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Review of Regular Languages: Definitions and Theorems

This comprehensive review covers Regular Languages (RL) defined by Regular Expressions (RE) and explores properties, including Kleen’s Theorem. Topics include RE definitions, Kleen's Theorem, RL properties, and proofs of key theorems. Learn how to define languages with regular expressions and understand the intersections of RLs.

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Review of Regular Languages: Definitions and Theorems

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  1. Regular Languages Definitions Review of RE, RL, and Kleen’s Theorem Properties of RL

  2. Definitions Review • RE : • 1. ,  , and a  are RE’s. • 2. If r1 and r2 are regular expressions, so are r1 + r2 , r1 • r2 , r1* , and (r1). • 3. Those are the only RE’s. • Kleen’s Theorem: L(FA) = L(TG) = L(RE) • RL: A language that can be defined by a RE

  3. Theorem 10 • If L1 and L2 are RL’s, then L1+ L2, L1 L2, and L1* are also RL’s. • Proof: (outline) L1+ L2 : r1 + r2 L1 L2 : r1 • r2 L1* : r1*

  4. Theorem 11 • If L is a RL, then L’ is also a RL. • Proof: (outline) Change all final states --> nonfinal states & change all nonfinal states --> final states

  5. Theorem 12 • If L1 and L2 are RL’s, then L1 L2 is also RL. • Proof: (outline) L1 L2 = (L1’ + L2’)’

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