finite automaton that accepts any binary string that contains either 00 or 11.
• What is Q? (a finite set of states)
• What is Σ? (a finite set called the alphabet)
• What is δ? (Q × Σ_ε —> P(Q) is the transition function)
• What is q₀? (the start state)
• What is F? (the set of accept states)
• Note: A nondeterministic finite automaton (NFA) is a 5-tuple (Q, Σ, δ, q₀, F).

Active Learning Problem

Construct a three-state NFA that accepts any binary number that consists of any number of 0s or any number of 1s.

Definition

Machine M = (Q, Σ, δ, q₀, F) accepts string w = w₁...w_n where each w_i ∈ Σ_ε if a sequence of states r₀...r_n exists with

• r₀ = q₀
• r_i+1 ∈ δ(r_i, w_i+1)
• r_n ∈ F

Active Learning

Theorem: Every NFA has an equivalent DFA.

Proof Sketch (assuming no ε transitions): Let NFA N = (Q, Σ, δ, q₀, F). Construct DFA M = (Q', Σ, δ', q₀', F').

Exercise: Convert the three-state NFA above to a DFA using the construction given in the proof.

In-Class Notes