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41 Slides--Regular Expressions

# 41 Slides--Regular Expressions - CS103 HO#41 Slides-Regular...

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CS103 HO#41 Slides--Regular Expressions May 5, 2010 1 Theorem 1.39 : Every nondeterministic finite automaton has an equivalent deterministic finite automaton. To convert an NFA N = (Q, Σ , δ , q 0 , F) recognizing A into a DFA M = (Q', Σ , δ ' , q 0 ', F'): 1. Q' = P (Q) 2. For R Q' and a ∈ Σ , let δ ' (R, a) = { q Q | q ∈ δ (r, a) for some r R }. Alternately, we may write δ ' (R, a) = δ (r, a) r R Theorem 1.39 : Every nondeterministic finite automaton has an equivalent deterministic finite automaton. To convert an NFA N = (Q, Σ , δ , q 0 , F) recognizing A into a DFA M = (Q', Σ , δ ' , q 0 ', F'): 1. Q' = P (Q) 2. For R Q' and a ∈ Σ , let δ ' (R, a) = { q Q | q ∈ δ (r, a) for some r R }. Alternately, we may write δ ' (R, a) = δ (r, a) 3. q 0 ' = { q 0 } 4. F' = { R Q' | R contains an accepting state of N } This does not handle ε -transitions. r R without ε -transitions. Theorem 1.39 : Every nondeterministic finite automaton has an equivalent deterministic finite automaton. To convert an NFA N = (Q, Σ , δ , q 0 , F) recognizing A into a DFA M = (Q', Σ , δ ' , q 0 ', F): 1. Q' = P (Q) 2. For R Q' and a ∈ Σ , let δ ' (R, a) = { q Q | q E( δ (r, a)) for some r R }. 3. q 0 ' = E({ q 0 }) 4. F' = { R Q' | R contains an accepting state of N } For an NFA with ε -transitions, make the changes shown in red, where E(R) = { q Q | q can be reached by following 0 or more ε -arrows }. This procedure is known as The Subset Construction Sipser has a good example of the Subset Construction on pp. 57 – 58. For the DFA, Q' = P ({1, 2, 3}), so there are 8 states. A Bad Case for the Subset Construction q 0 q 1 q 2 q n ... 0, 1 0, 1 0, 1 0, 1 1 0, 1 If we build an NFA like this for some value of n, the language accepted is the set of all strings of 0's and 1's such that the n th symbol from the end is a 1. If we build a DFA to accept the same language, it will have to "remember" the last n symbols it has read, and be prepared for any combination of 0's and 1's after the critical 1.

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• Fall '09
• Formal language, Regular expression, Nondeterministic finite state machine, Automata theory, nondeterministic ﬁnite automaton

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