# 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
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- Prof. Busch - LSU5 Derivation Tree Repeated variable
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- Prof. Busch - LSU6
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- Slide 9
- 9 Putting all together
- Slide 10
- Prof. Busch - LSU10 We can remove the middle part
- Slide 11
- Prof. Busch - LSU11
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- Prof. Busch - LSU12 We can repeated middle part two times 1 2
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- Prof. Busch - LSU13
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- Prof. Busch - LSU14 We can repeat middle part three times 1 2 3
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- Prof. Busch - LSU15
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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
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- 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
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- 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:
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- Prof. Busch - LSU33 Possible derivations
- Slide 34
- Prof. Busch - LSU34 Example:
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- Prof. Busch - LSU35 Remove Middle Part Yield:
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- Prof. Busch - LSU36 Repeat Middle part two times 1 2 Yield:
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- 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
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- Prof. Busch - LSU49 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma
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- Prof. Busch - LSU50 Let be the critical length of the pumping lemma Pick any string with length We pick:
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- Prof. Busch - LSU51 we can write: with lengths and From pumping lemma:
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- Prof. Busch - LSU52 Pumping Lemma says: for all
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- Prof. Busch - LSU53 We examine all the possible locations of string in
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- Prof. Busch - LSU54 Case 1: is in
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- Prof. Busch - LSU55
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- Prof. Busch - LSU56
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- Prof. Busch - LSU57 From Pumping Lemma: However: Contradiction!!!
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- Prof. Busch - LSU58 Case 2: is in Similar to case 1
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- Prof. Busch - LSU59 Case 3: is in Similar to case 1
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- Prof. Busch - LSU60 Case 4: overlaps and
- Slide 61
- Prof. Busch - LSU61 Sub-case 1:contains only
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- Prof. Busch - LSU62
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- Prof. Busch - LSU63
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- 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
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- Prof. Busch - LSU67
- Slide 68
- Prof. Busch - LSU68 From Pumping Lemma: However: Contradiction!!!
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- Prof. Busch - LSU69 Sub-case 3:contains only contains and Similar to sub-case 2
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- Prof. Busch - LSU70 Case 5: overlaps and Similar to case 4
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- Prof. Busch - LSU71 Case 6: overlaps, and Impossible!
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- 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