Abstract. The ISO standard for Standard Generalized Markup Language (SGML) provides a syntactic

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1 Deterministic Regulr Lnguges Anne Bruggemnn-Klein Institut fur Informtik Universitt Freiburg Rheinstr. 10{12, 7800 Freiburg Germny Derick Wood Deprtment of Computer Science University of Wterloo Wterloo, Ontrio N2L 3G1 Cnd Abstrct The ISO stndrd for Stndrd Generlized Mrkup Lnguge (SGML) provides syntctic met-lnguge for the denition of textul mrkup systems. In the stndrd the right hnd sides of productions re clled content models nd they re bsed on regulr expressions. The llowble regulr expressions re those tht re \unmbiguous" s dened by the stndrd. Unfortuntely, the stndrd's use of the term \unmbiguous" does not correspond to the two well known notions, since not ll regulr lnguges re denoted by \unmbiguous" expressions. Furthermore, the stndrd's denition of \unmbiguous" is somewht vgue. Therefore, we provide precise denition of \unmbiguous expressions" nd renme them deterministic regulr expressions to void ny confusion. A regulr expression E is deterministic if the cnonicl -free nite utomton ME recognizing L(E) is deterministic. A regulr lnguge is deterministic if there is deterministic expression tht denotes it. We give Kleene-like theorem for deterministic regulr lnguges nd we chrcterize them in terms of the structurl properties of the miniml deterministic utomt recognizing them. The ltter result enbles us to decide if given regulr expression denotes deterministic regulr lnguge nd, if so, to construct n equivlent deterministic expression. Clssiction: Automt nd forml lnguges, esp. forml models in document processing 1 Introduction Document processing systems like editors, formtters, nd retrievl systems del with mny different types of documents, like books, rticles, memos, dictionries, or letters, in ddition to user-dened document types or customized versions of \public" types. Recently, the Stndrd Generlized Mrkup Lnguge (SGML) [ISO86] hs been estblished s common pltform for the syntctic speciction of document types nd conforming documents. SGML is n ISO stndrd nd hs been endorsed by number of publishing houses throughout North Americ nd Europe, by the Europen Community, nd by the U.S. Deprtment of Defense. Document types in SGML re dened by context free grmmrs tht re mixture of Hrrison nd Ginsburg's brcketed grmmrs [GH67] nd LLonde's regulr right side grmmrs [LL77]. Regulr expressions form the right-hnd sides of productions, but not ll regulr expressions re llowed, only those tht re \unmbiguous" in the sense of Cluse of the stndrd. The intent of the stndrd is to mke it esier for humn to write regulr expressions tht cn be interpreted unmbiguously. To chieve this the stndrd requires ech regulr expression to be \unmbiguous" in the sense tht \n element : : : tht occurs in the document instnce must be

2 ble to stisfy only one primitive content token without looking hed in the document instnce." In other words, only such regulr expressions re vlid tht permit us to uniquely determine which ppernce of symbol in n expression should mtch symbol in n input word without looking beyond tht symbol in the input word. This requirement species exctly the clss of deterministic expressions tht we investigte here. An lterntive motivtion for our study is tht the theory of regulr lnguges hs become cornerstone in prcticl pplictions involving e.g. speciction, pttern mtching, nd the construction of scnners nd prsers. Perhps the most frequently occurring tsk is to construct, to speciction in the form of regulr expression, n utomton tht cn recognize the specied objects. Usully, one rst constructs non-deterministic nite utomton with -trnsitions (-NFA) in time liner in the size of E, elimintes the -trnsitions in qudrtic time, nd nlly converts the resulting NFA into deterministic nite utomton (DFA) [HU79]. The intermedite step cn be voided by directly constructing n -free utomton [BEGO71, ASU86]. It hs been climed [BS86] tht this NFA is the cnonicl representtion becuse it hs nturl connection with the derivtives [Brz64] of the originl expression. Since it tkes exponentil time in the worst cse to convert n NFA into DFA, it is nturl to sk for which regulr expressions E the cnonicl NFA M E is lredy deterministic. Such expressions re exctly wht we hve clled deterministic bove. It cn be tested in liner time whether regulr expression is deterministic, nd if so, the cnonicl deterministic utomton cn lso be constructed in liner time [Bru92]. In this pper, we rst give rigorous denition of deterministic regulr expressions. Then, we investigte the deterministic regulr lnguges, i.e. regulr lnguges tht cn be denoted by deterministic expression. As we will see, the deterministic regulr lnguges re proper subclss of the regulr lnguges; for exmple, for ech n 1, the expression (0 + 1) 0(0 + 1) n denotes regulr lnguge tht is not deterministic. First, we stte Kleene-like theorem for the clss of deterministic regulr lnguges. Next, we chrcterize the deterministic regulr lnguges in terms of the miniml deterministic utomt tht recognize them. To ech regulr lnguge L, the miniml deterministic nite utomton M L recognizing L is uniquely determined. We show tht deterministic regulr lnguges L cn be symbolized by structurl properties of M L. For stte q of M L, let the orbit O(q) of q denote the strongly connected component of q, i.e. the sttes of M L tht cn be reched from q nd vice vers. Some sttes of O(q), clled gtes, connect the orbit to the outside world. Now L is deterministic if nd only if ll orbits of M L dene deterministic regulr lnguges nd if for ech orbit O(q), ll gtes of O(q) hve identicl connections to the outside. Then we show tht ny deterministic regulr lnguge dened by DFA M with single orbit is of the form vnl, where L is deterministic nd vnl denotes the set of words w such tht vw is in L (the Brzozowski derivtive of L by v). Furthermore, miniml DFA recognizing L cn be constructed from M. Together, these results yield n lgorithm tht decides, given DFA M, whether its lnguge is deterministic nd, if so, constructs n equivlent deterministic expression. The decision lgorithm runs in time qudrtic in the size of M, but the corresponding deterministic expression cn be exponentil in the size of M. To give n exmple, for ech word w, the lnguge w of ll words over contining w s subword is deterministic regulr lnguge. Most proofs in this pper re just sketches. The complete proofs cn be found in the full version [BW91].

3 Figure 1 The Glushkov utomt corresponding to ( + b) + = ( 1 + b 2 ) 3 + nd (b ) = (b 1 2 ). 2 Deterministic regulr expressions The notion of symbol in word being stised or mtched by symbol in regulr expression hs been explined by number of uthors [BEGO71, ASU86, Hen68]. We prphrse here the description of Hennie [Hen68]. If word is denoted by n expression, it must be possible to spell out tht word by trcing n pproprite \pth" through the expression. If we indicte positions in expressions by subscripts, then the word bb is denoted by the expression (+b) + = ( 1 +b 2 ) 3 + becuse it corresponds to the pth tht strts t 1, visits b 2 twice, nd nlly rrives t 3. The set of subscripted symbols in n expression E is denoted by pos(e). Of course, the structure of the expression restricts the positions djcent symbols of word cn be mtched with. For instnce, if symbol in word is mtched by 3 in ( 1 + b 2 ) 3 +, then no further symbol of the word cn be mtched with symbol in the expression. These restrictions hve rst been formlized by Glushkov [Glu61]. This description suggests viewing regulr expression E s n utomton M E whose sttes correspond to the positions or occurrences of symbols in E nd whose trnsitions connect positions tht cn be consecutive on pth through E. We cll M E the Glushkov utomton of E. Figure 1 shows the two Glushkov utomt corresponding to the expressions (+b) + = ( 1 +b 2 ) 3 + nd (b ) = (b 1 2 ). In ddition to the sttes of M E tht correspond to positions in E, the true sttes, there is one unnmed stte in the Glushkov utomt of Figure 1 which cts s the initil stte. In generl, Glushkov utomt re non-deterministic, s (+b) + = ( 1 +b 2 ) 3 + illustrtes. After mtching n input symbol with 1, further cn either be mtched by 1 or 3. Thus, there is trnsition on from 1 to 1 nd to 3 in the Glushkov utomton. This exmple leds nturlly to precise denition of wht the SGML stndrd mens by deterministic expression. Denition 2.1 A regulr expression E is deterministic if M E is deterministic, i.e., if M E is DFA. A regulr lnguge is deterministic if there is some deterministic expression tht denotes it. Figure 1 illustrtes tht ( + b) + is not deterministic expression. Nevertheless, the lnguge denoted by ( + b) + is deterministic regulr lnguge, since it is lso denoted by (b ), which is deterministic expression. We dene M E inductively, rther thn in terms of the formlism introduced by Glushkov [Glu61] tht hs been used by number of uthors [BEGO71, ASU86, BS86].

4 M E hs the form M E = (Q E _[fq I g; ; E ; q I ; F E ), with Q E comprising the true sttes corresponding to positions in E, q I the new initil stte, the input lphbet, E : (Q E [ fq I g)?! 2 Q E the trnsition function, nd F E Q E [ fq I g the set of nl sttes. To simplify the discussion, we follow the generl convention tht regulr expressions re built from symbols in nd the empty string symbol, but not the empty set symbol ;. (Nevertheless, we consider the empty set to be deterministic regulr lnguge.) To identify two sttes in n utomton mens to replce them by new stte tht hs exctly the trnsitions tht both of the old ones hd. Denition 2.2 Given regulr expression E, we dene the construction of M E, illustrted in Figure 2, inductively s follows. [E = or ] M nd M re illustrted in Figure 2. [E = F + G] In M E the initil sttes of M F nd M G re identied. Let Q E = Q F _[ Q G ; (disjoint union, possibly fter renming sttes) F E = F F [ F G ; E (q; ) = 8 >< >: F (q; ) if q 2 Q F G (q; ) if q 2 Q G F (q I ; ) [ G (q I ; ) if q = q I : [E = F G] In M E copy of the initil stte of M G is identied with ech nl stte of M F. Let Q E = Q F _[ Q G ; (disjoint union, possibly fter renming sttes) ( FF [ (F F E = G n fq I g) if q I 2 F G, otherwise, F G E (q; ) = 8 >< >: F (q; ) if q 2 (Q F [ fq I g) n F F, F (q; ) [ G (q I ; ) if q 2 F F, G (q; ) if q 2 Q G. [E = F ] In M E ll trnsitions from q I in M F re dded to the nl sttes of M E. Let Q E = Q F ; F E = F F [ fq I g; ( F (q; ) [ E (q; ) = F (q I ; ) if q 2 F F, F (q; ) otherwise. Proposition 2.1 M E recognizes the lnguge denoted by E. Glushkov utomt hve some peculir structurl properties tht re worth investigting. First, the initil stte of M E hs no incoming trnsitions. This mkes the construction correct in the sense of Proposition 2.1. Furthermore, the sttes directly connected to the initil stte correspond exctly to the positions of E tht cn mtch the rst chrcter of word of E, nd the nl sttes in M E (besides s I ) correspond exctly to the ones tht mtch the lst chrcter of word of E. Thus, the initil stte hs trnsitions to rst positions in E, nd the nl sttes in M E (besides s I ) re nl positions of E. Second, for subexpression F of E, the structure of M F is retined in M E. Ech true stte of M F is lso true stte of M E, nd ll trnsitions between true sttes in M F belong lso to M E. Furthermore, mong ll positions of F, exctly the nl ones re nl sttes of M E or hve trnsitions in M E to positions outside of F. These conditions re even fullled uniformly, mening tht either ll or none of the nl positions of F re nl in E, nd tht either ll or

5 Figure 2 The inductive denition of M E.

6 none hve trnsition in M E on n 2 to position y of E outside of F. Hence, the nl positions of F re n interfce of M F to the surrounding prts of M E. Finlly, we will be especilly interested in mximl strred subexpressions of E, i.e. subexpressions of the form F tht re not subexpressions of nother strred subexpression G of E. For such n F, ny trnsitions in M E between positions of F re lredy trnsitions in M F. In this sense, M F is closed within M E. Proposition 2.2 Given regulr expression E, we cn decide if it is deterministic in time liner in the size of E. Proof The Glushkov utomton cn be constructed from E in such wy tht new trnsition is introduced t ech computtion step [Bru92]. As soon s we encounter trnsition tht mkes the utomton under construction non-deterministic, we stop nd report E to be non-deterministic. At this point, only time liner in the size of E hs been spent. 2 Book et l. [BEGO71] hve dened regulr expression E to be unmbiguous if the Glushkov utomton M E is unmbiguous, i.e. if for ech word w there is t most one computtion of M E tht ccepts w, or, equivlently, t most one pth through E tht spells out w. They hve shown tht ech regulr lnguge cn be denoted by n unmbiguous regulr expression. A deterministic expression E is n unmbiguous one where for ech word w the corresponding pth through E cn be computed incrementlly from w with just one symbol of look-hed. Thus, deterministic regulr expressions re relted to unmbiguous ones in the sme wy tht LL(1) grmmrs re relted to unmbiguous context-free grmmrs. This nlogy cn be mde precise: It is possible to trnslte regulr expression E in nturl wy into n equivlent context-free grmmr G E such tht E is deterministic if nd only if G E is LL(1). 3 The chrcteriztion theorem We rst stte without proof Kleene-like theorem for deterministic regulr lnguges. We then consider the cyclic structure of M E tht we cpture in terms of orbits. The structure of the orbits is essentilly preserved under minimiztion, nd, hence, orbits turn out to be exctly the right tool for chrcterizing deterministic regulr lnguges. We begin by dening three functions for lnguges tht cn lso be dpted to pply to regulr expressions. These functions re: rst(l), the set of symbols tht pper s the rst symbol of some word in L; lst(l), the set of symbols tht pper s the lst symbol of some word in L; nd followlst (L), the set of symbols tht follow prex of some word in L, where the prex is lso word in L. More formlly we hve: Denition 3.1 For L, let rst(l) = f 2 j w is in L for some word wg; lst(l) = f 2 j w is in L for some word wg; followlst (L) = f 2 j vw is in L; for some word v in L n fg nd some word wg: Theorem 3.1 stises the following conditions. 1. ;,, nd fg re in D. The deterministic regulr lnguges re the smllest clss D of lnguges tht 2. If A; B 2 D nd rst(a) \ rst(b) = ;, then A [ B 2 D. 3. If A; B 2 D, =2 A, nd followlst (A) \ rst(b) = ;, then AB 2 D.

7 4. If A 2 D, then A n fg 2 D. 5. If A 2 D nd followlst (A) \ rst(a) = ;, then A 2 D. It is well known tht, for ech regulr lnguge L, the minimum-stte deterministic utomton M L recognizing L is uniquely determined. We rgue tht, by exmining the cyclic structure of M L, we cn decide whether L is deterministic. Furthermore, for deterministic regulr lnguge L, we cn construct deterministic expression for L from M L. For ech DFA M = (Q; ; ; q 0 ; F ) recognizing L, the equivlence clss construction [ASU86] results in DFA M = (Q; ; ; [q 0 ]; F) which is isomorphic to M L. 1 For deterministic expression E denoting lnguge L, minimum-stte DFA M L cn be constructed directly from M E vi the equivlence clss construction. For non-deterministic expression E, however, the NFA M E hs rst to be converted to DFA vi the subset construction [ASU86]. Thus, we re looking for properties of M E tht re preserved under stte minimiztion, but not under subset construction. We strt from the structurl properties of M E noted in Section 2. Denition 3.2 Let M = (Q; ; ; q I ; F ) be n NFA. For q 2 Q, the strongly connected component of q, i.e. the sttes of M tht cn be reched from q nd from which q cn be reched s well, is clled the orbit of q nd denoted by O(q). We consider the orbit of q to be trivil if O(q) = fqg nd there re no trnsitions from q to itself in M. Denition 3.3 A stte q in n NFA M = (Q; ; ; q I ; F ) is clled gte of its orbit if either q is nl stte or there re q 0 2 Q n O(q) nd 2 with q?! q 0. The NFA M hs the orbit property if ech orbit of M is homogeneous with respect to its gtes, i.e. if, for ll gtes q 1 nd q 2 with O(q 1 ) = O(q 2 ), we hve: q 1 is nl stte if nd only if q 2 is nl stte. q 1?! q if nd only if q 2?! q, for ll q 2 Q n O(q 1 ) = Q n O(q 2 ) nd for ll 2. In Figure 3, both utomt hve three orbits, nmely f1g, f2; 3g, nd f4g. The singleton orbits re trivil, nd ech stte is gte of its orbit. The left utomton fullls the orbit property, the right one does not. Now we cn formulte necessry condition for regulr lnguge to be deterministic. Theorem 3.2 The miniml DFA M L recognizing deterministic regulr lnguge L hs the orbit property. This gives us our rst exmple of regulr lnguge tht is not deterministic. The rightmost utomton in Figure 3 is the miniml DFA for the lnguge denoted by ( + b) (c + bd), nd it does not hve the orbit property. Thus, this lnguge cnnot be deterministic. The proof of Theorem 3.2 is in two steps. First, we describe the orbits of the Glushkov NFA M E constructed in Denition 2.2 nd show tht M E hs the orbit property for ll regulr expressions. Next, we show tht the orbit property is preserved under stte minimiztion. Thus, if lnguge L is symbolized by deterministic regulr expression E, the miniml utomton M L = M E hs the orbit property. Lemm 3.3 Let E be regulr expression nd x 2 pos(e). If there is no strred subexpression F of E with x 2 pos(f ), then O(x) = fxg nd O(x) is trivil. On the other hnd, if F is the mximl strred subexpression of E with x 2 pos(f ), then O(x) = pos(f ), nd O(x) is not trivil. Finlly, the orbit of the initil stte q I is trivil. 1 Some minor techniclities re involved here, becuse the trnsition functions of M nd M re only prtilly dened. The detils re in the full version.

8 1 b 4 1 b 4 ( + b) 2 ( + b) (c + bd) 2 c Figure 3 Two NFAs, one fullls the orbit property, the other one does not. Lemm 3.4 Let E be regulr expression. Then, 1. M E hs the orbit property nd 2. if F is mximl strred subexpression of E with pos(f ) 6= ;, then the lst positions of F re the gtes of the orbit pos(f ). Figure 4 shows the Glushkov utomton for (bc ) = 1 (b 2 c 3 4. The gtes of orbit f2; 3; 4g re ) the sttes 3 nd 4, i.e. the lst position of the mximl strred subexpression (bc ) = (b 2 c 3 4). Now, consider deterministic utomton M nd its minimiztion M. If sttes p 1 ; : : : ; p n of M form n orbit in M, then the equivlence clsses [p 1 ]; : : : ; [p n ] belong to the sme orbit in M, which my, however, contin further elements. Figure 4 shows the Glushkov utomton M E for E = (bc ) = 1 (b 2 c 3 4) nd the miniml utomton M E. All sttes of M E besides stte 2 re equivlent, f1g is n orbit of M E, but [1] does not form complete orbit in M E. Nevertheless, the orbit of [1] in M E is completely generted by nother orbit of M E, nmely f2; 3; 4g. This is generl phenomenon, nmely, for ech orbit K of M, there is n orbit C of M tht fully genertes K, i.e. K = f[q] j q 2 Cg. C is clled lift of K. Now, if lifted orbit C is homogeneous with respect to its gtes, then so is K. Thus, we hve the following lemm, which concludes the proof of Theorem 3.2. Lemm 3.5 The orbit property is preserved under stte minimiztion. The orbit property hs gined us necessry condition for regulr lnguge to be deterministic. Another necessry condition evolves if we exmine the orbits themselves in isoltion. Denition 3.4 Let M be DFA. For q 2 Q M, let the orbit utomton M q of q, be the utomton obtined by restricting the stte set to O(q) with initil stte q nd nl sttes the gtes of O(q) in M. We sy the orbit of q is deterministic if L(M q ) is deterministic. A regulr lnguge L is sid to be n orbit lnguge if nd only if there is DFA M with single orbit tht recognizes L. Theorem 3.6 deterministic orbit lnguges. For ech deterministic lnguge L, the miniml DFA recognizing L hs only For the proof, gin we rst look t the orbit lnguges of Glushkov utomt nd then consider stte minimiztion. Let E be deterministic expression, nd let q be stte of M E with nontrivil orbit. Then, q is position of mximl strred subexpression F of E. Let L be the

9 Figure 4 The Glushkov utomton for the expression (bc ) = 1 (b 2 c 3 4 ) nd its minimiztion. lnguge of F, nd let v be word tht leds from the initil stte to q in M F. Since M F is closed within M E, the orbit lnguge of q in M E is lso the orbit lnguge of q in M F, which in turn is vnl := fw 2 j vw 2 Lg: The lnguge vnl is known s the derivtive of L by v [Brz64]. Thus, non-trivil orbit lnguge of M E is derivtive of lnguge denoted by mximl strred subexpression of E. The proof of the next proposition is in the full pper. Proposition 3.7 The derivtive of deterministic regulr lnguge is lso deterministic. As corollry, we hve: Lemm 3.8 Let E be deterministic regulr expression. Then, ll orbit lnguges of M E re deterministic regulr lnguges. Agin, this property is preserved under minimiztion, s cn be seen from the next lemm. Lemm 3.9 Let M be DFA nd M be its reduction. Then, for ech stte of M there is n equivlent stte q in M such tht 1. the orbit of q in M is lift of the orbit of [q] in M, nd 2. the orbit lnguges of q in M nd [q] in M re identicl. This concludes the proof of Theorem 3.6. The necessry conditions for miniml DFA to recognize deterministic regulr lnguge s given in Theorems 3.2 nd 3.6 re lso sucient: Theorem 3.10 Let L be regulr lnguge nd M be the miniml DFA recognizing L. Then, L is deterministic regulr lnguge if nd only if M hs the orbit property nd ll orbits of M re deterministic. Proof We show the impliction from right to left by induction on the number of orbits of M. Let M = (Q; ; ; q I ; F ) hve more thn one orbit. Furthermore, let q 1 ; : : : ; q n be the distinct sttes outside O(q I ) tht re rechble in one step from gte of O(q I ). All gtes of O(q I ) hve n i -trnsition to q i, nd no other outgoing trnsitions from gtes of O(q I ) to the outside exist. The i re pirwise distinct nd M qi hs no i -trnsition from nl stte. Let M i be the utomton whose sttes re the sttes of M tht re rechble from q i s the initil stte. Becuse M i hs fewer orbits thn M, M i is deterministic. Furthermore, L(M) = L(M qi )( 1 L(M 1 ) [ : : : [ n L(M n ))

10 Figure 5 An ; b-consistent DFA nd its ; b-cut. or L(M) = L(M qi )( 1 L(M 1 ) [ : : : [ n L(M n ) [ fg); nd deterministic expression for M cn be constructed from deterministic expressions for M qi nd M 1 ; : : : ; M n. 2 Theorem 3.10 is the rst step of decision lgorithm for deterministic regulr lnguges: Theorem 3.11 Given DFA M, we cn decide in time qudrtic in the size of M whether the lnguge of M is deterministic. If so, n equivlent deterministic expression cn be constructed. If miniml DFA M hs the orbit property, then its orbit utomt re lso miniml. This precludes to pply Theorem 3.10 directly for the orbit utomt. On the other hnd, we know lredy tht ech deterministic orbit lnguge of M hs the form vnl(f ), where F is deterministic expression. To conclude the proof of Theorem 3.11, we show how miniml DFA for L(F ) cn be constructed from M. Denition 3.5 A DFA M is -consistent, for 2, if there is stte f M () in M such tht ll nl sttes of M hve n -trnsition to f M (). Denition 3.6 Let M be i -consistent, for i 2, 1 i n, n 1. The 1 ; : : : ; n -cut M( 1 ; : : : ; n ) of M is constructed s follows. 1. A new stte q 0 is dded to M nd it is connected to f M ( i ) with n i -trnsition, for ll i. 2. All trnsitions with i from ech nl stte re removed from M. 3. Finlly, q 0 is mde initil nd nl. Figure 5 gives n exmple of n ; b-consistent DFA nd its ; b-cut. Theorem 3.12 Let M be miniml DFA recognizing lnguge L. Assume tht M consists of single, non-trivil orbit. Let 1 ; : : : ; n be the elements of for which M is consistent. Then, L is deterministic if nd only if 1. n L(M( 1 ; : : : ; n )) is deterministic. If L is deterministic nd 1 ; : : : ; n 2 re chosen s bove, we cn construct word v 2 L = L(M) = vnl(m( 1 ; : : : ; n )) ; with

11 Figure 6 The miniml DFA for (0 + 1) 0(0 + 1). nd deterministic expression for lnguge L cn be constructed from deterministic expression for lnguge L(M( 1 ; : : : ; n )). In lieu of proof, we illustrte Theorem 3.12 with two exmples. The lnguge recognized by the left utomton in Figure 5 is deterministic, becuse the ; b-cut is denoted by the deterministic regulr expression + b( + cc) +. One deterministic expression for the whole lnguge is c( + b( + cc)). Figure 6 shows the miniml DFA recognizing (0 + 1) 0(0 + 1). It consists of single orbit with two gtes, 00 nd 01, but is neither 0- nor 1-consistent. Thus, (0 + 1) 0(0 + 1) does not denote deterministic lnguge. The sme is true for (0 + 1) 0(0 + 1) n for ech n 1. References [ASU86] Alfred V. Aho, Rvi Sethi, nd Jerey D. Ullmn. Compilers: Principles, Techniques, nd Tools. Addison-Wesley Series in Computer Science, Addison-Wesley, Reding, Msschusetts, [BEGO71] Ronld Book, Shimon Even, Sheil Greibch, nd Gene Ott. Ambiguity in grphs nd expressions. IEEE Trnsctions on Computers, C-20(2):149{153, Februry [Bru92] Anne Bruggemnn-Klein. Regulr expressions into nite utomt. In Imre Simon, editor, Ltin '92, pges 87{98, Springer-Verlg, Berlin, Lecture Notes in Computer Scien0ce 583. [Brz64] Jnusz A. Brzozowski. Derivtives of regulr expressions. Journl of the ACM, 11(4):481{494, October [BS86] [BW91] [GH67] Gerrd Berry nd Rvi Sethi. From regulr expressions to deterministic utomt. Theoreticl Computer Science, 48:117{126, Anne Bruggemnn-Klein nd Derick Wood. On the expressive power of SGML document grmmrs. In preprtion, Seymour Ginsburg nd Michel M. Hrrison. Brcketed context-free lnguges. Journl of Computer nd System Sciences, 1(1):1{23, Mrch [Glu61] V.M. Glushkov. The bstrct theory of utomt. Russin Mthemticl Surveys, 16:1{53, [Hen68] Frederick C. Hennie. Finite-Stte Models for Logicl Mchines. John Wiley, New York, [HU79] John E. Hopcroft nd Jerey D. Ullmn. Introduction to Automt Theory, Lnguges nd Computtion. Addison-Wesley Series in Computer Science, Addison-Wesley, Reding, Msschusetts, 1979.

12 [ISO86] ISO Informtion processing text nd oce systems stndrd generlized mrkup lnguge (SGML). October Interntionl Orgniztion for Stndrdiztion. [LL77] Wilf R. LLonde. Regulr right prt grmmrs nd their prsers. Communictions of the ACM, 20(10):731{741, October 1977.