# Lecture 21 - PowerPoint PPT Presentation

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Lecture 21. Regular languages review Several ways to define regular languages Two main types of proofs/algorithms Relative power of two computational models proofs/constructions Closure property proofs/constructions Language class hierarchy Applications of regular languages.

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Lecture 21

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### Lecture 21

• Regular languages review

• Several ways to define regular languages

• Two main types of proofs/algorithms

• Relative power of two computational models proofs/constructions

• Closure property proofs/constructions

• Language class hierarchy

• Applications of regular languages

### Three definitions

• LFSA

• A language L is in LFSA iff there exists an FSA M s.t. L(M) = L

• LNFA

• A language L is in LNFA iff there exists an NFA M s.t. L(M) = L

• Regular languages

• A language L is regular iff there exists a regular expression R s.t. L(R) = L

• Conclusion

• All these language classes are equivalent

• Any language which can be represented using any one of these models can be represented using either of the other two models

### Relative power proofs

• These proofs work between two language classes and two computational models

• The crux of these proofs are algorithms which behave as follows:

• Input: One program from the first computational model

• Output: A program from the second computational model that is equivalent in function to the first program

### Closure property proofs

• These proofs work within a single language class and typically within a single computational model

• The crux of these proofs are algorithms which behave as follows:

• Input: 1 or 2 programs from a given computational model

• Output: A third program from the same computational model that accepts/describes a third language which is a combination of the languages accepted/described by the two input programs

L

L1

L1 intersect L2

L

LNFA

L2

LFSA

LFSA

M1

M3

M

M2

M’

NFA’s

FSA’s

FSA’s

### Comparison

REC

H

?

RE

All languages over alphabet S

H

regular

### Three remaining topics

• Myhill-Nerode Theorem

• Provides technique for proving a language is not regular

• Also represents fundamental understanding of what a regular language is

• Decision problems about regular languages

• Most are solvable in contrast to problems about recursive languages

• Pumping lemma

• Provides technique for proving a language is not regular

### Review Problems

• We will cover one example of converting a regular expression into an NFA

• We will work on a new closure property proof

• regular languages are closed under language reversal