COMS W3261
Computer Science Theory
Lecture 4: September 21, 2009
Nondeterministic Finite Automata

1. Outline

• Review
• Nondeterministic finite automata
• The subset construction
• DFAs and NFAs are equivalent

2. Review

• Deterministic finite automata
• If L = L(A) for some DFA A, then L is a regular language.

3. Nondeterministic Finite Automata

• A nondeterministic finite automaton (NFA) consists of
A finite set of states Q.
• An input alphabet consisting of a finite set of symbols Σ.
• A transition function δ that maps Q × &Sigma to P(Q), the set of subsets of Q.
Like a DFA, this transition function can be represented by a transition diagram in which the nodes are labeled by states and arcs by symbols. Unlike a DFA, an NFA may have transitions to zero or more states from a given state on a given input symbol.
• A start state that is one of the states in Q.
• A set of final (or accepting) states F. F is a subset of Q.
• An NFA accepts an input string x if there is a path of transitions from the initial state to a final state that spells out x. Note that, unlike for a DFA, in an NFA there may be more than one path from the initial state to a final state that spells out x.
• Note the definition of an extended transition function for an NFA. See Sect 2.3.3 of HMU.
• L(A), the language defined by an NFA A is the set of strings accepted by A.

4. Equivalence of DFAs and NFAs: the Subset Construction

• From an NFA N = (Q_N, Σ, δ_N, q₀, F_N), we can construct a DFA D = (Q_D, Σ, δ_D, {q₀}, F_D) that simulates all possible moves of N on any given input.
Q_D is the set of all subsets of Q_N.
• The input alphabet of D is the same as that of N.
• For each state S in Q_N and each input symbol a in Σ,
δ_D(S,a) is the union of all states in δ_N(p,a) for all p in S.
• The start state of D is the state {q₀}.
• F_D, the final states of D, are those states of Q_D that contain a state in F_N.
• We can prove by induction on the length of an input string w that D can reach deterministic state S in Q_D on w iff N can reach each nondeterministic state p in S on w. See Sect. 2.3.5 of HMU.

5. Bad Case for the Subset Construction

• Let L be the language (a+b)*a(a+b)^n-1, that is, all strings of a's and b's in which the n^th character from the end is an "a". The smallest DFA that accepts L must have at least 2ⁿ states.

6. Practice Problems

1. Construct an NFA that accepts all strings of a's and b's ending in abb.
2. Show all sequences that your NFA can make on the input string ababb.
3. Use the subset construction to convert your NFA into an equivalent DFA.
4. Construct an NFA that accepts (a+b)*a(a+b)(a+b).
5. Use the subset construction to convert your NFA into an equivalent DFA.

7. Reading Assignment

• HMU: Sects. 2.3-2.4