Lecture 21

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# Lecture 21 - PowerPoint PPT Presentation

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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Presentation Transcript
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

Language class hierarchy

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