Harvard CS 121 and CSCI E-121 Lecture 4: Nondeterministic Finite Automata and Closure Properties

Transcription

1 Hrvrd CS 121 nd CSCI E-121 Lecture 4: Nondeterministic Finite Automt nd Closure Properties Hrry Lewis September 11, 2014 Reding: Sipser, 1.1 nd 1.2.

2 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 } 1

3 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. 2

4 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. 3

5 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? 4

6 NFAs vs. DFAs NFAs seem more powerful thn DFAs. Are they? Theorem: For every NFA N, there exists DFA M such tht L(M) = L(N). Proof Outline: Given ny NFA N, to construct DFA M such tht L(M) = L(N): 5

7 NFAs vs. DFAs NFAs seem more powerful thn DFAs. Are they? Theorem: For every NFA N, there exists DFA M such tht L(M) = L(N). Proof Outline: Given ny NFA N, to construct DFA M such tht L(M) = L(N): Hve the DFA keep trck, t ll times, of ll possible sttes the NFA could be in fter reding the sme initil prt of the input string. I.e., the sttes of M re sets of sttes of N, nd δ M (R, w) is the set of ll sttes N could rech fter reding w, strting from stte in R. 6

8 Exmple of the SUBSET CONSTRUCTION NFA N for {x 1 x 2 x k : k 0 nd ech x i {b, b, }}. N : 0 b 1 2 b 3 N strts in stte 0 so we will construct DFA M strting in stte {0}. 7

9 Exmple of the SUBSET CONSTRUCTION NFA N for {x 1 x 2 x k : k 0 nd ech x i {b, b, }}. N : b b 3 N strts in stte 0 so we will construct DFA M strting in stte {0}. Here it is: b b b b b All other trnsitions re to the ded stte. The other sttes re unrechble, though techniclly must be defined. Finl sttes re ll those contining 0. 8

10 Forml Construction of DFA M from NFA N = (Q, Σ, δ, q 0, F ) On the ssumption tht δ(p, ε) = for ll sttes p. (i.e., we ssume no ε-trnsitions, just to simplify things bit) M = (Q, Σ, δ, q 0, F ) where Q = P (Q) q 0 = {q 0 } F = {R Q : R F } (tht is, R Q ) δ (R, σ) = {q Q : q δ(r, σ) for some r R} = δ(r, σ) r R 9

11 Proving tht the construction works Clim: For every string w, running M on input w ends in the stte {q Q : some computtion of N on input w ends in stte q}. Pf: By induction on w. Cn be extended to work even for NFAs with ε-trnsitions. THE SUBSET CONSTRUCTION 10

12 Rbin & Scott, Finite Automt nd Their Decision Problems, Michel O. Rbin See the ACM Author Profile in the Digitl Librry Cittion For their joint pper "Finite Automt nd Their Decision Problem," which introduced the ide of nondeterministic mchines, which hs proved to be n enormously vluble concept. Their (Scott & Rbin) clssic pper hs been continuous source of inspirtion for subsequent work in this field. Biogrphicl Informtion Michel O. Rbin (born 1931 in Breslu, Germny) is noted computer scientist nd recipient of the Turing Awrd, the most prestigious wrd in the field. Rbin ws born s the son of rbbi in wht ws then known s Breslu (it becme Wroclw, nd prt of Polnd, fter the Second World Wr). He received n M. Sc. from Hebrew University of Jeruslem in 1953, nd PhD from Princeton University in The cittion for the Turing Awrd, wrded in 1976 jointly to Rbin nd Dn Scott for pper written in 1959, sttes tht the wrd ws grnted: For their joint pper "Finite Automt nd Their Decision Problem," which introduced the ide of nondeterministic mchines, which hs proved to be n enormously vluble concept. Their (Scott & Rbin) clssic pper hs been continuous source of inspirtion for subsequent work in this field. Nondeterministic mchines hve become key concept in computtionl complexity theory, prticulrly with the description of complexity clsses P nd NP, s the most well-known exmple. In 1975, Rbin lso invented rndomized lgorithm, the Miller-Rbin primlity test, tht could determine very quickly, but with tiny probbility of error, whether number ws prime number. Fst primlity testing is key in the successful implementtion of most public-key cryptogrphy. 11

13 Nondeterminism gives us new progrmming tool Strings tht begin with b 12

14 Nondeterminism gives us new progrmming tool Strings tht begin with b Strings tht end with b 13

15 Nondeterminism gives us new progrmming tool Strings tht begin with b Strings tht end with b Strings tht begin or end with b 14

16 Nondeterminism gives us new progrmming tool Strings tht begin with b Strings tht end with b Strings tht begin or end with b Strings tht hve b s substring nywhere 15

17 Closure Properties Theorem: The clss of regulr lnguges is closed under: Union: L 1 L 2 Conctention: L 1 L 2 = {xy : x L 1 nd y L 2 } Kleene *: L 1 = {x 1 x 2 x k : k 0 nd ech x i L 1 } Complement: L 1 Intersection: L 1 L 2 16

18 Union Union: If L 1 nd L 2 re regulr, then L 1 L 2 is regulr. M 1 M 2 M ε ε M hs the sttes nd trnsitions of M 1 nd M 2 plus new strt stte ε- trnsitioning to the old strt stte 17

19 Conctention, Kleene *, Complementtion Conctention: L(M) = L(M 1 ) L(M 2 ) Kleene *: L(M) = L(M 1 ) Complement: L(M) = L(M 1 ) 18

20 Closure under Intersection Intersection S T = S T S T = S = T Hence closure under union nd complement implies closure under intersection 19

21 A more constructive nd direct proof of closure under intersection Better wy ( Cross Product Construction ): From DFAs M 1 = (Q 1, Σ, δ 1, q 1, F 1 ) nd M 2 = (Q 2, Σ, δ 2, q 2, F 2 ), construct M = (Q, Σ, δ, q 0, F ): Q = Q 1 Q 2 F = F 1 F 2 δ( r 1, r 2, σ) = δ 1 (r 1, σ), δ 2 (r 2, σ) q 0 = q 1, q 2 Then L(M 1 ) L(M 2 ) = L(M) 20

22 Some Efficiency Considertions The subset construction shows tht ny n-stte NFA cn be implemented s 2 n -stte DFA. NFA Sttes DFA Sttes the number of prticles in the universe How to implement this construction on ordinry digitl computer? NFA sttes DFA stte bit vector 1,..., n n 21

23 Is this construction the best we cn do? Could there be construction tht lwys produces n n 2 stte DFA for exmple? Theorem: For every n 1, there is lnguge L n such tht 1. There is n (n + 1)-stte NFA recognizing L n. 2. There is no DFA recognizing L n with fewer thn 2 n sttes. Conclusion: For finite utomt, nondeterminism provides n exponentil svings over determinism (in the worst cse). 22

24 Proving tht exponentil blowup is sometimes unvoidble (Could there be construction tht lwys produces n n 2 stte DFA for exmple?) Consider (for some fixed n=17, sy) L n = {w {, b} : the nth symbol from the right end of w is n } There is n (n + 1)-stte NFA tht ccepts L n. There is no DFA tht ccepts L n nd hs < 2 n sttes 23

25 A Fooling Argument Suppose DFA M hs < 2 n sttes, nd L(M) = L n There re 2 n strings of length n. By the pigeonhole principle, two such strings x y must drive M to the sme stte q. Suppose x nd y differ t the k th position from the right end (one hs, the other hs b) (k = 1, 2,..., or n) M must tret x n k nd y n k identiclly (ccept both or reject both). These strings differ t position n from the right end. So L(M) L n, contrdiction. QED. 24

26 Illustrtion of the fooling rgument M is in stte q 0 M is in stte q x y b n k x n k y n k n b M in stte q 0 M in stte q Different symbols n positions from right M in sme stte p, x nd y re different strings (so there is position k where one hs nd the other hs b) But both strings drive M from s to the sme stte q 25

27 Wht the rgument proves This shows tht the subset construction is within fctor of 2 of being optiml In fct it is optiml, i.e., s good s we cn do in the worst cse. Still, in mny cses, the generte-sttes-s-needed method yields DFA with 2 n sttes (e.g. if the NFA ws deterministic to begin with!) 26