Lecture 3

More documents

Recommendations

Info

Another example If L is regular, so is Cycle(L) = {a2a1a4a3..a2na2n-1 | a1a2a3a4..a2n-1a2n ∊ L} Let M be a DFA for L. M’ will be a DFA for Cycle(L) that uses a state to “remember” what symbol (a) it just read. Then, when it reads the next symbol b, M’ will simulate M on input ba. Formally Q’ = Q ∪ (Q×Σ) δ’(q,a) = (q,a) δ’((q,a),b) = δ(q,ba) Prove by induction on n that δ’(q,a2a1a4a3..a2na2n-1) = δ(q,a1a2a3a4..a2n-1a2n) Clearly true when n=0 since δ’(q,ε) = q = δ(q,ε) Assume true for n-1, so δ’(q,a2a1a4a3..a2n-2a2n-3) = δ(q,a1a2a3a4..a2n-3a2n-2) = p, say Then δ’(q,a2a1a4a3..a2na2n-1) = δ’(δ’(q,a2a1a4a3..a2n-2a2n-3), a2na2n-1) = δ(p, a2n-1a2n) = δ(δ(q,a1a2a3a4..a2n-3a2n-2), a2n-1a2n) = δ(q,a1a2a3a4..a2n-1a2n) We have proved that a2a1a4a3..a2na2n-1 is accepted by M’ if and only if a1a2a3a4..a2n-1a2n is accepted by M. So Cycle(L) = L(M’) is regular. Friday, February 8, 13
Similar examples Under which of these are reg langs closed? MAX(L) = {x∊L | xy ∊ L ⇒ y=ε} MIN(L) = {x∊L | (x=yz and y∊L) ⇒ z=ε} INIT(L) = {x∊L | ∃y and xy∊L} SQRT(L) = {x | for some y with |y|=|x| 2 xy∊L} Friday, February 8, 13
Page 1 and 2: CSC 341 Automata etc Rhys Price Jon
Page 3 and 4: Proving a Language is Regular Metho
Page 5 and 6: Proving a Language is Regular Metho
Page 7 and 8: I like closure properties!!! How ca
Page 9 and 10: Proving new closure properties Meth
Page 11: Sipser Problem 1.42 A,B regular wit
Page 15 and 16: Quotient of languages If L1 and L2
Page 17 and 18: Proving language no regular In gene
Page 19 and 20: Theorem (Pumping Lemma) If A is a r
Page 21 and 22: The Battle Continues EvilOne must g
Page 23: Myhill-Nerode Theorem do by hand de

Lecture 3

Create successful ePaper yourself

Delete template?

Save as template?