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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
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
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
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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