The Pumping Lemma for ContextFree Languages Costas Busch

The Pumping Lemma for Context-Free Languages Costas Busch - RPI 1

Take an infinite context-free language Generates an infinite number of different strings Example: Costas Busch - RPI 2

In a derivation of a long string, variables are repeated A derivation: Costas Busch - RPI 3

Derivation tree string Costas Busch - RPI 4

Derivation tree string repeated Costas Busch - RPI 5

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Repeated Part Costas Busch - RPI 7

Another possible derivation from Costas Busch - RPI 8

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A Derivation from Costas Busch - RPI 10

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A Derivation from Costas Busch - RPI 13

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A Derivation from Costas Busch - RPI 18

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In General: Costas Busch - RPI 22

Consider now an infinite context-free language Let Take be the grammar of so that I has no unit-productions no -productions Costas Busch - RPI 23

Let = (Number of productions) x (Largest right side of a production) Let Example : Costas Busch - RPI 24

Take a string with length We will show: in the derivation of a variable of is repeated Costas Busch - RPI 25

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maximum right hand side of any production Costas Busch - RPI 27

Number of productions in grammar Costas Busch - RPI 28

Number of productions in grammar Some production must be repeated Repeated variable Costas Busch - RPI 29

Derivation of string Some variable is repeated Costas Busch - RPI 30

Derivation tree of string Last repeated variable repeated Strings of terminals Costas Busch - RPI 31

Possible derivations: Costas Busch - RPI 32

We know: This string is also generated: Costas Busch - RPI 33

We know: This string is also generated: The original Costas Busch - RPI 34

We know: This string is also generated: Costas Busch - RPI 35

We know: This string is also generated: Costas Busch - RPI 36

We know: This string is also generated: Costas Busch - RPI 37

Therefore, any string of the form is generated by the grammar Costas Busch - RPI 38

Therefore, knowing that we also know that Costas Busch - RPI 39

Observation: Since is the last repeated variable Costas Busch - RPI 40

Observation: Since there are no unit or Costas Busch - RPI -productions 41

The Pumping Lemma: For infinite context-free language there exists an integer such that for any string we can write with lengths and it must be: Costas Busch - RPI 42

Applications of The Pumping Lemma Costas Busch - RPI 43

Non-context free languages Context-free languages Costas Busch - RPI 44

Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages Costas Busch - RPI 45

Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma Costas Busch - RPI 46

Pumping Lemma gives a magic number such that: Pick any string with length We pick: Costas Busch - RPI 47

We can write: with lengths and Costas Busch - RPI 48

Pumping Lemma says: for all Costas Busch - RPI 49

We examine all the possible locations of string in Costas Busch - RPI 50

Case 1: is within Costas Busch - RPI 51

Case 1: and consist from only Costas Busch - RPI 52

Case 1: Repeating and Costas Busch - RPI 53

Case 1: From Pumping Lemma: Costas Busch - RPI 54

Case 1: From Pumping Lemma: However: Contradiction!!! Costas Busch - RPI 55

Case 2: is within Costas Busch - RPI 56

Case 2: Similar analysis with case 1 Costas Busch - RPI 57

Case 3: is within Costas Busch - RPI 58

Case 3: Similar analysis with case 1 Costas Busch - RPI 59

Case 4: overlaps Costas Busch - RPI and 60

Case 4: Possibility 1: contains only Costas Busch - RPI 61

Case 4: Possibility 1: contains only Costas Busch - RPI 62

Case 4: From Pumping Lemma: Costas Busch - RPI 63

Case 4: From Pumping Lemma: However: Contradiction!!! Costas Busch - RPI 64

Case 4: Possibility 2: contains and contains only Costas Busch - RPI 65

Case 4: Possibility 2: contains and contains only Costas Busch - RPI 66

Case 4: From Pumping Lemma: Costas Busch - RPI 67

Case 4: From Pumping Lemma: However: Contradiction!!! Costas Busch - RPI 68

Case 4: Possibility 3: contains only contains and Costas Busch - RPI 69

Case 4: Possibility 3: contains only contains and Similar analysis with Possibility 2 Costas Busch - RPI 70

Case 5: overlaps Costas Busch - RPI and 71

Case 5: Similar analysis with case 4 Costas Busch - RPI 72

There are no other cases to consider (since overlap , string , and Costas Busch - RPI cannot at the same time) 73

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 - RPI 74