Prof. Busch - LSU1 Pumping Lemma for Context-free Languages

Prof. Busch - LSU2 Take an infinite context-free language Example: Generates an infinite number of different strings

Prof. Busch - LSU3 A possible derivation: In a derivation of a “long” enough string, variables are repeated

Prof. Busch - LSU4 Derivation Tree Repeated variable

Prof. Busch - LSU5 Derivation Tree Repeated variable

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7

8

9 Putting all together

Prof. Busch - LSU10 We can remove the middle part

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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

Prof. Busch - LSU17 For any

Prof. Busch - LSU18 From Grammar and given string We inferred that a family of strings is in for any

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)

Prof. Busch - LSU20 Let be the number of variables Let be the maximum right-hand size of any production Example:

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

Prof. Busch - LSU22 Derivation tree of some variable is repeated We will show:

Prof. Busch - LSU23 First we show that the tree of has at least levels of nodes Suppose the opposite: Levels At most

Prof. Busch - LSU24 Level 0: Level 1: nodes Maximum number of nodes per level nodes The maximum right-hand side of any production

Prof. Busch - LSU25 Level 0: Level 1: nodes Level 2: Maximum number of nodes per level

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

Prof. Busch - LSU27 Therefore, maximum length of string : However we took, Contradiction!!! Therefore, the tree must have at least levels

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)

Prof. Busch - LSU29 Since there are at most different variables, some variable is repeated Pigeonhole principle END OF CLAIM PROOF

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:

Prof. Busch - LSU31 Strings of terminals yield We can write

Prof. Busch - LSU32 Example:

Prof. Busch - LSU33 Possible derivations

Prof. Busch - LSU34 Example:

Prof. Busch - LSU35 Remove Middle Part Yield:

Prof. Busch - LSU36 Repeat Middle part two times 1 2 Yield:

Prof. Busch - LSU37

Prof. Busch - LSU38 Repeat Middle part times 1 i Yield:

Prof. Busch - LSU39

Prof. Busch - LSU40 If we know that: then we also know: Therefore, For all since

Prof. Busch - LSU41 Observation 1: Since has no unit and -productions At least one of or is not

Prof. Busch - LSU42 Observation 2: since in subtree only variable is repeated subtree of Explanation follows….

Prof. Busch - LSU43 subtree of Various yields since no variable is repeated in Maximum right-hand side of any production

Prof. Busch - LSU44 Thus, if we choose critical length then, we obtain the pumping lemma for context-free languages

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:

Prof. Busch - LSU46 Applications of The Pumping Lemma

Prof. Busch - LSU47 Context-free languages Non-context free languages

Prof. Busch - LSU48 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages

Prof. Busch - LSU49 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma

Prof. Busch - LSU50 Let be the critical length of the pumping lemma Pick any string with length We pick:

Prof. Busch - LSU51 we can write: with lengths and From pumping lemma:

Prof. Busch - LSU52 Pumping Lemma says: for all

Prof. Busch - LSU53 We examine all the possible locations of string in

Prof. Busch - LSU54 Case 1: is in

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Prof. Busch - LSU56

Prof. Busch - LSU57 From Pumping Lemma: However: Contradiction!!!

Prof. Busch - LSU58 Case 2: is in Similar to case 1

Prof. Busch - LSU59 Case 3: is in Similar to case 1

Prof. Busch - LSU60 Case 4: overlaps and

Prof. Busch - LSU61 Sub-case 1:contains only

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Prof. Busch - LSU63

Prof. Busch - LSU64 From Pumping Lemma: However: Contradiction!!!

Prof. Busch - LSU65 Sub-case 2:contains and contains only

Prof. Busch - LSU66 By assumption

Prof. Busch - LSU67

Prof. Busch - LSU68 From Pumping Lemma: However: Contradiction!!!

Prof. Busch - LSU69 Sub-case 3:contains only contains and Similar to sub-case 2

Prof. Busch - LSU70 Case 5: overlaps and Similar to case 4

Prof. Busch - LSU71 Case 6: overlaps, and Impossible!

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