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Unformatted text preview: Nonregular languages (Pumping Lemma) Regular languages b a * a c b + * ... etc * ) ( b a c b + + Nonregular languages } : { n b a n n }*} , { : { b a v vv R How can we prove that a language is not regular? L Prove that there is no DFA or NFA or RE that accepts L Difficulty: this is not easy to prove ( since there is an infinite number of them) Solution: use the Pumping Lemma !!! The Pigeonhole Principle pigeons pigeonholes 4 3 A pigeonhole must contain at least two pigeons ........... ........... pigeons pigeonholes n m m n The Pigeonhole Principle ........... pigeons pigeonholes n m m n There is a pigeonhole with at least 2 pigeons The Pigeonhole Principle and DFAs Consider a DFA with states 4 1 q 2 q 3 q a b 4 q b a b b a a Consider the walk of a long string: 1 q 2 q 3 q a b 4 q b b b a a a aaaab 1 q 2 q 3 q 2 q 3 q 4 q a a a a b A state is repeated in the walk of (length at least 4) aaaab aaaab 1 q 2 q 3 q 2 q 3 q 4 q a a a a b 1 q 2 q 3 q 4 q Pigeons: Nests: (Automaton states) Are more than Walk of The state is repeated as a result of the pigeonhole principle (walk states) Repeated state Consider the walk of a long string: 1 q 2 q 3 q a b 4 q b b b a a a aabb 1 q 2 q 3 q 4 q 4 q a a b b A state is repeated in the walk of (length at least 4) Due to the pigeonhole principle: aabb aabb 1 q 2 q 3 q 4 q Automaton States Pigeons: Nests: (Automaton states) Are more than Walk of The state is repeated as a result of the pigeonhole principle (walk states) 1 q 2 q 3 q 4 q 4 q a a b b Repeated state i q ..........
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This note was uploaded on 12/02/2011 for the course AR 107 taught by Professor Gracegraham during the Fall '11 term at Montgomery College.
 Fall '11
 GraceGraham

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