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Coursepack3-3 - Module 26 Pumping Lemma A technique for...

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1 1 Module 26 Pumping Lemma – A technique for proving a language L is NOT regular – What does the Pumping Lemma mean? – Proof of Pumping Lemma 2 Pumping Lemma How do we use it? 3 Pumping Condition A language L satisfies the pumping condition if: – there exists an integer n > 0 such that – for all strings x in L of length at least n – there exist strings u, v, w such that x = uvw and • |uv| n and • |v| 1 and For all k 0, uv k w is in L
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2 4 Pumping Lemma All regular languages satisfy the pumping condition All languages over {a,b} Regular languages “Pumping Languages” 5 Implications We can use the pumping lemma to prove a language L is not regular – How? We cannot use the pumping lemma to prove a language is regular – How might we try to use the pumping lemma to prove that a language L is regular and why does it fail? Regular Pumping 6 Pumping Lemma What does it mean?
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3 7 Pumping Condition A language L satisfies the pumping condition if: there exists an integer n > 0 such that for all strings x in L of length at least n there exist strings u, v, w such that x = uvw and • |uv| n and • |v| 1 and For all k 0, uv k w is in L 8 v can be pumped Let x = abcdefg be in L Then there exists a substring v in x such that v can be repeated (pumped) in place any number of times and the resulting string is still in L – u v k w is in L for all k 0 For example v = cde • uv 0 w = uw = abfg is in L • uv 1 w = uvw = ab cde fg is in L • uv 2 w = uvvw = ab cde cde fg is in L • uv 3 w = uvvvw = ab cde cde cde fg is in L • … 1) x in L 2) x = u v w 3) For all k 0, u v k w is in L 9 What the other parts mean A language L satisfies the pumping condition if: there exists an integer n > 0 such that defer what n is till later for all strings x in L of length at least n x must be in L and have sufficient length there exist strings u, v, w such that x = uvw and • |uv| n and v occurs in the first n characters of x • |v| 1 and v is not λ For all k 0, uv k w is in L
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4 10 Example 1 Let L be the set of even length strings over {a,b} Let x = abaa Let n = 2 What are the possibilities for v? – a baa, a b aa – ab aa Which one satisfies the pumping lemma? 11 Examples 2 * Let L be the set of strings over {a,b} where the number of a’s mod 3 is 1 Let x = abbaaa Let n = 3 What are the possibilities for v? – a bbaaa, a b baaa, ab b aaa – ab baaa, a bb aaa – abb aaa Which ones satisfy the pumping lemma? 12 Pumping Lemma Proof
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5 13 High Level Outline Let L be an arbitrary regular language Let M be an FSA such that L(M) = L – M exists by definition of LFSA and the fact that regular languages and LFSA are identical Show that L satisfies the pumping condition – Use M in this part 14 First step: n+1 prefixes of x Let n be the number of states in M Let x be an arbitrary string in L of length at least n Let x i
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