COMS W3261
Computer Science Theory
Lecture 5: September 23, 2009
NFAs with Epsilon-Transitions

1. Outline

• Review
• NFA with epsilon-transitions

2. Review

• Nondeterministic finite automata
• The subset construction

3. NFA with Epsilon-Transitions

• An ε-NFA is an NFA (Q, Σ, δ, q₀, F) whose transition function δ is a mapping from Σ ∪ {&epsilon} to the subsets of Q.
• The language of an ε-NFA is the set of all strings that spell out a path from the start state to a final state. There can be ε-transitions along this path.

4. Epsilon-Closures

• We define ECLOSE(q), the ε-closure of state q in Q, recursively as follows:
• State q is in ECLOSE(q).
• If state p is in ECLOSE(q), then all states in δ(p, ε) are also in ECLOSE(q).
• We can compute the ε-closure of a set of states S by taking the union of the ε-closures of all the individual states in S.

5. Converting an ε-NFA to a DFA

• We can eliminate all ε-transitions from an ε-NFA by converting it into an equivalent DFA using the subset construction.
• Given an ε-NFA E = (Q_E, Σ, δ_E, q₀, F_E), we construct the DFA D = (Q_D, Σ, δ_D, q_D, F_D) as follows:
• Q_D = P(Q_E).
• δ_D is computed as follows:
Let S = {p₁, p₂,..., p_k} and let {r₁, r₂,..., r_m} be the union of δ_E(p_i, a) for i = 1, 2, ..., k and all a in Σ.
Then, δ_D(S, a) = ECLOSE({r₁, r₂,..., r_m}).
• q_D = ECLOSE(q₀).
• F_D = { S | S is in Q_D and S contains a state in F_E }.
• As with the subset construction, we can prove by induction that L(D) = L(E).

6. Practice Problems

1. HMU, Exercise 2.5.1.
2. HMU, Exercise 2.5.3.
3. (Hard)If L is a language, let FirstHalf(L) = { x | xy is in L and length(x) = length(y)}. Let D be a DFA that accepts L. From D construct an NFA that accepts FirstHalf(L).

7. Reading Assignment

• HMU: Sects. 2.5-2.8