NFA and regex. the Boolean algebra of languages. non-deterministic machines. regular expressions

Transcription

1 NFA nd regex l the Boolen lger of lnguges non-deterministi mhines regulr expressions Informtis

2 The intersetion of two regulr lnguges is regulr Run oth mhines in prllel? Build one mhine tht simultes two mhines running in prllel! Keep trk of the stte of eh mhine. Informtis

3 The intersetion of two regulr lnguges is regulr = Informtis 3

4 The intersetion of two regulr lnguges is regulr = Informtis 4

5 The intersetion of two regulr lnguges is regulr = Informtis 5

6 The regulr lnguges A Σ* form Boolen Alger Sine they re losed under intersetion nd omplement. Informtis 6

7 Two exmples + Even inry numers Input sequene is epted if it ends with zero. + Input sequene is epted if it ends with one. Odd inry numers Informtis 7

8 Three exmples Whih inry numers re epted? + 3 Informtis 8

9 By three or not y three? divisile y three not divisile y three Informtis 9

10 The omplement of regulr lnguge is regulr If A Σ* is reognised y M then A = Σ* \ A is reognised y M where M nd M re identil exept tht the epting sttes of M re the nonepting sttes of M nd vie-vers Informtis

11 The intersetion of two regulr lnguges is regulr divisile y 6 divisile y nd divisile y 3 Informtis

12 The intersetion of two regulr lnguges is regulr Run oth mhines in prllel? Build one mhine tht simultes two mhines running in prllel! Keep trk of the stte of eh mhine. Informtis

13 The intersetion of two regulr lnguges is regulr = Informtis 3

14 The intersetion of two regulr lnguges is regulr = Informtis 4

15 The intersetion of two regulr lnguges is regulr = Informtis 5

16 The regulr lnguges A Σ* form Boolen Alger Sine they re losed under intersetion nd omplement. Informtis 6

17 Is there regulr expression for every FSM? L = L = L = Informtis 7

18 Is there regulr expression for every FSM? L = L = ε Informtis 8 L =

19 Is there regulr expression for every FSM? L = L L L = ε Informtis 9 L = L

20 Is there regulr expression for every FSM? L = L L = L L = ε L = L Informtis

21 Is there regulr expression for every FSM? L = L L = L L = ()* L = ε L = L Informtis

22 Arden s Lemm If R nd S re regulr expressions then the eqution S X = R X S hs solution X = R S* R X If ε L(S) then this solution is unique. Informtis

23 Is there regulr expression for every FSM? L L L Let Li e the lnguge epted if i is the epting stte L = ε L = L L = L L L = L ε L = ε ε L = Informtis 3

24 Is there regulr expression for every FSM? L = L L = L3 L L3 = ε L = ε L L = (ε L ) L 3 = L L = L ( ) Informtis 4

25 Is there regulr expression for every FSM? L = L ( ) L = ( )* 3 L = L = ( )* L3 = ε L = ε ( )* Informtis 5

26 Arden s Lemm If R nd S re regulr expressions then the eqution X = R X S hs solution X = R S* If ε L(S) then this solution is unique. L = L ( ) L = ( )* Informtis 6

27 regulr expressions ny hrter is regexp mthes itself if R nd S re regexps, so is RS mthes mth for R followed y mth for S if R nd S re regexps, so is R S mthes ny mth for R or S (or oth) if R is regexp, so is R* mthes ny sequene of or more mthes for R The lger of regulr expressions lso inludes elements nd mthes nothing; mthes the empty string Kleene *, + * + Stephen Cole Kleene

28 regulr expressions denote regulr sets ny hrter is regexp {<>} if R nd S re regexs, so is RS { r s r R nd s S } if R nd S re regexps, so is R S R S if R is regexp, so is R* { rn n N nd r R S = S = S empty set S = S = S {<>} singleton empty sequene: Kleene *, + * + Stephen Cole Kleene