Lecture 3

Proving new closure properties
Method 2
Adapt a machine
Example: If L is regular so is L’ = {a1a3a5..a2n-1 | a1..a2n∊L}
Let M be a DFA for L
Define δ’(p,a) = {δ(p,ab) |
b∊Σ}
I extended δ:Q×Σ* → Q in the
obvious way
Make δ’ : 2 Q → 2 Q using union
∀P⊆Q, δ’(P,a) = ∪p∊Pδ’(p,ab)
F’ is subsets of Q containing
a state of F
M’ = (Q,Σ,δ’,{q0},F’) is an NFA for L’
Proof ?
Prove by induction on n that
δ’(P,a1..an) = ∪q∊P{δ(q,a1b1a2b2..bn-1anbn) | bi∊Σ}
Clearly if n=0, δ’(P,ε)={δ(q,ε) | q∊P}=P
Assume ind.hyp. true ∀i