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Lecture 21

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- 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

- 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

- 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

- 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

REC

H

?

RE

All languages over alphabet S

H

regular

- 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

- 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