costas busch - lsu1 pumping lemma for context-free languages

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Costas Busch - LSU3 A possible derivation: In a derivation of a long enough string, variables are repeated

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Costas Busch - LSU1 Pumping Lemma for Context-free Languages Costas Busch - LSU2 Take an infinite context-free language Example: Generates an infinite number of different strings Costas Busch - LSU3 A possible derivation: In a derivation of a long enough string, variables are repeated Costas Busch - LSU4 Derivation Tree Repeated variable Costas Busch - LSU5 Derivation Tree Repeated variable Costas Busch - LSU6 7 8 9 Putting all together Costas Busch - LSU10 We can remove the middle part Costas Busch - LSU11 Costas Busch - LSU12 We can repeated middle part two times 1 2 Costas Busch - LSU13 Costas Busch - LSU14 We can repeat middle part three times 1 2 3 Costas Busch - LSU15 Costas Busch - LSU16 Repeat middle part times i 1 Costas Busch - LSU17 For any Costas Busch - LSU18 From Grammar and given string We inferred that a family of strings is in for any Costas 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) Costas Busch - LSU20 Let be the number of variables Let be the maximum right-hand size of any production Example: Costas 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 Costas Busch - LSU22 Derivation tree of some variable is repeated We will show: Costas Busch - LSU23 First we show that the tree of has at least levels of nodes Suppose the opposite: Levels At most Costas Busch - LSU24 Level 0: Level 1: nodes Maximum number of nodes per level nodes The maximum right-hand side of any production Costas Busch - LSU25 Level 0: Level 1: nodes Level 2: Maximum number of nodes per level Costas Busch - LSU26 nodesLevel : Levels nodesLevel : Maximum possible string length = max nodes at level = Maximum number of nodes per level Level 0:nodes At most Costas Busch - LSU27 Therefore, maximum length of string : However we took, Contradiction!!! Therefore, the tree must have at least levels Costas Busch - LSU28 Thus, there is a path from the root to a leaf with at least nodes Levels At least symbol Variables (root) (leaf) Costas Busch - LSU29 Since there are at most different variables, some variable is repeated Pigeonhole principle END OF CLAIM PROOF Costas 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: Costas Busch - LSU31 Strings of terminals yield We can write Costas Busch - LSU32 Example: Costas Busch - LSU33 Possible derivations Costas Busch - LSU34 Example: Costas Busch - LSU35 Remove Middle Part Yield: Costas Busch - LSU36 Repeat Middle part two times 1 2 Yield: Costas Busch - LSU37 Costas Busch - LSU38 Repeat Middle part times 1 i Yield: Costas Busch - LSU39 Costas Busch - LSU40 If we know that: then we also know: Therefore, For all since Costas Busch - LSU41 Observation 1: Since has no unit and -productions At least one of or is not Costas Busch - LSU42 Observation 2: since in subtree only variable is repeated subtree of Explanation follows. Costas Busch - LSU43 subtree of Various yields since no variable is repeated in Maximum right-hand side of any production Costas Busch - LSU44 Thus, if we choose critical length then, we obtain the pumping lemma for context-free languages Costas 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: Costas Busch - LSU46 Applications of The Pumping Lemma Costas Busch - LSU47 Context-free languages Non-context free languages Costas Busch - LSU48 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages Costas Busch - LSU49 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma Costas Busch - LSU50 Let be the critical length of the pumping lemma Pick any string with length We pick: Costas Busch - LSU51 we can write: with lengths and From pumping lemma: Costas Busch - LSU52 Pumping Lemma says: for all Costas Busch - LSU53 We examine all the possible locations of string in Costas Busch - LSU54 Case 1: is in Costas Busch - LSU55 Costas Busch - LSU56 Costas Busch - LSU57 From Pumping Lemma: However: Contradiction!!! Costas Busch - LSU58 Case 2: is in Similar to case 1 Costas Busch - LSU59 Case 3: is in Similar to case 1 Costas Busch - LSU60 Case 4: overlaps and Costas Busch - LSU61 Sub-case 1:contains only Costas Busch - LSU62 Costas Busch - LSU63 Costas Busch - LSU64 From Pumping Lemma: However: Contradiction!!! Costas Busch - LSU65 Sub-case 2:contains and contains only Costas Busch - LSU66 By assumption Costas Busch - LSU67 Costas Busch - LSU68 From Pumping Lemma: However: Contradiction!!! Costas Busch - LSU69 Sub-case 3:contains only contains and Similar to sub-case 2 Costas Busch - LSU70 Case 5: overlaps and Similar to case 4 Costas Busch - LSU71 Case 6: overlaps, and Impossible! Costas 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

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