Applications of Regular Closure

The intersection of a context-free language and a regular language is a context-free language Regular Closure context free regular context-free

An Application of Regular Closure Prove that: is context-free

We know: is context-free

We also know: is regular is regular

context-free regular (regular closure) is context-free is context-free

Another Application of Regular Closure Prove that: is not context-free

Impossible!!! If is context-free Then context-free regular (regular closure) Then context-free regular context-free Impossible!!! Therefore, is not context free

Decidable Properties of Context-Free Languages

Membership Question: for context-free grammar find if string Membership Algorithms: Parsers Exhaustive search parser CYK parsing algorithm

Empty Language Question: for context-free grammar find if Algorithm: Remove useless variables Check if start variable is useless

Infinite Language Question: for context-free grammar find if is infinite Algorithm: 1. Remove useless variables 2. Remove unit and productions 3. Create dependency graph for variables 4. If there is a loop in the dependency graph then the language is infinite

Example: Infinite language Dependency graph

The Pumping Lemma for Context-Free Languages

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

Variables are repeated A derivation:

Derivation tree string

Derivation tree string repeated

Repeated Part

Another possible derivation

Therefore, the string is also generated by the grammar

We know: We also know this string is generated:

We know: Therefore, this string is also generated:

We know: Therefore, this string is also generated:

We know: Therefore, this string is also generated:

Therefore, knowing that is generated by grammar , we also know that is generated by

In general: We are given an infinite context-free grammar Assume has no unit-productions no -productions

> Take a string with length bigger than (Number of productions) x (Largest right side of a production) Consequence: Some variable must be repeated in the derivation of

String Last repeated variable repeated

Possible derivations:

We know: This string is also generated:

We know: This string is also generated: The original

We know: This string is also generated:

We know: This string is also generated:

We know: This string is also generated:

Therefore, any string of the form is generated by the grammar

Therefore, knowing that we also know that

Observation: Since is the last repeated variable

Observation: Since there are no unit or productions

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:

Applications of The Pumping Lemma

Non-context free languages

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

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

Pumping Lemma gives a magic number such that: Pick any string with length We pick:

We can write: with lengths and

Pumping Lemma says: for all

We examine all the possible locations of string in

Case 1: is within

Case 1: and consist from only

Case 1: Repeating and

Case 1: From Pumping Lemma:

Case 1: From Pumping Lemma: However: Contradiction!!!

Case 2: is within

Case 2: Similar analysis with case 1

Case 3: is within

Case 3: Similar analysis with case 1

Case 4: overlaps and

Case 4: Possibility 1: contains only contains only

Case 4: Possibility 1: contains only contains only

Case 4: From Pumping Lemma:

Case 4: From Pumping Lemma: However: Contradiction!!!

Case 4: Possibility 2: contains and contains only

Case 4: Possibility 2: contains and contains only

Case 4: From Pumping Lemma:

Case 4: From Pumping Lemma: However: Contradiction!!!

Case 4: Possibility 3: contains only contains and

Case 4: Possibility 3: contains only contains and Similar analysis with Possibility 2

Case 5: overlaps and

Case 5: Similar analysis with case 4

There are no other cases to consider (since , string cannot overlap , and at the same time)

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