Pumping Lemma for Context-free Languages Costas Busch - LSU

Take an infinite context-free language Generates an infinite number of different strings Example: Costas Busch - LSU

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

Derivation Tree Repeated variable Costas Busch - LSU

Derivation Tree Repeated variable Costas Busch - LSU

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Costas Busch - LSU

Costas Busch - LSU

Putting all together Costas Busch - LSU

We can remove the middle part Costas Busch - LSU

Costas Busch - LSU

We can repeated middle part two times 1 2 Costas Busch - LSU

Costas Busch - LSU

We can repeat middle part three times 1 2 3 Costas Busch - LSU

Costas Busch - LSU

Repeat middle part times 1 i Costas Busch - LSU

For any Costas Busch - LSU

We inferred that a family of strings is in From Grammar and given string We inferred that a family of strings is in for any Costas Busch - LSU

Arbitrary Grammars Consider now an arbitrary infinite context-free language Let be the grammar of Take so that it has no unit-productions and no -productions (remove them) Costas Busch - LSU

Let be the number of variables Let be the maximum right-hand size of any production Example: Costas Busch - LSU

Claim: Proof: 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 Proof: Proof by contradiction Costas Busch - LSU

Derivation tree of We will show: some variable is repeated Costas Busch - LSU

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

Maximum number of nodes per level The maximum right-hand side of any production Costas Busch - LSU

Maximum number of nodes per level Costas Busch - LSU

Maximum number of nodes per level At most Level : nodes Levels Level : nodes Maximum possible string length = max nodes at level = Costas Busch - LSU

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

Thus, there is a path from the root to a leaf with at least nodes Variables Levels symbol (leaf) Costas Busch - LSU

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

Take to be the deepest, so that only is repeated in subtree Take now a string with From claim: some variable is repeated subtree of Take to be the deepest, so that only is repeated in subtree Costas Busch - LSU

We can write Strings of terminals yield yield yield yield yield Costas Busch - LSU

Example: Costas Busch - LSU

Possible derivations Costas Busch - LSU

Example: Costas Busch - LSU

Remove Middle Part Yield: Costas Busch - LSU

Repeat Middle part two times 1 2 Yield: Costas Busch - LSU

Costas Busch - LSU

Repeat Middle part times 1 i Yield: Costas Busch - LSU

Costas Busch - LSU

Therefore, If we know that: then we also know: For all since Costas Busch - LSU

Observation 1: At least one of or is not Since has no unit and -productions At least one of or is not Costas Busch - LSU

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

since no variable is repeated in subtree of Various yields since no variable is repeated in Maximum right-hand side of any production Costas Busch - LSU

Thus, if we choose critical length then, we obtain the pumping lemma for context-free languages Costas Busch - LSU

For any infinite context-free language The Pumping Lemma: For any infinite context-free language there exists an integer such that for any string we can write with lengths and it must be that: Costas Busch - LSU

Applications of The Pumping Lemma Costas Busch - LSU

Non-context free languages Costas Busch - LSU

for context-free languages Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages Costas Busch - LSU

Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma Costas Busch - LSU

Let be the critical length of the pumping lemma Pick any string with length We pick: Costas Busch - LSU

From pumping lemma: we can write: with lengths and Costas Busch - LSU

Pumping Lemma says: for all Costas Busch - LSU

We examine all the possible locations of string in Costas Busch - LSU

Case 1: is in Costas Busch - LSU

Costas Busch - LSU

Costas Busch - LSU

From Pumping Lemma: However: Contradiction!!! Costas Busch - LSU

Case 2: is in Similar to case 1 Costas Busch - LSU

Case 3: is in Similar to case 1 Costas Busch - LSU

Case 4: overlaps and Costas Busch - LSU

Sub-case 1: contains only contains only Costas Busch - LSU

Costas Busch - LSU

Costas Busch - LSU

From Pumping Lemma: However: Contradiction!!! Costas Busch - LSU

Sub-case 2: contains and contains only Costas Busch - LSU

By assumption Costas Busch - LSU

Costas Busch - LSU

From Pumping Lemma: However: Contradiction!!! Costas Busch - LSU

Sub-case 3: contains only contains and Similar to sub-case 2 Costas Busch - LSU

Case 5: overlaps and Similar to case 4 Costas Busch - LSU

Case 6: overlaps , and Impossible! Costas Busch - LSU

In all cases we obtained a contradiction Therefore: the original assumption that is context-free must be wrong Conclusion: is not context-free Costas Busch - LSU