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.

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Lecture 21

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Lecture 21

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


Defining regular languages

Defining regular languages


Three definitions

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


Two types of proofs constructions

Two types of proofs/constructions


Relative power proofs

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

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


Comparison

L

L1

L1 intersect L2

L

LNFA

L2

LFSA

LFSA

M1

M3

M

M2

M’

NFA’s

FSA’s

FSA’s

Comparison


Language class hierarchy

REC

H

?

RE

All languages over alphabet S

H

Language class hierarchy

regular


Three remaining topics

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

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


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