NFA Closure Properties

1. NFA Closure Properties

2. NFAs also have closure properties • We have given constructions for showing that DFAs are closed under • Complement • Intersection • Difference • Union • We will now establish that NFAs are closed under • Reversal • Kleene star • Concatenation

3. Reversal of L-NFAs • Closure under reversal is easy using L-NFAs. If you take such an automaton for L, you need to make the following changes to transform it into an automaton for LRev: • Reverse all arcs • The old start state becomes the only new final state. • Add a new start state, and an L-arc from it to all old final states.

4. Example • Reverse all arcs • The old start state becomes the only new final state. • Add a new start state, and an L-arc from it to all old final states.

5. Concatentation • L  R = { x  y | x in L and y in R} • To form a new L-NFA that recognizes the concatenation of two other L-NFAs with the same alphabet do the following • Union the states (you might have to rename them) • Add an L transition from each final state of the first to the start state of the second.

6. Formally • Let • L =(QL,A,sL,FL,TL) • R =(QR,A,sR,FR,TR) • L  R = =(QLQR,A,sL,FR,T) Where T s L | sЄFL = SR T s c | sЄQL = TL s c T s c | sЄQR = TR s c