Regular expressions Regular languages

1. Regular expressionsRegular languages Sipser 1.3 (pages 63-76)

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3. Regular expressions • Definition 1.52: Say that R is a regular expression if R is • a for some a in the alphabet Σ • ε • Ø • (R1∪R2), where R1 and R2 are regular exressions • (R1°R2), where R1 and R2 are regular expressions • (R1*), where R1is a regular expression

4. Regular expressions and NFAs • Theorem 1.54: A language is regular if and only if some regular expression describes it. • Proof (⇐) • If a ∈ Σ, then a is regular. • ε is regular. • Ø is regular. • If R1 and R2 are regular, then (R1∪R2) is regular. • If R1 and R2 are regular, then (R1°R2) is regular. • If R1 is a regular, then (R1*) is regular

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6. Going forward • Theorem 1.54: A language is regular if and only if some regular expression describes it. • (⇒) 3-state DFA 5-state GNFA 4-state GNFA regular expression 2-state GNFA 3-state GNFA

7. Generalized NFA Transition to every other state, but no transitions in Transitions in from every other state, but no transitions out Exactly one arrow to and from every state (including itself) other than start and accept

8. Proof DFA to GNFA… • Step 1: Add a unique start state with an εjump to the original one • Step 2: Add a unique accept state with εjumps from the previous accept states • Step 3: Convert multiple labels to ∪ • Step 4: Add Ø jumps for any transition that’s missing

9. Induction step: rip a state

10. A simple example