Harvard CS 121 and CSCI E-121 Lecture 3: Finite Automata

Transcription

1 Hrvrd CS 121 nd CSCI E-121 Lecture 3: Finite Automt Hrry Lewis September 9, 2014 Reding: Sipser, 1.1 nd 1.2.

2 Lnguges A lnguge L over lphbet Σ is set of strings over Σ (i.e. L Σ ) Computtionl problem: given x Σ, is x L? Any YES/NO problem cn be cst s lnguge. Exmples of simple lnguges: All words in the Americn Heritge Dictionry {, h, rdvrk,..., zyzzv}., Σ, Σ {x Σ : x = 3} = {, b, b, bb, b, bb, bb, bbb} 1

3 More complicted lnguges The set of strings x {, b} such tht x hs more s thn b s. The set of strings x {0, 1} such tht x is the binry representtion of prime number. All C progrms tht do not go into n infinite loop. L 1 L 2, L 1 L 2, L 1 L 2 if L 1 nd L 2 re lnguges.. 2

4 The highly bstrct nd metphoricl term lnguge A lnguge cn be either finite or infinite A lnguge need not hve ny internl structure 3

5 Be creful to distinguish ε The empty string ( string) The empty set ( set, possibly lnguge) {ε} The set contining one element, which is the empty string ( lnguge) { } The set contining one element, which is the empty set ( set of sets, mybe set of lnguges) 4

6 Opertions on Lnguges Set opertions Conctention of Lnguges L 1 L 2 = {xy : x L 1, y L 2 } e.g. {, b}{, bb} = {, b, bb, bbb} e.g. {ε}l = L e.g. L =? 5

7 Kleene str: L = {w 1 w n : n 0, w 1,..., w n L} e.g. {} = {ε,,,...} e.g. {b, b,, bb} = ll even length strings e.g. Σ = Kleene Str of Σ e.g. =? 6

8 (Deterministic) Finite Automt Exmple: Home Stereo P = power button (ON/OFF) S = source button (CD/Rdio/TV), only works when stereo is ON, but source remembered when stereo is OFF. Strts OFF, in CD mode. A computtionl problem: does given sequence of button presses w {P, S} leve the system with the rdio on? Does this relly model the sitution dequtely? - This bstrction rules out pushing 2 buttons t once, holding one down while pushing the other, etc. 7

9 The Home Stereo DFA 8

10 Forml Definition of DFA A DFA M is 5-Tuple (Q, Σ, δ, q 0, F ) Q : Finite set of sttes Σ : Alphbet δ : Trnsition function, Q x Σ Q q 0 : Strt stte, q 0 Q F : Accept (or finl) sttes, F Q If δ(p, σ) = q, then if M is in stte p nd reds symbol σ Σ then M enters stte q (while moving to next input symbol) Home Stereo exmple: 9

11 Another Visuliztion b b b Input tpe Reding hed moves left to right, one squre t time Strt stte mrked with < Double-circled sttes re ccepting or finl Finite-stte control chnges stte depending on: current stte next symbol M ccepts string x if After strting M in the strt[initil] stte with hed on first squre, when ll of x hs been red, M winds up in finl stte. 10

12 Exmples Bounded Counting: A DFA for {x : x hs n even # of s nd n odd # of b s} q 0 q 1 b b b b q 2 q 3 Trnsition function δ: b q 0 q 1 q 2 q 1 q 0 q 3 q 2 q 3 q 0 q 3 q 2 q 1. i.e. δ(q 0, ) = q 1, etc. = strt stte = finl stte Q = {q 0, q 1, q 2, q 3 } Σ = {, b} F = {q 2 } 11

13 Another Exmple, to work out together Pttern Recognition: A DFA tht ccepts { x : x hs b s substring}. 12

14 Another Exmple Pttern Recognition: A DFA tht ccepts { x : x hs bb s substring}. 13

15 Another Exmple A DFA tht ccepts { x : x hs bb s substring}. You re going through constructive process string DFA tht is utomted in every text editor! Relly compiler tht genertes DFA code from n input string pttern 14

16 DFAs re very simple but gret del is still unknown! How big is the smllest utomton tht cn tell two given strings prt? 15

17 DFAs re very simple but gret del is still unknown! How big is the smllest utomton tht cn tell two given strings prt? Tht is, for ny strings x, y, let S(x, y) be the number of sttes in the smllest DFA M tht ccepts x nd rejects y. (Doesn t mtter wht M does when given other strings s input.) For n 0, wht is S(n) = mx{s(x, y) : x, y n}? 16

18 Forml Definition of Computtion M = (Q, Σ, δ, q 0, F ) ccepts w = w 1 w 2 w n Σ (where ech w i Σ) if there exist r 0,..., r n Q such tht 1. r 0 = q 0, 2. δ(r i, w i+1 ) = r i+1 for ech i = 0,..., n 1, nd 3. r n F. The lnguge recognized (or ccepted) by M, denoted L(M), is the set of ll strings ccepted by M. Exmple: 17

19 Trnsition function on n entire string More forml (not necessry for us, but nottion sometimes useful): Inductively define δ : Q Σ Q by δ (q, ε) = q, δ (q, wσ) = δ(δ (q, w), σ), for ny w Σ nd σ Σ. Intuitively, δ (q, w) = stte reched fter strting in q nd reding the string w. M ccepts w if δ (q 0, w) F. Determinism: Given M nd w, the sttes r 0,..., r n re uniquely determined. Or in other words, δ (q, w) is well defined for ny q nd w: There is precisely one stte to which w drives M if it is strted in given stte. 18

20 Trnsition function on n entire string More forml (not necessry for us, but nottion sometimes useful): Inductively define δ : Q Σ Q by δ (q, ε) = q, δ (q, wσ) = δ(δ (q, w), σ). Intuitively, δ (q, w) = stte reched fter strting in q nd reding the string w. M ccepts w if δ (q 0, w) F. Determinism: Given M nd w, the sttes r 0,..., r n re uniquely determined. Or in other words, δ (q, w) is well defined for ny q nd w: There is precisely one stte to which w drives M if it is strted in given stte. 19

21 The impulse for nondeterminism A lnguge for which it is hrd to design DFA: {x 1 x 2 x k : k 0 nd ech x i {b, b, }}. But it is esy to imgine device to ccept this lnguge if there sometimes cn be severl possible trnsitions! b OR b OR b ε b b 20

22 Nondeterministic Finite Automt An NFA is 5-tuple (Q, Σ, δ, q 0, F ), where Q, Σ, q 0, F re s for DFAs δ : Q (Σ {ε}) P (Q). When in stte p reding symbol σ, cn go to ny stte q in the set δ(p, σ). there my be more thn one such q, or there my be none (in cse δ(p, σ) = ). Cn jump from p to ny stte in δ(p, ε) without moving the input hed. 21

23 Computtions by n NFA N = (Q, Σ, δ, q 0, F ) ccepts w Σ if we cn write w = y 1 y 2 y m where ech y i Σ {ε} nd there exist r 0,..., r m Q such tht 1. r 0 = q 0, 2. r i+1 δ(r i, y i+1 ) for ech i = 0,..., m 1, nd 3. r m F. Nondeterminism: Given N nd w, the sttes r 0,..., r m re not necessrily determined. 22

24 Exmple of n NFA N : q 0 b q 1 q 2 b q 3 N = ({q 0, q 1, q 2, q 3 }, {, b}, δ, q 0, {q 0 }), where δ is given by: b ε q 0 {q 1 } q 1 {q 2 } q 2 {q 0 } {q 0, q 3 } q 3 {q 0 } Work out the tree of ll possible computtions on bb 23

25 Tree of computtions Tree of computtions of NFA N on string bb: 24

26 How to simulte NFAs? NFA ccepts w if there is t lest one ccepting computtionl pth on input w But the number of pths my grow exponentilly with the length of w! Cn exponentil serch be voided? 25

27 NFAs vs. DFAs NFAs seem more powerful thn DFAs. Are they? 26