# costas busch - rpi1 the pumping lemma for context-free languages

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• Slide 1
• Costas Busch - RPI1 The Pumping Lemma for Context-Free Languages
• Slide 2
• Costas Busch - RPI2 Take an infinite context-free language Example: Generates an infinite number of different strings
• Slide 3
• Costas Busch - RPI3 A derivation: In a derivation of a long string, variables are repeated
• Slide 4
• Costas Busch - RPI4 Derivation treestring
• Slide 5
• Costas Busch - RPI5 repeated Derivation treestring
• Slide 6
• Costas Busch - RPI6
• Slide 7
• 7 Repeated Part
• Slide 8
• Costas Busch - RPI8 Another possible derivation from
• Slide 9
• Costas Busch - RPI9
• Slide 10
• 10 A Derivation from
• Slide 11
• Costas Busch - RPI11
• Slide 12
• Costas Busch - RPI12
• Slide 13
• Costas Busch - RPI13 A Derivation from
• Slide 14
• Costas Busch - RPI14
• Slide 15
• Costas Busch - RPI15
• Slide 16
• Costas Busch - RPI16
• Slide 17
• Costas Busch - RPI17
• Slide 18
• Costas Busch - RPI18 A Derivation from
• Slide 19
• Costas Busch - RPI19
• Slide 20
• Costas Busch - RPI20
• Slide 21
• Costas Busch - RPI21
• Slide 22
• Costas Busch - RPI22 In General:
• Slide 23
• Costas Busch - RPI23 Consider now an infinite context-free language Take so that I has no unit-productions no -productions Let be the grammar of
• Slide 24
• Costas Busch - RPI24 (Number of productions) x (Largest right side of a production) = Let Example : Let
• Slide 25
• Costas Busch - RPI25 Take a string with length We will show: in the derivation of a variable of is repeated
• Slide 26
• Costas Busch - RPI26
• Slide 27
• Costas Busch - RPI27 maximum right hand side of any production
• Slide 28
• Costas Busch - RPI28 Number of productions in grammar
• Slide 29
• Costas Busch - RPI29 Number of productions in grammar Some production must be repeated Repeated variable
• Slide 30
• Costas Busch - RPI30 Some variable is repeated Derivation of string
• Slide 31
• Costas Busch - RPI31 Last repeated variable repeated Strings of terminals Derivation tree of string
• Slide 32
• Costas Busch - RPI32 Possible derivations:
• Slide 33
• Costas Busch - RPI33 We know: This string is also generated:
• Slide 34
• Costas Busch - RPI34 This string is also generated: The original We know:
• Slide 35
• Costas Busch - RPI35 This string is also generated: We know:
• Slide 36
• Costas Busch - RPI36 This string is also generated: We know:
• Slide 37
• Costas Busch - RPI37 This string is also generated: We know:
• Slide 38
• Costas Busch - RPI38 Therefore, any string of the form is generated by the grammar
• Slide 39
• Costas Busch - RPI39 knowing that we also know that Therefore,
• Slide 40
• Costas Busch - RPI40 Observation: Since is the last repeated variable
• Slide 41
• Costas Busch - RPI41 Observation: Since there are no unit or -productions
• Slide 42
• Costas Busch - RPI42 The Pumping Lemma: there exists an integer such that for any string we can write For infinite context-free language with lengths and it must be:
• Slide 43
• Costas Busch - RPI43 Applications of The Pumping Lemma
• Slide 44
• Costas Busch - RPI44 Context-free languages Non-context free languages
• Slide 45
• Costas Busch - RPI45 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages
• Slide 46
• Costas Busch - RPI46 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma
• Slide 47
• Costas Busch - RPI47 Pumping Lemma gives a magic number such that: Pick any string with length We pick:
• Slide 48
• Costas Busch - RPI48 We can write: with lengths and
• Slide 49
• Costas Busch - RPI49 Pumping Lemma says: for all
• Slide 50
• Costas Busch - RPI50 We examine all the possible locations of string in
• Slide 51
• Costas Busch - RPI51 Case 1: is within
• Slide 52
• Costas Busch - RPI52 Case 1: and consist from only
• Slide 53
• Costas Busch - RPI53 Case 1: Repeating and
• Slide 54
• Costas Busch - RPI54 Case 1: From Pumping Lemma:
• Slide 55
• Costas Busch - RPI55 Case 1: From Pumping Lemma: However: Contradiction!!!
• Slide 56
• Costas Busch - RPI56 Case 2: is within
• Slide 57
• Costas Busch - RPI57 Case 2: Similar analysis with case 1
• Slide 58
• Costas Busch - RPI58 Case 3: is within
• Slide 59
• Costas Busch - RPI59 Case 3: Similar analysis with case 1
• Slide 60
• Costas Busch - RPI60 Case 4: overlaps and
• Slide 61
• Costas Busch - RPI61 Case 4: Possibility 1:contains only
• Slide 62
• Costas Busch - RPI62 Case 4: Possibility 1:contains only
• Slide 63
• Costas Busch - RPI63 Case 4: From Pumping Lemma:
• Slide 64
• Costas Busch - RPI64 Case 4: From Pumping Lemma: However: Contradiction!!!
• Slide 65
• Costas Busch - RPI65 Case 4: Possibility 2:contains and contains only
• Slide 66
• Costas Busch - RPI66 Case 4: Possibility 2:contains and contains only
• Slide 67
• Costas Busch - RPI67 Case 4: From Pumping Lemma:
• Slide 68
• Costas Busch - RPI68 Case 4: From Pumping Lemma: However: Contradiction!!!
• Slide 69
• Costas Busch - RPI69 Case 4: Possibility 3:contains only contains and
• Slide 70
• Costas Busch - RPI70 Case 4: Possibility 3:contains only contains and Similar analysis with Possibility 2
• Slide 71
• Costas Busch - RPI71 Case 5: overlaps and
• Slide 72
• Costas Busch - RPI72 Case 5: Similar analysis with case 4
• Slide 73
• Costas Busch - RPI73 There are no other cases to consider (since, string cannot overlap, and at the same time)
• Slide 74
• Costas Busch - RPI74 In all cases we obtained a contradiction Therefore: The original assumption that is context-free must be wrong Conclusion:is not context-free

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