12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016
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1 CS125 Lecture 12 Fll Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple nondeterministic polynomil-time lgorithm to decide whether numer N is composite would nondeterministiclly guess fctoriztion L, M of the numer, nd then verify tht L M = N. (It turns out tht there is lso deterministic polynomil-time lgorithm for deciding compositeness, discovered in 2002, ut it is much more complicted.) Nondeterminism is not relistic or physicl computtionl resource, ut turns out to e very useful for cpturing mny computtionl prolems of interest nd etter-understnding relistic deterministic models of computtion. Just like introducing the imginry numer i = 1 turns out to e very useful in nswering questions out the ordinry rel numers Nondeterministic Finite Automt A lnguge for which it is hrd to design DFA: L = {,,} = {x 1 x 2 x k : k 0 nd ech x i {,,}}. But it is esy to imgine device to recognize this lnguge if there sometimes cn e severl possile trnsitions! OR OR ε Def: 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). 12-1
2 Lecture When in stte p reding symol σ, cn go to ny stte q in the set δ(p,σ). there my e more thn one such q, or there my e none (in cse δ(p,σ) = /0). Cn jump from p to ny stte in δ(p,ε) without moving the input hed. Computtions y 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. Exmple of n NFA N : q 0 q 1 q 2 q 3 N = ({q 0,q 1,q 2,q 3 },{,},δ,q 0,{q 0 }), where δ is given y: ε q 0 {q 1 } /0 /0 q 1 {q 2 } /0 /0 q 2 {q 0 } {q 0,q 3 } /0 q 3 {q 0 } /0 /0
3 Lecture Tree of computtions Tree of computtions of NFA N on string : An NFA N ccepts w if there is t lest one ccepting computtion pth on input w, so we could check ll computtion pths to determine whether N ccepts w. But the numer of pths my grow exponentilly with the length of w! Cn the exponentil serch e voided? NFAs vs. DFAs NFAs seem more powerful thn DFAs. Are they? Theorem 12.1 For every NFA N, there exists DFA M such tht L(M) = L(N). Proof y Construction: Given ny NFA N, we construct DFA M such tht L(M) = L(N). The ide is to hve the DFA M keep trck of the set of sttes tht N could e in fter hving red the input string so fr. N : Before writing it down formlly, we illustrte with n exmple. Recll our NFA N for L = {,,}. q 0 q 1 q 2 q 3
4 Lecture N strts in stte q0 so we will construct DFA M strting in stte {q0 }: Forml Description of the Suset Construction Given n NFA N = (Q, Σ, δ, q0, F), we construct DFA M = (Q0, Σ, δ0, q00, F 0 ) where Q0 = P(Q) q00 = E({q0 }) / (tht is, R Q0 ) F 0 = {R Q : R F 6= 0} δ0 (R, σ) = E({q Q : q δ(r, σ) for some r R}) = [ E(δ(r, σ)), r R where for set S Q, E(S) is the set of sttes tht cn e reched strting from stte in S nd following 0 or more ε trnsitions. It cn e shown y induction on w tht 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}. Rin & Scott, Finite Automt nd Their Decision Prolems, 1959
5 Lecture Michel O. Rin See the ACM Author Profile in the Digitl Lirry Cittion For their joint pper "Finite Automt nd Their Decision Prolem," which introduced the ide of nondeterministic mchines, which hs proved to e n enormously vlule concept. Their (Scott & Rin) clssic pper hs een continuous source of inspirtion for susequent work in this field. Biogrphicl Informtion Michel O. Rin (orn 1931 in Breslu, Germny) is noted computer scientist nd recipient of the Turing Awrd, the most prestigious wrd in the field. Rin ws orn s the son of ri in wht ws then known s Breslu (it ecme Wroclw, nd prt of Polnd, fter the Second World Wr). He received n M. Sc. from Herew University of Jeruslem in 1953, nd PhD from Princeton University in The cittion for the Turing Awrd, wrded in 1976 jointly to Rin nd Dn Scott for pper written in 1959, sttes tht the wrd ws grnted: For their joint pper "Finite Automt nd Their Decision Prolem," which introduced the ide of nondeterministic mchines, which hs proved to e n enormously vlule concept. Their (Scott & Rin) clssic pper hs een continuous source of inspirtion for susequent work in this field. Nondeterministic mchines hve ecome key concept in computtionl complexity theory, prticulrly with the description of complexity clsses P nd NP, s the most well-known exmple. In 1975, Rin lso invented rndomized lgorithm, the Miller-Rin primlity test, tht could determine very quickly, ut with tiny proility of error, whether numer ws prime numer. Fst primlity testing is key in the successful implementtion of most pulic-key cryptogrphy. In 1987, Rin, together with Richrd Krp, creted one of the most well-known efficient string serch lgorithms, the Rin-Krp string serch lgorithm, known for its rolling hsh. Rin's more recent reserch hs concentrted on computer security. He is currently the Thoms J. Wtson Sr. Professor of Computer Science t Hrvrd University. Using NFAs for Pttern Recognition NFAs cn express quite complicted pttern-recognition prolems. Indeed, it is esy to construct n NFA N tht ccepts exctly the strings generted y ny given regulr expression, such s R = (( ) (c d)( ) (c d)( ) ). This regulr expression R genertes the set L(R) of strings over lphet Σ = {,,c,d} tht hve n even numer of occurrences of c or d. We cn esily convert R (or ny regulr expression, for tht mtter) into n NFA N such L(N) = L(R): Turing Pper Additionl Links Michel O. Rin DEAS Reserch Profile Short Description in Informtion Science Hll of Fme t University of Pittsurgh. Formlly, regulr expression is lnguge defined inductively vi the following rules: (1) {σ} is regulr expression for ny σ Σ (2) {ε} is regulr expression, where ε is the empty string
6 Lecture (3) /0 is regulr expression (4) If R is regulr expression, then so is R = {x 1...x k : k 0, i x i R} (5) If R 1,R 2 re regulr expression, then so is R 1 R 2 = {x 1 x 2 : x 1 R 1,x 2 R 2 } (6) If R 1,R 2 re regulr expressions, then so is R 1 R 2 = {x : x R 1, or x R 2 } It turns out tht L is regulr expression iff L is regulr (i.e. some DFA ccepts it). Since DFAs cn simulte NFAs, it is equivlent to sy tht L is regulr expression iff some NFA ccepts it. One direction of the proof is more strightforwrd: nmely to show tht ny regulr expression is the lnguge ccepted y some NFA N. Those NFAs cn e creted s in the imge elow (for (4)-(6), we show how to use NFAs for R,R 1,R 2 to construct new NFA fter pplying some rule). The converse is lso true (ut hrder to prove): for every DFA M, one cn construct regulr expression R such tht L(R) = L(M). So DFAs, NFAs, nd Regulr Expressions ll descrie exctly the sme set of lnguges! If you re interested in the full proof, see the recommended text y Sipser. The sic ide is to define something clled
7 Lecture GNFA (generlized nondeterministic finite utomton), which is like n NFA except tht edges cn e leled with ritrry regexps nd not just Σ {ε}. We insist upon GNFA of specific formt: There is only one strt nd one ccept stte. The strt stte hs no incoming edges, nd hs outgoing edges to every other stte. The ccept stte hs no outgoing edges, nd hs incoming edges from every other strt stte. All sttes other thn the strt nd ccept sttes hve ll the possile ( Q 2) 2 edges to ech other, in ddition to ech one of them hving self loop. Given DFA M, we cn convert it to this formt quite esily. First, we mke new ccept stte such tht every other ccept stte hs n ε-trnsition to it, nd ll other non-ccept sttes hve /0 trnsitions to it. We lso mke new strt stte with n ε-trnsition to the originl strt stte, nd with /0 trnsitions to ll other sttes. Then for ech other stte, we dd self-loop with ε-trnsition. Also if sttes q,q other thn the ccept/strt sttes hd no edge from q to q efore, then we dd one with edge lel /0. Now we otin GNFA M ccepting exctly L(M). The min ide is this: if M hs exctly k = 2 sttes, then we re done! This is ecuse there is single edge from the strt to ccept stte, nd we cn simply red off the regexp written s its lel. Otherwise, if k > 2, there is some stte q which is neither the ccept nor the strt stte. Then we will remove q from the GNFA to otin one with one less stte. We hve to lter existing edges to mke sure the lnguge ccepted y the GNFA doesn t chnge. Suppose q,q re two other sttes (we my hve q = q ). Suppose the edge from q to q is leled R 4, nd q to q is leled R 1, q hs self-loop leled R 2 nd q to q is leled R 3. Then we replce the edge from q to q with new edge leled R 1 R 2 R 3 R 4, since we could either get from q to q through q, or not using q. Induction shows tht this works. So to decide whether given string w Σ mtches given regulr expression R, we cn convert R to n NFA N, convert N to DFA M, nd then run M on R. Q: Wht s the prolem with this pproch? How cn we do etter? Theorem 12.2 Given n NFA N = (Q,Σ,δ,q 0,F) nd string w, we cn decide whether w L(N) in time O( Q 2 w ). Proof: For suset R Q of sttes, recll the definition of E(R) ove s the set of sttes rechle from those in R using only ε-trnsitions. Note E(R) cn e computed in time O( Q 2 ) using, sy, redth-first serch, since the underlying grph hs Q vertices nd t most Q 2 edges. We initilize R 0 = E({q 0 }). Then for ech i = 1,..., w,
8 Lecture we cn compute R i = q Ri 1 E(δ(q,w i )), which is the set of sttes our NFA could possily e in fter processing the prefix w 1...w i. Then N ccepts w iff R w F is non-empty.
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