prof. busch - lsu1 pumping lemma for context-free languages

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  • Slide 1
  • Prof. Busch - LSU1 Pumping Lemma for Context-free Languages
  • Slide 2
  • Prof. Busch - LSU2 Take an infinite context-free language Example: Generates an infinite number of different strings
  • Slide 3
  • Prof. Busch - LSU3 A possible derivation: In a derivation of a long enough string, variables are repeated
  • Slide 4
  • Prof. Busch - LSU4 Derivation Tree Repeated variable
  • Slide 5
  • Prof. Busch - LSU5 Derivation Tree Repeated variable
  • Slide 6
  • Prof. Busch - LSU6
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  • 9 Putting all together
  • Slide 10
  • Prof. Busch - LSU10 We can remove the middle part
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  • Prof. Busch - LSU11
  • Slide 12
  • Prof. Busch - LSU12 We can repeated middle part two times 1 2
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  • Prof. Busch - LSU14 We can repeat middle part three times 1 2 3
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  • Prof. Busch - LSU16 Repeat middle part times i 1
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  • Prof. Busch - LSU17 For any
  • Slide 18
  • Prof. Busch - LSU18 From Grammar and given string We inferred that a family of strings is in for any
  • Slide 19
  • Prof. Busch - LSU19 Arbitrary Grammars Consider now an arbitrary infinite context-free language Take so that it has no unit-productions and no -productions Let be the grammar of (remove them)
  • Slide 20
  • Prof. Busch - LSU20 Let be the number of variables Let be the maximum right-hand size of any production Example:
  • Slide 21
  • Prof. Busch - LSU21 Take string with. Then in the derivation tree of there is a path from the root to a leaf where a variable of is repeated Claim: Proof: Proof by contradiction
  • Slide 22
  • Prof. Busch - LSU22 Derivation tree of some variable is repeated We will show:
  • Slide 23
  • Prof. Busch - LSU23 First we show that the tree of has at least levels of nodes Suppose the opposite: Levels At most
  • Slide 24
  • Prof. Busch - LSU24 Level 0: Level 1: nodes Maximum number of nodes per level nodes The maximum right-hand side of any production
  • Slide 25
  • Prof. Busch - LSU25 Level 0: Level 1: nodes Level 2: Maximum number of nodes per level
  • Slide 26
  • Prof. Busch - LSU26 nodesLevel : Levels nodesLevel : Maximum possible string length = max nodes at level = Maximum number of nodes per level Level 0:nodes At most
  • Slide 27
  • Prof. Busch - LSU27 Therefore, maximum length of string : However we took, Contradiction!!! Therefore, the tree must have at least levels
  • Slide 28
  • Prof. Busch - LSU28 Thus, there is a path from the root to a leaf with at least nodes Levels At least symbol Variables (root) (leaf)
  • Slide 29
  • Prof. Busch - LSU29 Since there are at most different variables, some variable is repeated Pigeonhole principle END OF CLAIM PROOF
  • Slide 30
  • Prof. Busch - LSU30 Take now a stringwith some variable is repeated Take to be the deepest, so that only is repeated in subtree subtree of From claim:
  • Slide 31
  • Prof. Busch - LSU31 Strings of terminals yield We can write
  • Slide 32
  • Prof. Busch - LSU32 Example:
  • Slide 33
  • Prof. Busch - LSU33 Possible derivations
  • Slide 34
  • Prof. Busch - LSU34 Example:
  • Slide 35
  • Prof. Busch - LSU35 Remove Middle Part Yield:
  • Slide 36
  • Prof. Busch - LSU36 Repeat Middle part two times 1 2 Yield:
  • Slide 37
  • Prof. Busch - LSU37
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  • Prof. Busch - LSU38 Repeat Middle part times 1 i Yield:
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  • Prof. Busch - LSU39
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  • Prof. Busch - LSU40 If we know that: then we also know: Therefore, For all since
  • Slide 41
  • Prof. Busch - LSU41 Observation 1: Since has no unit and -productions At least one of or is not
  • Slide 42
  • Prof. Busch - LSU42 Observation 2: since in subtree only variable is repeated subtree of Explanation follows.
  • Slide 43
  • Prof. Busch - LSU43 subtree of Various yields since no variable is repeated in Maximum right-hand side of any production
  • Slide 44
  • Prof. Busch - LSU44 Thus, if we choose critical length then, we obtain the pumping lemma for context-free languages
  • Slide 45
  • Prof. Busch - LSU45 The Pumping Lemma: there exists an integer such that for any string we can write For any infinite context-free language with lengths and it must be that:
  • Slide 46
  • Prof. Busch - LSU46 Applications of The Pumping Lemma
  • Slide 47
  • Prof. Busch - LSU47 Context-free languages Non-context free languages
  • Slide 48
  • Prof. Busch - LSU48 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages
  • Slide 49
  • Prof. Busch - LSU49 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma
  • Slide 50
  • Prof. Busch - LSU50 Let be the critical length of the pumping lemma Pick any string with length We pick:
  • Slide 51
  • Prof. Busch - LSU51 we can write: with lengths and From pumping lemma:
  • Slide 52
  • Prof. Busch - LSU52 Pumping Lemma says: for all
  • Slide 53
  • Prof. Busch - LSU53 We examine all the possible locations of string in
  • Slide 54
  • Prof. Busch - LSU54 Case 1: is in
  • Slide 55
  • Prof. Busch - LSU55
  • Slide 56
  • Prof. Busch - LSU56
  • Slide 57
  • Prof. Busch - LSU57 From Pumping Lemma: However: Contradiction!!!
  • Slide 58
  • Prof. Busch - LSU58 Case 2: is in Similar to case 1
  • Slide 59
  • Prof. Busch - LSU59 Case 3: is in Similar to case 1
  • Slide 60
  • Prof. Busch - LSU60 Case 4: overlaps and
  • Slide 61
  • Prof. Busch - LSU61 Sub-case 1:contains only
  • Slide 62
  • Prof. Busch - LSU62
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  • Slide 64
  • Prof. Busch - LSU64 From Pumping Lemma: However: Contradiction!!!
  • Slide 65
  • Prof. Busch - LSU65 Sub-case 2:contains and contains only
  • Slide 66
  • Prof. Busch - LSU66 By assumption
  • Slide 67
  • Prof. Busch - LSU67
  • Slide 68
  • Prof. Busch - LSU68 From Pumping Lemma: However: Contradiction!!!
  • Slide 69
  • Prof. Busch - LSU69 Sub-case 3:contains only contains and Similar to sub-case 2
  • Slide 70
  • Prof. Busch - LSU70 Case 5: overlaps and Similar to case 4
  • Slide 71
  • Prof. Busch - LSU71 Case 6: overlaps, and Impossible!
  • Slide 72
  • Prof. Busch - LSU72 In all cases we obtained a contradiction Therefore: the original assumption that is context-free must be wrong Conclusion:is not context-free