2.4 Theoretical Foundations

Transcription

1 2 Progrmming Lnguge Syntx 2.4 Theoretil Fountions As note in the min text, snners n prsers re se on the finite utomt n pushown utomt tht form the ottom two levels of the Chomsky lnguge hierrhy. At eh level of the hierrhy, mhines n e either eterministi or noneterministi. A eterministi utomton lwys performs the sme opertion in given sitution. A noneterministi utomton n perform ny of set of opertions. A noneterministi mhine is si to ept string if there exists hoie of opertion in eh sitution tht will eventully le to the mhine sying yes. As it turns out, noneterministi n eterministi finite utomt re eqully powerful: every DFA is, y efinition, egenerte NFA, n the onstrution in Exmple 2.14 emonstrtes tht for ny NFA we n rete DFA tht epts the sme lnguge. The sme is not true of push-own utomt: there re ontext-free lnguges tht re epte y n NPDA ut not y ny DPDA. Fortuntely, DPDAs suffie in prtie to ept the syntx of rel progrmming lnguges. Prtil snners n prsers re lwys eterministi Finite Automt Preisely efine, eterministi finite utomton (DFA) M onsists of (1) finite set Q of sttes, (2) finite lphet Σ of input symols, (3) istinguishe initil stte q 1 Q, (4) set of istinguishe finl sttes F Q, n(5)trnsition funtion δ : Q Σ Q tht hooses new stte for M se on the urrent sttentheurrentinputsymol.m egins in stte q 1. One y one it onsumes its input symols, using δ to move from stte to stte. When the finl symol hs een onsume, M is interprete s sying yes if it is in stte in F; otherwise it is interprete s sying no. We n exten δ in the ovious wy to tke strings, rther thn symols, s inputs, llowing us to sy tht M epts string x if δ(q 1, x) F. WenthenefineL(M), the lnguge epte y M, toetheset C 13

2 C 14 Chpter 2 Progrmming Lnguge Syntx Strt q 1 q 2.. q 3 q 4 Figure 2.33 Miniml DFA for the lnguge onsisting of ll strings of eiml igits ontining single eiml point. Apte from Figure 2.10 in the min text. The symol here is short for 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. EXAMPLE 2.56 Forml DFA for *(.. ) * {x δ(q 1, x) F}. In noneterministi finite utomton (NFA), the trnsition funtion, δ, is multivlue: the utomton n move to ny of set of possile sttes from given stte on given input. In ition, it my move from one stte to nother spontneously ; suh trnsitions re si to tke input symol ɛ. We n illustrte these efinitions with n exmple. Consier the irles-nrrows utomton of Figure C 2.33 (pte from Figure 2.10 in the min text). This is the miniml DFA epting strings of eiml igits ontining single eiml point. Σ={0, 1, 2, 3, 4, 5, 6, 7, 8, 9,.} is the mhine s input lphet. Q = {q 1, q 2, q 3, q 4 } is the set of sttes; q 1 is the initil stte; F = {q 4 } ( singleton in this se) is the set of finl sttes. The trnsition funtion n e represente y set of triples δ = {(q 1, 0, q 2 ),..., (q 1, 9, q 2 ), (q 1,., q 3 ), (q 2, 0, q 2 ),...,(q 2, 9, q 2 ), (q 2,., q 4 ), (q 3, 0, q 4 ),...,(q 3, 9, q 4 ), (q 4, 0, q 4 ),...,(q 4, 9, q 4 )}. In eh triple (q i,, q j ), δ(q i, ) =q j. Given the onstrutions of Exmples 2.12 n 2.14, we know tht there exists n NFA tht epts the lnguge generte y ny given regulr expression, n DFA equivlent to ny given NFA. To show tht regulr expressions n finite utomt re of equivlent expressive power, ll tht remins is to emonstrte tht there exists regulr expression tht genertes the lnguge epte y ny given DFA. We illustrte the require onstrution elow for our eiml strings exmple (Figure C 2.33). More forml n generl tretment of ll the regulr lnguge onstrutions n e foun in stnr utomt theory texts [HMU07, Sip13]. FromDFAtoRegulrExpression To onstrut regulr expression equivlent to given DFA, we employ ynmi progrmming lgorithm tht uils solutions to suessively more omplite suprolems from tle of solutions to simpler suprolems. We egin with set of simple regulr expressions tht esrie the trnsition funtion, δ. Forll sttes i,weefine r 0 ii = m ɛ

3 2.4.1 Finite Automt C 15 where { m } = { δ(q i, ) =q i } is the set of hrters leling the self-loop from stte q i k to itself. If there is no suh self-loop, r 0 ij = ɛ. Similrly, for i j,we efine r 0 ij = m EXAMPLE 2.57 Reonstruting regulr expression for the eiml string DFA where { m } = { δ(q i, ) =q j } is the set of hrters leling the r from q i to q j.ifthereisnosuhr,rij 0 is the empty regulr expression. (Note the ifferene here: we n sty in stte q i y not epting ny input, so ɛ is lwys one of the lterntives in rii 0,utnotinr0 ij when i j.) Given these r 0 expressions, the ynmi progrmming lgorithm inutively lultes expressions rij k with lrger supersripts. In eh, k nmes the highestnumere stte through whih ontrol my pss on the wy from q i to q j. We ssume tht sttes re numere strting with q 1,sowhenk = 0wemusttrnsition iretly from q i to q j, with no intervening sttes. In our smll exmple DFA, r11 0 = r0 33 = ɛ, nr0 22 = r0 44 = ɛ, whih we will revite ɛ. Similrly, r13 0 = r0 24 =., n r12 0 = r0 34 =. Expressions r0 14, r0 21, r0 23, r0 31, r0 32, r0 41, r0 42,nr0 43 re ll empty. For k > 0, the rij k expressions will generlly generte multihrter strings. At eh step of the ynmi progrmming lgorithm, we let r k ij = r k 1 ij r k 1 ik r k 1 kk *r k 1 kj Tht is, to get from q i to q j without going through ny sttes numere higher thn k, we n either go from q i to q j without going through ny stte numere higher thn k 1 (whih we lrey know how to o), or else we n go from q i to q k (without going through ny stte numere higher thn k 1), trvel out from q k n k gin n ritrry numer of times (never visiting stte numere higher thn k 1 in etween), n finlly go from q k to q j (gin without visiting stte numere higher thn k 1). If ny of the onstituent regulr expressions is empty, we omit its term of the outermost lterntion. At the en, our overll nswer is r1f n 1 r1f n 2... r1f n t,wheren = Q is the totl numer of sttes n F = {q f1, q f2,...,q ft } is the set of finl sttes. Beuse r11 0 = ɛ n there re no trnsitions from Sttes 2, 3, or 4 to Stte 1, nothing hnges in the first inutive step in our exmple; tht is, i [ rii 1 = rii 0 ]. The seon step is it more interesting. Sine we re now llowe to go through Stte2,wehver22 2 = r2 22 r2 22 *r2 22 = ( ɛ ) ( ɛ )( ɛ )*( ɛ ) = *. Beuse r21 1, r1 23, r1 32,nr1 42 re empty, however, r2 11, r2 33,nr2 44 remin the sme s r11 1, r1 33,nr1 44.Insimilrvein,wehve r 2 12 = ( ɛ )*( ɛ ) = + r 2 14 = ( ɛ )*. = +. r 2 24 =. ( ɛ )( ɛ )*. = *.

4 C 16 Chpter 2 Progrmming Lnguge Syntx Missing trnsitions n empty expressions from the previous step leve r13 2 = r1 13 =. n r34 2 = r1 34 =. Expressions r2 21, r2 23, r2 31, r2 32, r2 41, r2 42,nr2 43 remin empty. Inthethirinutivestep,wehve r 3 13 =.. ɛ*ɛ =. r14 3 r34 3 = +.. ɛ* = +.. = ɛɛ* = All other expressions remin unhnge from the previous step. Finlly, we hve r 4 14 = ( +.. ) ( +.. )( ɛ )*( ɛ ) = ( +.. ) ( +.. ) * = ( +.. ) * = +. *. + Sine F hs single memer (q 4 ), this expression is our finl nswer. EXAMPLE 2.58 A regulr lnguge with lrge miniml DFA EXAMPLE 2.59 Exponentil DFA low-up Spe Requirements In Setion we note without proof tht the onversion from n NFA to DFA my le to exponentil low-up in the numer of sttes. Certinly this i not hppen in our eiml string exmple: the NFA of Figure 2.8 hs 14 sttes, while the equivlent DFA of Figure 2.9 hs only 7, n the miniml DFA of Figures 2.10 n C 2.33 hs only 4. Consier, however, the suset of ( )* in whih some letter ppers t lest three times. The miniml DFA for this lnguge hs 28 sttes. As shown in Figure C 2.34, 27 of these re sttes in whih we hve seen i, j, nk s, s, n s, respetively. The 28th (n only finl) stte is rehe one we hve seen t lest three of some speifi hrter. By ontrst, there exists n NFA for this lnguge with only eight sttes, s shown in Figure C It requires tht we guess, t the outset, whether we will see three s, three s, or three s. It mirrors the struture of the nturl regulr expression ( )* ( )* ( )* ( )* ( )* ( )* ( )* ( )* ( )* ( )* ( )* ( )*. Of ourse, the eight-stte NFA oes not emerge iretly from the onstrution of Figure 2.7; tht onstrution proues 52-stte mhine with ertin mount of reunny, n with mny extrneous sttes n epsilon proutions. But onsier the similr suset of ( )* in whih some igit ppers t lest ten times. The miniml DFA for this lnguge hs 10,000,000,001 sttes: non-finl stte for eh omintion of zeros through nines with less thn ten of eh, n single finl stte rehe one ny igit hs ppere t lest ten times. One possile regulr expression for this lnguge is

5 2.4.1 Finite Automt C , Strt , ,, ,,, ,,, Figure 2.34 DFA for the lnguge onsisting of ll strings in ( )* in whih some letter ppers t lest three times. Stte nme ijk inites tht i s, j s, n k s hve een seen so fr. Within the ui portion of the figure, most ege lels re elie: trnsitions move to the right, trnsitions go k into the pge, n trnsitions move own. (( )* 0 ( )* 0 ( )* 0 ( )* 0 ( )* 0 ( )* 0 ( )* 0 ( )* 0 ( )* 0 ( )* 0 ( )*) (( )* 1 ( )* 1 ( )* 1 ( )* 1 ( )* 1 ( )* 1 ( )* 1 ( )* 1 ( )* 1 ( )* 1 ( )*)... (( )* 9 ( )* 9 ( )* 9 ( )* 9 ( )* 9 ( )* 9 ( )* 9 ( )* 9 ( )* 9 ( )* 9 ( )*)

6 C 18 Chpter 2 Progrmming Lnguge Syntx Strt,,,,,,,,,, Figure 2.35 NFA orresponing to the DFA of Figure C Our onstrution woul yiel very lrge NFA for this expression, ut lerly mny orers of mgnitue smller thn ten illion sttes! Push-Down Automt A eterministi push-own utomton (DPDA) N onsists of (1) Q, (2) Σ, (3) q 1,n(4)F, sindfa,plus(6)finitelphetγ of stk symols, (7) istinguishe initil stk symol Z 1 Γ, n(5 ) trnsition funtion δ : Q Γ {Σ {ɛ}} Q Γ,whereΓ is the set of strings of zero or more symols from Γ. N egins in stte q 1,withsymolZ 1 in n otherwise empty stk. It repetely exmines the urrent stte q n top-of-stk symol Z. If δ(q,ɛ, Z) is efine, N moves to stte r n reples Z with α in the stk, where (r,α)=δ(q,ɛ, Z). In this se N oes not onsume its input symol. If δ(q,ɛ, Z) is unefine, N exmines n onsumes the urrent input symol. It then moves to stte s n reples Z with β, where(s,β)=δ(q,, Z). N is interprete s epting string of input symols if n only if it onsumes the symols n ens in stte in F. As with finite utomt, noneterministi push-own utomton (NPDA) is istinguishe y multivlue trnsition funtion: n NPDA n hoose ny of set of new sttes n stk symol replements when fe with given stte, input, n top-of-stk symol. If δ(q,ɛ, Z) is nonempty, N n lso hoose new stte n stk symol replement without inspeting or onsuming its urrent input symol. While we hve seen tht noneterministi n eterministi finite utomt re eqully powerful, this orresponene oes not rry over to pushown utomt: there re ontext-free lnguges tht re epte y n NPDA ut not y ny DPDA. The proof tht CFGs n NPDAs re equivlent in expressive power is more omplex thn the orresponing proof for regulr expressions n finite utomt. The proof is lso of limite prtil importne for ompiler onstrution; we o not present it here. While it is possile to rete n NPDA for ny