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Reverse of a Regular LanguagePowerPoint Presentation

Reverse of a Regular Language

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## PowerPoint Slideshow about ' Reverse of a Regular Language' - hollie

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### Regular Grammars GenerateRegular Languages

Construct NFA that accepts :

invert the transitions of the NFA

that accepts

Theorem:

The reverse of a regular language

is a regular language

is regular

Grammars

- Grammars express languages
- Example: the English language

- A derivation of “the boy walks”:

- A derivation of “a dog runs”:

L = { “a boy runs”,

“a boy walks”,

“the boy runs”,

“the boy walks”,

“a dog runs”,

“a dog walks”,

“the dog runs”,

“the dog walks” }

Another Example

- Grammar:
- Derivation of sentence :

- Grammar:
- Derivation of sentence :

More Notation

- Grammar

Set of variables

Set of terminal symbols

Start variable

Set of Production rules

Example

- Grammar :

More Notation

- Sentential Form:
- A sentence that contains
- variables and terminals
- Example:

Sentential Forms

sentence

- We write:
- Instead of:

Linear Grammars

- Grammars with
- at most one variable at the right side
- of a production
- Examples:

A Non-Linear Grammar

Grammar :

Another Linear Grammar

- Grammar :

Regular Grammars

- A regular grammaris any
- right-linear or left-linear grammar
- Examples:

Observation

- Regular grammars generate regular languages
- Examples:

Theorem - Part 1

Languages

Generated by

Regular Grammars

Regular

Languages

Any regular grammar generates

a regular language

Theorem - Part 2

Languages

Generated by

Regular Grammars

Regular

Languages

Any regular language is generated

by a regular grammar

Proof – Part 1

Languages

Generated by

Regular Grammars

Regular

Languages

The language generated by

any regular grammar is regular

The case of Right-Linear Grammars

- Let be a right-linear grammar
- We will prove: is regular
- Proof idea: We will construct NFA
- with

Example:

Grammar

- For each production:
- we add transitions and intermediate nodes

………

- For each production:
- we add transitions and intermediate nodes

………

It holds that:

The case of Left-Linear Grammars

- Let be a left-linear grammar
- We will prove: is regular
- Proof idea:
- We will construct a right-linear
- grammar with

the productions look like:

- It is easy to see that:
- Since is right-linear, we have:

Regular

Language

Regular

Language

Regular

Language

Proof - Part 2

Languages

Generated by

Regular Grammars

Regular

Languages

Any regular language is generated

by some regular grammar

Any regular language is generated

by some regular grammar

Proof idea:

Let be the NFA with .

Construct from a regular grammar

such that

- Since is regular
- there is an NFA such that

Example:

Add production:

- Since is right-linear grammar
- is also a regular grammar
- with

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