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Automaton (NFA) (with ε-transitions) is a 5-tuple: (Q,Σ, δ, q0, Fi where Q, Σ, q0 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.
In Exercise 9 you showed that the recognition problem and universal recognition problem for SL2 are decidable. We can use the structure of Myhill graphs to show that other problems
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State & prove pumping lemma for regular set. Show that for the language L={ap |p is a prime} is not regular
The Emptiness Problem is the problem of deciding if a given regular language is empty (= ∅). Theorem 4 (Emptiness) The Emptiness Problem for Regular Languages is decidable. P
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We represented SLk automata as Myhill graphs, directed graphs in which the nodes were labeled with (k-1)-factors of alphabet symbols (along with a node labeled ‘?' and one labeled
We developed the idea of FSA by generalizing LTk transition graphs. Not surprisingly, then, every LTk transition graph is also the transition graph of a FSA (in fact a DFA)-the one
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