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Regular_Pumping-1

# Regular_Pumping-1 - Nonregularlanguages(PumpingLemma nn...

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Fall 2006 Costas Busch - RPI 1 Non-regular languages (Pumping Lemma)

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Fall 2006 Costas Busch - RPI 2 Regular languages b a * a c b + * ... etc * ) ( b a c b + + Non-regular languages } 0 : { n b a n n }*} , { : { b a v vv R
Fall 2006 Costas Busch - RPI 3 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 !!!

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Fall 2006 Costas Busch - RPI 4   The Pigeonhole Principle
Fall 2006 Costas Busch - RPI 5   pigeons pigeonholes 4 3

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Fall 2006 Costas Busch - RPI 6   A pigeonhole must contain at least two pigeons
Fall 2006 Costas Busch - RPI 7   ........... ........... pigeons pigeonholes n m m n

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Fall 2006 Costas Busch - RPI 8 The Pigeonhole Principle   ........... pigeons pigeonholes n m m n There is a pigeonhole  with at least 2 pigeons
Fall 2006 Costas Busch - RPI 9 The Pigeonhole Principle and  DFAs

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Fall 2006 Costas Busch - RPI 10   Consider a DFA with       states  4 1 q 2 q 3 q a b 4 q b a b b a a
Fall 2006 Costas Busch - RPI 11 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

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Fall 2006 Costas Busch - RPI 12 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
Fall 2006 Costas Busch - RPI 13 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

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Fall 2006 Costas Busch - RPI 14 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
Fall 2006 Costas Busch - RPI 15 i q ...... ......

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Regular_Pumping-1 - Nonregularlanguages(PumpingLemma nn...

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