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As de?ned the powerset construction builds a DFA with many states that can never be reached from Q′0. Since they cannot be reached from Q′0 there is no path from Q′0 to a state in F′ which passes through them and they can be deleted from the automaton without changing the language it accepts. In practice it is much easier to build Q′ as needed, only including those state sets that actually are needed.
To see how this works, lets carry out an example. For maximum generality, let's start with the NFA with ε-transitions given above, repeated here:
Because it is simpler to write the transition function (δ) out as a table than it is to write out the transition relation (T) as a set of tuples, we will work with the δ representation. When given a transition graph of an NFA with ε-transitions like this there are 6 steps required to reduce it to a DFA:
1. Write out the transition function and set of ?nal states of the NFA.
2. Convert it to an NFA without ε-transitions.
(a) Compute the ε-Closure of each state in the NFA.
(b) Compute the transition function of the equivalent NFA without ε-transitions.
(c) Compute the set of ?nal states of the equivalent NFA without ε- transitions.
dfa for (00)*(11)*
(c) Can you say that B is decidable? (d) If you somehow know that A is decidable, what can you say about B?
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
Construct a Mealy machine that can output EVEN or ODD According to the total no. of 1''s encountered is even or odd.
Paths leading to regions B, C and E are paths which have not yet seen aa. Those leading to region B and E end in a, with those leading to E having seen ba and those leading to B no
Theorem The class of recognizable languages is closed under Boolean operations. The construction of the proof of Lemma 3 gives us a DFA that keeps track of whether or not a give
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The Recognition Problem for a class of languages is the question of whether a given string is a member of a given language. An instance consists of a string and a (?nite) speci?cat
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