31+Slides--NFAs%2C+Regular+Expressions

31+Slides--NFAs%2C+Regular+Expressions - CS103 HO#31...

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CS103 HO#31 Slides--NFAs, Regular Expressions 4/25/11 1 Theorem 1.39 : Every nondeterministic finite automaton has an equivalent deterministic finite automaton. To convert an NFA N = (Q, , , q 0 , F) recognizing A into a DFA M = (Q', , ' , q 0 ', F ' ): 1. Q' = P (Q) 2. For R Q' and a   , let ' (R, a) = { q Q | q   (r, a) for some r R }. Alternately, we may write ' (R, a) = (r, a) 3. q 0 ' = { q 0 } 4. F' = { R Q' | R contains an accepting state of N } This does not handle -transitions. r R without -transitions. Theorem 1.39 : Every nondeterministic finite automaton has an equivalent deterministic finite automaton. To convert an NFA N = (Q, , , q 0 , F) recognizing A into a DFA M = (Q', , ' , q 0 ', F): 1. Q' = P (Q) 2. For R Q' and a   , let ' (R, a) = { q Q | q E( (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 -transitions, make the changes shown in red, where { E(R) = { q Q | q can be reached by following 0 or more -arrows }. 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. Although converting an NFA to a DFA often results in a machine with roughly the same number of states as the NFA, this one needs at least 2 n states. Reference: Hopcroft, Motwani, and Ullman, Automata Theory, Languages, and Computation, 3 rd Edition . Another Bad Case for the Subset Construction Let = {a, b, c, ..., z} Let L Missing = {w | there is a symbol from not in w} The NFA would look much like the one we had for the
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  • Spring '11
  • PLUMMER
  • SEPTA Regional Rail, Formal language, Regular expression, Nondeterministic finite state machine, Automata theory, regular expressions

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