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Automaton (NFA) (with ε-transitions) is a 5-tuple: (Q,Σ, δ, q_{0}, F_{i} where Q, Σ, q_{0} and F are as in a DFA and T ⊆ Q × Q × (Σ ∪ {ε}).
We must also modify the de?nitions of the directly computes relation and the path function to allow for the possibility that ε-transitions may occur anywhere in a computation or path. The ε-transition from state 1 to state 3 in the example, for instance, allows the automaton on input ‘a' to go from state 0 not only to state 1 but also to immediately go on to state 3. Similarly, it allows the automaton, when in state 1 with input ‘b', to move ?rst to state 3 and then take the ‘b' edge to state 0 or, when in state 0 with input ‘a', to move ?rst to state 2 and then take the ‘a' edge to state 3. Thus, on a given input ‘σ', the automaton can take any sequence of ε-transitions followed by exactly one σ-transition and then any sequence of ε-transitions. To capture this in the de?nition of δ we start by de?ning the function ε-Closure which, given a state, returns the set of all states reachable from it by any sequence of ε-transitions.
Suppose A = (Q,Σ, T, q 0 , F) is a DFA and that Q = {q 0 , q 1 , . . . , q n-1 } includes n states. Thinking of the automaton in terms of its transition graph, a string x is recogn
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Kleene called this the Synthesis theorem because his (and your) proof gives an effective procedure for synthesizing an automaton that recognizes the language denoted by any given r
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