Path function of a nfa, Theory of Computation

The path function δ : Q × Σ*→ P(Q) is the extension of δ to strings:

1132_Path Function of a NFA1.png

Again, this just says that to ?nd the set of states reachable by a path labeled w from a state q in an NFA with ε-transitions start by ?nding the set of states reachable from q using only ε-transitions and then, for each symbol σ of w (in order) ?nd the set of states reachable from those by an edge labeled σ and then the set of states reachable from those by any sequence of ε-transitions, etc.

Nothing else in the de?nitions need change. The automaton still accepts w if there is any computation on (q0,w) that terminates in a ?nal state after scanning the entire input. Equivalently, it accepts w if there is a path labeled w from the initial state to a ?nal state, which is to say, if δ(q0,w) includes any member of F. Note that the automaton of the example above will accept ‘ε' since state 2 is in ε-Closure(0) and, therefore in δ(0, ε).

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