Main Goal I basic concepts of automata and process theory regular languages

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1 Cure verview Main Gal I baic cncept f autmata and prce thery regular language determinitic finite autmatn DFA nn-determinitic finite autmatn NFA regular exprein finite memry cntext-free language puh-dwn autmatn PDA cntext-free grammar CFG pare tree recurive enumerable language reactive Turing machine rtm claical Turing machine ctm unretricted grammar elf-reference and circularity

2 Cure verview (cnt.) Main Gal II mathematical prf and frmal reaning tutr grup hift t prf at final exam n cheduled tuitin n exercie examinatin interim tet n Chapter 2 (20%) prgramming aignment: Turing machine prgram (20%) participatin tutr grup (10%) final exam (50%) plan learning activitie with peer tudent

3 Mathematical inductin fr each prperty P N and natural number n,k 0: ( P(0) k N: P(k) = P(k +1) ) = n N: P(n) bai P(0) inductin tep P(n+1) inductin hypthei P(n) r k, 0 k n: P(k) thi cure: natural number; in general: well-funded tructure n N: (2n 1) = n 2 fr all n 0, if x 1,x 2,...,x n > 0 then (x 1 x 2... x n ) 1/n (x 1 +x x n )/n 3 n 1 i even

4 Structural inductin (fr tring) fr each prperty P Σ and natural number n,k 0: ( P(ε) a Σ, w Σ : P(w) = P(aw) ) = w Σ : P(w) bai P(ε) inductin tep P(aw) inductin hypthei P(w)

5 A play f tenni game, et, match lve, 15, 30, 40, game winning al require at leat tw cre mre deuce, advantage-in, advantage-ut

6 A tenni autmatn lve 15-lve lve lve 15-all lve lve lve-40 game-in game-ut all deuce adv-in adv-ut

7 A tenni autmatn (cnt.) lve 15-lve lve lve 15-all lve lve lve all game-in game-ut adv-in deuce adv-ut ingle initial tate ne r mre final tate many labeled tranitin

8 A tenni autmatn (cnt.) lve 15-lve lve lve 15-all lve lve lve all game-in game-ut adv-in deuce adv-ut accepted tring like and

9 Clicker quetin 1 Which tring i nt accepted by the tenni autmatn? A. B. C. lve 15-lve lve lve 15-all lve lve lve all game-in game-ut adv-in deuce adv-ut D. all are accepted

10 Determinitic Finite Autmata 2IT70 Finite Autmata and Prce Thery Techniche Univeriteit Eindhven April 18, 2016

11 Determinitic finite autmatn DFA D = (Q, Σ, δ, q 0, F) Q finite et f tate Σ finite alphabet δ : Q Σ Q tranitin functin q 0 Q initial tate F Q et f final tate 2IT70 (2016) Sectin /22

12 The tenni example lve 15-lve lve lve 15-all lve lve lve all game-in game-ut adv-in deuce adv-ut 2IT70 (2016) Sectin /22

13 The tenni example (cnt.) lve 15-lve lve lve 15-all lve lve lve all game-in game-ut adv-in deuce adv-ut et f tate {lve, 15 lve, lve 15,... } game-in, game-ut, deuce, adv-in, adv-ut} alphabet {, } tranitin lve 15 lve, lve lve 15,... initial tate lve et f final tate {game-in, game-ut} 2IT70 (2016) Sectin /22

14 Clicker quetin 3 Why de the fllwing nt repreent a DFA? q 0 0 q 0 1 q 2 A. The alphabet ha mre than 2 letter. B. It accept the empty tring ε. C. It ha a tranitin relatin, but nt a tranitin functin. D. It de repreent a DFA. 2IT70 (2016) Sectin /22

15 ne-tep and multi-tep yield cnfiguratin (q,w) fr tate q and tring w ne-tep yield (q,w) D (q,w ) iff a: w = aw, δ(q,a) = q multi-tep yield (q,w) D (q,w ) iff n 0 w 0,...,w n Σ q 0,...,q n Q : (q,w) = (q 0,w 0 ), (q i 1,w i 1 ) D (q i,w i ), fr i = 1..n (q n,w n ) = (q,w ) (q,w) = (q 0,w 0 ) D (q 1,w 1 ) D D (q n,w n ) = (q,w ) fr uitable n, w 0,...,w n, q 0,...,q n 2IT70 (2016) Sectin /22

16 Anther example DFA b a a,b a q 0 q a 1 q b 2 q 3 b (q 0,abaa) (q 1,baa) (q 0,aa) (q 1,a) (q 2,ε) (q 0,bbaa) (q 0,baa) (q 0,aa) (q 1,a) (q 2,ε) (q 1,aa) (q 2,ε) and (q 1,aaaa) (q 2,ε) (q 0,aab) (q 3,ε), (q 0,baab) (q 3,ε), and (q 0,baaaabaabb) (q 3,ε) 2IT70 (2016) Sectin /22

17 Language accepted by DFA L(D) = {w Σ q F: (q 0,w) D (q,ε)} b a a,b a q 0 q a 1 q b 2 q 3 b accepted language {w {a,b} w ha a ubtring aab} 2IT70 (2016) Sectin /22

18 Clicker quetin 2 Which language i the language accepted by thi autmatn? ee a a e b b b b a e a A. {a,b,aba,bab} B. {a(bb) n n 0} {b(aa) n n 0} C. {w {a,b} # a (w) i dd} D. {w {a,b} # a (w)+# b (w) i dd} 2IT70 (2016) Sectin /22

19 Path et DFA D, tate q pathet D (q) = {w Σ (q 0,w) D (q,ε)} even a a dd pathet D (even) = {a n n 0, n even} pathet D (dd) = {a n n 0, n dd} 2IT70 (2016) Sectin /22

20 Yet anther example DFA 0 0,1 q 0 1 q 1 1 q 2 0 L = {w {0,1} w ha n ubtring 11} tate pathet regular exprein q 0 n ubtring 11 and n lat ymbl 1 0 (10 + ) q 1 n ubtring 11 and lat ymbl 1 0 (10 + ) 1 q 2 ubtring 11 (0+1) 11(0+1) regular exprein will be explained later 2IT70 (2016) Sectin /22

21 Anther example DFA (rev.) ee a e tate pathet a ee {w # a (w) even, # b (w) even} b b b b e {w # a (w) dd, # b (w) even} a e {w # a (w) even, # b (w) dd} e a {w # a (w) dd, # b (w) dd} L(D) = pathet D (e) pathet D (e) = {w # a (w) dd, # b (w) even} {w # a (w) even, # b (w) dd} = {w # a (w)+# b (w) dd} = {w w dd} 2IT70 (2016) Sectin /22

22 Language accepted by DFA (reviited) b a a,b a q 0 q a 1 q b 2 q 3 b tate pathet q 0 {w n ubtring aa, nt ending in a} q 1 {w n ubtring aa, ending in a} q 2 {w ending in aa} q 3 {w ubtring aab } accepted language {w {a,b} w ha a ubtring aab } 2IT70 (2016) Sectin /22