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Regular Languages. Definitions Review of RE, RL, and Kleen’s Theorem Properties of RL. Definitions Review. RE : 1. , , and a are RE’s. 2. If r 1 and r 2 are regular expressions, so are r 1 + r 2 , r 1 • r 2 , r 1 * , and (r 1 ). 3. Those are the only RE’s.
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Regular Languages Definitions Review of RE, RL, and Kleen’s Theorem Properties of RL
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
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*
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
Theorem 12 • If L1 and L2 are RL’s, then L1 L2 is also RL. • Proof: (outline) L1 L2 = (L1’ + L2’)’
