Lecture 3

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Proving new closure properties Method 1 Use already derived closure properties Example: Closure under intersection We know 1) Reg langs closed under ∪ and complement 2) De Morgan’s laws If L1 and L2 are regular, so are C(L1) and C(L2) ..... closure under complement C(L1) ∪ C(L2) is regular ...... closure under ∪ L1∩L2 = C(C(L1) ∪ C(L2)) ..... de Morgan and is regular by closure under complement Friday, February 8, 13
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
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Lecture 3

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