I F I G R e s e a r c h R e p o r t. Minimization, Characterizations, and Nondeterminism for Biautomata. IFIG Research Report 1301 April 2013

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1 I F I G R e s e r c h R e p o r t Institut für Informtik Minimiztion, Chrcteriztions, nd Nondeterminism for Biutomt Mrkus Holzer Sestin Jkoi IFIG Reserch Report 1301 April 2013 Institut für Informtik JLU Gießen Arndtstrße Giessen, Germny Tel: Fx: mil@informtik.uni-giessen.de

2 IFIG Reserch Report IFIG Reserch Report 1301, April 2013 Minimiztion, Chrcteriztions, nd Nondeterminism for Biutomt Mrkus Holzer 1 nd Sestin Jkoi 2 Institut für Informtik, Universität Giessen Arndtstrße 2, Giessen, Germny Astrct. We show how to minimize iutomt with Brzozowski-like lgorithm y pplying reversl nd powerset construction twice. Biutomt were recently introduced in [O. Klím, L. Polák: On iutomt. RAIRO Theor. Inf. Appl., 46(4), 2012] s generliztion of ordinry finite utomt, reding the input from oth sides. The correctness of the Brzozowski-like minimiztion lgorithm needs little it more rgumenttion thn for ordinry finite utomt since for iutomton its dul or reverse utomton, uilt y reversing ll trnsitions, does not necessrily ccept the reversl of the originl lnguge. To this end we first generlize the notion of iutomt to del with nondeterminism nd moreover, to tke structurl properties of the forwrd- nd ckwrd-trnsition of the utomton into ccount. This results in vriety of iutomt models, which ccepting power is chrcterized. As yproduct we give simple structurl chrcteriztion of cyclic regulr nd commuttive regulr lnguges in terms of deterministic iutomt. Ctegories nd Suject Descriptors: F.1.1 [Computtion y Astrct Devices]: Models of Computtion Automt; F.2.2 [Anlysis of Algorithms nd Prolem Complexity]: Nonnumericl Algorithms nd Prolems Computtions on discrete structures; F.4.3 [Mthemticl Logic nd Forml Lnguges]: Forml Lnguges Clsses defined y grmmrs or utomt; Additionl Key Words nd Phrses: deterministic nd nondeterministic iutomt, cyclic lnguges, commuttive lnguges, Brzozowski s minimiztion lgorithm. 1 E-mil: holzer@informtik.uni-giessen.de 2 E-mil: jkoi@informtik.uni-giessen.de Copyright c 2013 y the uthors

3 1 Introduction Biutomt were recently introduced in [5] s generliztion of ordinry deterministic finite utomt. Simply speking, iutomton is device consisting of deterministic finite control, red-only input tpe, nd two reding heds, one reding the input from left to right, nd the other hed reding the input from right to left. Similr two-hed finite utomt models were introduced, e.g., in [3, 6, 8]. An input word is ccepted y iutomton, if there is n ccepting computtion strting the heds on the two ends of the word meeting somewhere in n ccepting stte. Although the choice of reding symol y either hed is nondeterministic, the determinism of the iutomton is enforced y two properties: (i) The heds red input symols independently, i.e., if one hed reds symol nd the other reds nother, the resulting stte does not depend on the order in which the heds red these single letters. (ii) If in stte of the finite control one hed ccepts symol, then this letter is ccepted in this stte y the other hed s well. Lter we cll the former property the -property nd the ltter one the F-property. In [5] it ws shown tht iutomt shre lot of properties with ordinry deterministic finite utomt. For instnce, there is unique (up to isomorphism) miniml deterministic iutomton for every regulr lnguge, which oeys nice description in terms of two-sided derivtives or quotients cf. Brzozowski s construction for ordinry miniml deterministic finite utomt [1]. Moreover, simple structurl chrcteriztions sed on iutomt for lnguge fmilies such s the piecewise testle or prefix-suffix testle lnguges were given in [5]. Recently in [4] lso descriptionl complexity issues for iutomt were ddressed. This is the strting point for our investigtion. We focus on the minimiztion prolem for iutomt. For ordinry deterministic finite utomt minimiztion is efficiently solvle. While the lgorithm with the est running time of O(n log n) remins difficult to understnd, the most elegnt one is tht of Brzozowski [2], which minimizes n ordinry finite utomton A, regrdless whether it is deterministic or nondeterministic, y pplying the reversl nd powerset construction twice in sequence. Thus, it computes the utomton P([P(A R )] R ), to otin n equivlent miniml deterministic finite utomton here the superscript R refers to the reversl or dul opertion on utomt nd P denotes the powerset construction. Whether this elegnt minimiztion method cn lso e pplied to iutomton A is not completely cler, since the ove mentioned two properties to enforce determinism my e lost y computing A R or P(A R ). To this end we introduce nondeterministic iutomt. It is known tht these mchines lredy ccept non-regulr lnguges nd chrcterize the fmily of liner context-free lnguges [6], ut s side result we prove tht nondeterministic iutomt with the -property ccept regulr lnguges only. In the min line of reserch we show tht Brzozowski-like minimiztion of iutomt with - nd F-property, regrdless whether they re deterministic or nondeterministic, is possile. Since in Brzozowski s minimiztion the powerset construction is used this technique is exponentil. Note tht it is esy to see tht there re more efficient minimiztion lgorithms for deterministic iutomt with oth the - nd F-property, 2

4 y simply dpting other existing minimiztion lgorithms for ordinry deterministic finite utomt. As yproduct of our investigtions, we give simple structurl chrcteriztions of cyclic regulr lnguges nd commuttive regulr lnguges in terms of deterministic iutomt with - nd F-property. The pper is orgnized s follows: In the next section we introduce the necessry nottion on iutomt. In ddition we lso define n nlogous to the F-property, clled I-property, which will tke cre on symols tht re red from n initil stte. Then in Section 3 we show some sic properties on these devices. In prticulr we show tht nondeterministic iutomt with the - property ccept regulr lnguges only. Moreover there we lso give structurl iutomt chrcteriztions of the fmilies of cyclic nd of commuttive lnguges. In Section 4 we prove the sics on dul utomt, which will then e used in the ultimte section to prove the correctness of the Brzozowski-like minimiztion for deterministic iutomt with - nd F-property. 2 Preliminries We use more generl notion of iutomt thn in [5], ut it resemles tht of nondeterministic liner utomt s defined in [6], which chrcterize the fmily of liner context-free lnguges. A nondeterministic iutomton is sixtuple A = (Q, Σ,,, I, F), where Q is finite set of sttes, Σ is n lphet, : Q Σ 2 Q is the forwrd trnsition function, : Q Σ 2 Q is the ckwrd trnsition function, I Q is the set of initil sttes, nd F Q is the set of finl or ccepting sttes. The trnsition functions nd re extended to words in the following wy, for ll words v Σ nd letters Σ: q λ = {q}, q v = p v, nd q λ = {q}, q v = p v, p (q ) p (q ) nd further, oth nd cn e extended to sets of sttes S Q, nd w Σ y S w = p S p w, nd S w = p S p w. The iutomton A ccepts w Σ, if the word w cn e written s w = u 1 u 2...u k v k...v 2 v 1, for some u i, v i Σ with 1 i k, such tht [((...((((I u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k ] F. (1) The lnguge ccepted y A is defined s L(A) = { w Σ A ccepts w }. A iutomton A is deterministic, if I = 1, nd q = q = 1 for ll sttes q Q nd letters Σ. In this cse we simply write q = p, or q = p insted of q = {p}, or q = {p}, respectively, treting nd to e functions mpping Q Σ to Q. The utomton A hs the confluence or dimond property, for short -property, if (q ) = (q ), for every stte q Q nd, Σ. Further, A hs the equl cceptnce property, for short F-property, if q F if nd only if q F, for every stte q Q nd letter Σ. A deterministic iutomton tht hs oth the - nd the F-property is exctly wht is clled iutomton in [5]. Finlly, A hs the equl initil fn-out property, for short I-property, if I = I, for every letter Σ. Two iutomt A nd B re equivlent if they ccept the sme lnguge, which mens L(A) = L(B) 3

5 0 c 1 2 c c c 6 Fig. 1. A nondeterministic iutomton A, tht hs oth the - nd the F-property, ut not the I-property. holds. Further, we need some nottion on lnguges ssocited with sttes of iutomt. For iutomton A = (Q, Σ,,, I, F) nd stte q Q let qa = (Q, Σ,,, {q}, F) nd A q = (Q, Σ,,, I, {q}). We sy tht L( q A) is the right lnguge of stte q nd tht L(A q ) is the left lnguge of stte q. Two sttes p, q Q re equivlent, if nd only if L( p A) = L( p A). We illustrte these definitions y the following exmple. Exmple 1. Consider the nondeterministic iutomton A = (Q, Σ,,, I, F) with Q = {0, 1,...,6}, Σ = {,, c}, I = {0}, F = {6}, nd whose trnsition functions, nd re depicted in Figure 1 solid rrows denote forwrd trnsitions y, nd dshed rrows denote ckwrd trnsitions y. One cn check, tht A hs the -property, i.e., tht (q d) e = (q e) d, for ll inputs d, e Σ nd sttes q Q. For exmple we hve (0 ) c = {0, 1} c = {2, 4}, nd (0 c) = {2} = {2, 4}. Further, A hs the F-property, i.e., for ll sttes q Q, nd inputs d Σ we hve (q d) F if nd only if (q d) F. For exmple oth sets 1 = {3}, nd 1 = hve n empty intersection with F, nd the two sets 5 = {5, 6} nd 5 = {5, 6} oth contin the ccepting stte 6. However, the iutomton A does not hve the I-property, ecuse {0} = {0, 1} = {0}. If we removed the ckwrd trnsition loop on letter in stte 5, i.e., if 5 = {6}, insted of 5 = {5, 6}, then A would not hve the -property nymore (ecuse then (5 ) (5 ) ), ut it would still hve the F-property. One oserves tht the right lnguge of stte 2 is L( 2 A) =, the left lnguge of stte 2 is L(A 2 ) = c, nd the lnguge ccepted y A is L(A) = c. We will show in Section 3, tht iutomton with oth the -property, nd the F-property ccepts word w if nd only if reding w leds from some initil stte to finl stte, while only using forwrd trnsitions. With tht result, L(A) cn e esily determined in this exmple. Next we generlize the well known powerset construction of ordinry finite utomt to iutomt. For iutomton A = (Q, Σ,,, I, F), its powerset utomton is the deterministic iutomton P(A) = (Q, Σ,,, q 0, F ), where the stte set Q 2 Q consists of ll sttes tht re rechle from the initil 4

6 stte q 0 = I, the set of ccepting sttes is F = { P Q P F }, nd the forwrd nd ckwrd trnsition functions re defined s P = p, nd P = p, p P for every stte P Q nd letter Σ. To prove the correctness of this construction, we use the following simple fct: if A = (Q, Σ,,, I, F) is nondeterministic iutomton tht hs - property, then (S ) = (S ), nd if A hs the F-property, then (S ) F if nd only if (S ) F, for every S Q nd, Σ. Now we prove the following result. Lemm 2. If A is nondeterministic iutomton, then P(A) is equivlent to A, i.e., L(A) = L(P(A)). Furthermore, for ll X {, F, I}, if A hs the X-property, then the deterministic iutomton P(A) hs the X-property, too. Proof. Let A = (Q, Σ,,, I, F) e iutomton, nd let its powerset iutomton e B = P(A) = (Q, Σ,,, q 0, F ). For w Σ, we hve w L(A) if nd only if w = u 1 u 2...u k v k...v 2 v 1, with u i, v i Σ, 1 i k, nd p P [((...((((I u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k ] F, nd this in turn is equivlent to [((...((((I u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k ] F, which holds if nd only if w L(B). Thus, L(A) = L(B). Since the trnsition functions, nd of B re just the extensions of the functions, nd of A to sets of sttes, the -property, the F-property, nd the I-property re preserved y the powerset construction. We illustrte the construction in the following exmple. Exmple 3. Consider the iutomton A from Exmple 1, which is depicted in Figure 1. The powerset iutomton P(A), which is deterministic iutomton, is shown in Figure 2. Note tht since the iutomton A hs oth the -property, nd the F-property, lso the powerset iutomton P(A) hs oth these properties. 3 Bsic Properties of Biutomt In this section we study the effect of the previously defined properties of iutomt. On the one hnd, we lredy know tht the most generl model of iutomt, nmely nondeterministic iutomt without ny restrictions, chrcterizes the fmily of liner context-free lnguges [6], while on the other hnd, the most restricted iutomton model, tht is, deterministic iutomt with the - nd F-property, descries the fmily of regulr lnguges [5]. But wht else cn e sid out the ccepting power of these devices? To this end, we first tke closer look on the -property. At first glnce, we show tht it lso extends to words. Recll, tht if iutomton A with stte set Q hs the -property, then (S ) = (S ), for every suset S Q nd, Σ. 5

7 c c c c Fig. 2. The powerset iutomton P(A) for the nondeterministic iutomton A from Figure 1. The sink stte, nd trnsitions leding to it re not shown. Lemm 4. Let A = (Q, Σ,,, I, F) e nondeterministic iutomton with the -property. Then (S u) v = (S v) u, for every S Q nd u, v Σ. Proof. First we prove tht (S ) v = (S v) holds for every S Q, word w Σ, nd Σ y induction on the length of v. For v = 0, tht is, for v = λ, we hve (S ) v = S = (S v). Now let v 1. Assume tht word v cn e written s v = v, for some Σ nd v Σ. Then (S ) v = (S ) v = ((S ) ) v = ((S ) ) v. Now we cn use the inductive ssumption on v nd otin ((S ) ) v = ((S ) v ) = (S v ) = (S v). Thus, we hve shown (S ) v = (S v), for every S Q, Σ, nd v Σ. Now we cn prove the sttement of the lemm y performing induction on the length of the word u. The induction se strts with u = 0, tht is, u = λ. There we hve (S u) v = S v = (S v) u. Now let u 1. Assume tht u writes s u = u, for some Σ nd u Σ. Then (S u) v = (S u ) v = ((S ) u ) v, nd y using first the inductive ssumption on u, nd then the sttement from ove, we otin ((S ) u ) v = ((S ) v) u = ((S v) ) u = (S v) u = (S v) u, which proves the sttement of the lemm. By itertively using Lemm 4, it follows tht for iutomt with the - property the ccepting condition shown in Eqution (1) is equivlent to the condition [(I u 1 u 2... u k ) v k...v 2 v 1 ] F, i.e., such iutomton ccepts word w Σ if nd only if [(I u) v] F, for some words u, v Σ with w = uv. The cceptnce condition ecomes even simpler, if the iutomton dditionlly hs the F-property, which results from the following lemm. 6

8 Lemm 5. Let A = (Q, Σ,,, I, F) e nondeterministic iutomton with oth the -property, nd the F-property. Then [(S uv) w] F if nd only if [(S u) vw] F, for every S Q nd u, v, w Σ. Proof. We prove the sttement y induction on the length of v. Note tht the sttement holds for v = 0, since (S uv) w = (S u) vw, for v = λ. Now let v 1. In this cse the word v cn e written s v = v, for some v Σ nd letter Σ. Then y Lemm 4 it follows (S uv) w = ((S uv ) ) w = ((S uv ) w). Thus, [(S uv) w] F if nd only if [((S uv ) w) ] F. Since A hs the F-property, the ltter holds if nd only if [((S uv ) w) ] F. Oserve, tht [((S uv ) w) ] is equl to [(S uv ) w]. Now we use the inductive ssumption on v, nd see tht [(S uv ) w] F if nd only if [(S u) v w] F, which in turn is equivlent to [(S u) vw] F, nd therefore concludes the proof. By itertively using Lemm 5, it follows tht iutomton with oth the - property nd the F-property ccepts word w Σ if nd only if [I w] F, or equivlently, [I w] F. We summrize this in the following corollry. Corollry 6. If A is nondeterministic iutomton with oth the -property, nd the F-property, then L(A) = { w Σ [I w] F }. We cn pply Corollry 6 to the iutomton A from Exmple 1, nd esily see tht L(A) = c, y only considering the forwrd trnsitions. From Corollry 6, one cn see tht L(A) is regulr lnguge, if A hs oth the -property, nd the F-property. But the F-property is not essentil here, since lredy the -property lone gurntees the regulrity of the lnguge L(A), s we show in the following theorem. Theorem 7. Let A e nondeterministic iutomton with the -property. Then L(A) is regulr lnguge. Proof. Let A = (Q, Σ,,, I, F) e iutomton with the -property. Lemm 4 implies, tht A ccepts word w Σ if nd only if there re words u, v Σ, with w = uv, nd [(I u) v] F. This mens tht there re q 0, q, q f Q, such tht q 0 I, q q 0 u, q f q v, nd q f F. Thus, the lnguge L(A) cn e descried y L(A) = { u q I u } {v q v F }, q Q nd it remins to show tht oth sets L 1 (q) = { u q I u }, nd moreover L 2 (q) = { v q v F } re regulr, for every q Q. The lnguge L 1 (q) consists of ll the words tht, when red forwrd, led from some initil stte to stte q, nd lnguge L 2 consists of the words tht, when red ckwrds, led from stte q to some ccepting stte. Thus, lnguge L 1 (q) is ccepted y the nondeterministic finite utomton A 1 = (Q, Σ, δ 1, I, {q}) with δ 1 (p, ) = p, 7

9 for every p Q nd Σ this shows tht L 1 (q) is regulr lnguge. Further, the lnguge L 2 (q) R is ccepted y the nondeterministic finite utomton A 2 = (Q, Σ, δ 2, {q}, F), with δ 2 (p, ) = p, for every p Q nd Σ since regulr lnguges re closed under reversl, the set L 2 (q) is regulr lnguge, too. Since regulr lnguges re closed under conctention nd union, the proof is complete. The following exmple shows tht the lnguge ccepted y iutomton without the -property my lredy e non-regulr. Exmple 8. Let us consider the deterministic iutomton A, which is defined s A = ({0, 1, 2}, {, },,, {0}, {0}), with the trnsition functions 0 = 1, 1 = 0, nd ll other trnsitions go to the sink stte 2. The iutomton A is depicted in Figure 3, where the sink stte 2, nd ll trnsitions leding to it re not shown. This iutomton does not hve the -property, ecuse (0 ) = 0, while (0 ) = 2. The lnguge ccepted y A is L(A) = { n n n 0 }, which is well known to e liner context free ut not regulr. 0 1 Fig. 3. A deterministic iutomton without the -property, tht ccepts non-regulr lnguge. The sink stte 2, nd trnsitions leding to it re not shown. Of course there re lso iutomt tht ccept regulr lnguges, lthough they my e missing the -property. From descriptionl complexity point of view, iutomt without the -property llow more succinct representtion of regulr lnguge, when compred to iutomt with this property, or compred to deterministic finite utomt. The following exmple presents smll deterministic iutomton, such tht ny equivlent deterministic iutomton with oth the -property, nd the F-property, s well s ny equivlent deterministic finite utomton is t lest of exponentil size. Exmple 9. Consider the regulr lnguge L = (+) n (+) (+) n, which is ccepted y the deterministic iutomton A, tht is depicted in Figure 4. The iutomton A does neither hve the -property, nor the F-property.,, 0..., n 0..., n f Fig. 4. A liner-size deterministic iutomton B for the lnguge L. Undefined trnsitions led to non-ccepting trp stte, which is not shown here.,, 8

10 Further, A hs O(n) sttes, ut every equivlent deterministic finite utomton ccepting L needs Ω(2 n ) sttes. Since the miniml deterministic finite utomton is contined in the miniml deterministic iutomton with oth the - property, nd the F-property, lso every such iutomton needs Ω(2 n ) sttes. But if we stin from the F-property, the lnguge L cn lso e ccepted y deterministic iutomton B with the -property with O(n 2 ) sttes. The sttes of B re pirs (i, j), with i, j {0, 1,..., n, f}, nd dditionlly, sink stte s, the initil stte is (0, 0), the only ccepting stte is (f, f), nd the trnsition functions B, nd B re defined s follows, for ll i, j {0, 1,...,n, f}: { (i + 1, j) if i / {n, f}, (i, j) B = (f, j) if i {n, f}, (i + 1, j) if i / {n, f}, (i, j) B = s if i = n, (f, j) if i = f, { (i, j + 1) if j / {n, f}, (i, j) B = (i, f) if j {n, f}, (i, j + 1) if j / {n, f}, (i, j) B = s if j = n, (i, f) if j = f, nd the sink stte s goes to itself on every trnsition. This utomton counts the numer of symols it hs red from the left in the first component of the stte, nd the numer of symols it hs red from the right in the second component of the stte. If counter reches n, then the next symol (in the corresponding reding direction) must e n. Since the trnsition function only opertes on the first component of stte, nd the function only opertes on the second component of stte, one cn see tht B hs the -property. But since (f, n) B = (f, n) (f, f) = (f, n) B, the F-property is not present in B. This exmple shows tht, from descriptionl complexity point of view, it is expensive to trnsform iutomton tht does not hve oth the -property, nd the F-property into iutomton tht hs oth these two properties. It remins to discuss the I-property. If we consider iutomt with the - property, nd the I-property, then switching from to when reding suword induces circulr shift on the word, which cn e seen in the following lemm. Lemm 10. Let A = (Q, Σ,,, I, F) e nondeterministic iutomton with oth the -property, nd the I-property. Then (I uv) w = (I v) wu, for every u, v, w Σ. Proof. We use induction on the length of u. For u = 0, tht is, in cse u = λ we hve (I uv) w = (I v) w = (I v) wu. Now let u 1 nd ssume tht u = u, for some Σ nd u Σ. By Lemm 4, nd the I-property, we hve (I uv) w = ((I ) u v) w = ((I ) u v) w = ((I u v) ) w = (I u v) w, nd then the proof cn e concluded y using the inductive ssumption on u to otin (I u v) w = (I v) wu = (I v) wu, which concludes the proof. 9

11 Lemm 10 implies tht the left lnguge of every stte q, i.e., the set of words leding to q, in iutomton with oth the -property, nd the I-property is cyclic. Here lnguge L Σ is cyclic if nd only if L = (L), where (L) = { vu Σ uv L }. In prticulr, this implies the following result. Corollry 11. Let A e nondeterministic iutomton with oth the - nd the I-property, then L(A) is regulr cyclic lnguge. In fct, for the cnonicl iutomton, which is the miniml deterministic iutomton tht hs oth the -property, nd the F-property [5], lso the converse impliction holds. For regulr lnguge L Σ we define the cnonicl iutomton A L = (Q L, Σ, L, L, I L, F L ) with Q L = { u 1 Lv 1 u, v Σ }, initil sttes I L = {L}, finl sttes F L = { u 1 Lv 1 λ u 1 Lv 1 }, nd q L = 1 q nd q L = q 1, where. u 1 Lv 1 = { w Σ uwv L }, for u, v Σ. Thus we otin the following chrcteriztion of regulr cyclic lnguges. Theorem 12. A regulr lnguge is cyclic if nd only if its cnonicl iutomton hs the I-property. Proof. If A is iutomton, not necessrily the cnonicl one, ut with the I-property, then we know from Corollry 11, tht L(A) is regulr cyclic lnguge. For the converse impliction, let L Σ e regulr cyclic lnguge, nd let A L = (Q, Σ,,, {q 0 }, F) e its cnonicl iutomton with q 0 = L. Since L is cyclic, for every word v Σ nd Σ, we hve v L if nd only if v L. Thus, for every Σ, we otin q 0 = 1 L = { v Σ v L } which is equl to the set { v Σ v L } = L 1 = q 0, so A hs the I-property. This proves the stted clim. We cn lso chrcterize commuttive regulr lnguges y the structure of their cnonicl iutomton. A regulr lnguge L Σ is commuttive if for ll words u, v Σ nd symols, Σ we hve uv L if nd only if uv L. One cn see y induction tht this condition is equivlent to the condition tht for ll words u, v, x, y Σ we hve uxyv L if nd only if uyxv L. Theorem 13. Let L Σ e regulr lnguge nd A = (Q, Σ,,, I, F) its cnonicl iutomton. Then L is commuttive if nd only if q = q, for every q Q nd Σ. Proof. Let L Σ e regulr lnguge, nd let A = (Q, Σ,,, I, F) e the cnonicl iutomton of L, with q = q, for ll q Q nd Σ. Then, y Corollry 11, ll the right lnguges L( q A) of sttes q Q re cyclic, since q A hs the I-property. Since the right lnguges of sttes of the cnonicl iutomton re quotients u 1 Lv 1, for u, v Σ, ll these quotients re cyclic. Thus, for ll u, v, x, y Σ we hve xy u 1 Lv 1 if nd only if yx u 1 Lv 1. It follows tht L is commuttive. For the converse, ssume L is commuttive, nd consider symol Σ, nd stte q Q tht corresponds to quotient u 1 Lv 1, for some u, v Σ. 10

12 Then we hve q = [u 1 Lv 1 ] = (u) 1 Lv 1 = { x Σ uxv L }, q = [u 1 Lv 1 ] = u 1 L(v) 1 = { x Σ uxv L }, nd since L is commuttive, it follows q = q. Note tht the condition q = q together with the -property in prticulr implies tht (q ) = (q ) holds for the forwrd trnsition function of the cnonicl iutomton. This nicely shows the connection to commuttive finite utomt [7], where δ(q, ) = δ(q, ) holds for the trnsition function δ of the finite utomton. 4 The Dul of Biutomton For clssicl finite utomt, n utomton for the reversl of the ccepted lnguge cn e otined y constructing the reversl, or dul utomton, i.e., y reversing the trnsitions, nd interchnging initil nd finl sttes. For iutomt, one otins n utomton for the reversl of the lnguge y simply interchnging the trnsition functions nd. Nevertheless, it is interesting to see wht hppens, if we pply similr construction s for finite utomt to iutomt. We will see in Section 5, tht similr to finite utomt, the dul of iutomton cn e used to construct miniml iutomton. Now let us define the reversl, or dul of the iutomton A = (Q, Σ,,, I, F) s the iutomton A R = (Q, Σ, R, R, F, I), tht is otined from A y interchnging the initil nd finl sttes, nd y reversing ll trnsitions, such tht p q R if nd only if q p, nd p q R if nd only if q p. Note tht (A R ) R = A. Lemm 14. Let A = (Q, Σ,,, I, F) e nondeterministic iutomton, nd let A R = (Q, Σ, R, R, F, I) e the dul of A. Then for every sttes p, q Q nd words u, v Σ, we hve q (p u) v if nd only if p (q R v R ) R u R, nd q (p v) u if nd only if p (q R u R ) R v R. Proof. We prove the first prt of the sttement, nmely q (p u) v if nd only if p (q R v R ) R u R, y induction on uv. The second prt of the sttement then follows y symmetric rgumenttion, since (A R ) R = A. For uv = 0, i.e., for u = v = λ, we hve (p u) v = {p} = (p R v R ) R u R, so the sttement holds in this cse. Now let uv 1, then we hve uv = u v, with u, v Σ, nd Σ, such tht either u = u, or v = v. First consider the cse u = u. Then we hve q (p u) v if nd only if there re sttes p 1, p 2 Q, such tht p 1 p u, p 2 p 1, nd q p 2 v. Now we cn pply the inductive ssumption on the words u, nd v, since oth re shorter thn uv = u v, nd see tht p 1 p u holds if nd only if p p 1 R u R, nd q p 2 v holds if nd only if p 2 q R v R. Further, y definition of R, we hve p 2 p 1 if nd only if p 1 p 2 R. Putting ll this together, we hve q (p u) v if nd only if there re sttes p 1, p 2 Q, such tht p 2 q R v R, p 1 p 2 R, nd p p 1 R u R, i.e., if nd only if p (q R v R ) R u R. The other cse, v = v cn e shown similrly: q (p u) v = ((p u) v ) if nd only if there is stte p 1 Q, 11

13 such tht p 1 (p u) v, nd q p 1. By the inductive ssumption, we hve p 1 (p u) v if nd only if p (p 1 R v R ) R u R, nd y the definition of R, we hve q p 1 if nd only if p 1 q R R. Thus, we hve q (p u) v if nd only if p ((q R R ) R v R ) R u R = (q R v R ) R u R. Note tht if A hs the -property, then the sttement of Lemm 14 cn e simplified to q (p u) v if nd only if p (q R u R ) R v R. Lemm 15. Let A e nondeterministic iutomton, then the following holds: 1. A hs the -property if nd only if A R hs the -property. 2. A hs the F-property if nd only if A R hs the I-property. 3. A hs the I-property if nd only if A R hs the F-property. Proof. Let A = (Q, Σ,,, I, F) e iutomton, nd A R = (Q, Σ, R, R, F, I) e its dul. Assume A hs the -property, i.e., (q ) = (q ), for every q Q nd, Σ. To see tht A R hs the -property, consider stte q Q, nd note tht y Lemm 14 we hve (q R ) R = { p Q q (p ) }. Since A hs the -property, we know tht { p Q q (p ) } = { p Q q (p ) }, nd y using Lemm 14 gin, we otin { p Q q (p ) } = (q R ) R. Thus, we hve shown (q R) R = (q R ) R, for every q Q nd, Σ. We hve shown tht the -property of A implies the -property of A R. The reverse impliction immeditely follows, since (A R ) R = A, so the first sttement is proven. Now we show tht the F-property of A implies the I-property of A R. If A hs the F-property, then { q Q (q ) F } = {q Q (q ) F }, for every Σ. Since y Lemm 14 we hve F R = f R = { q Q f q } = { q Q (q ) F } f F nd F R = f F f F f R = { q Q f q } = { q Q (q ) F }, f F the dul A R hs the I-property note tht F is the set of initil sttes of A R. For the reverse impliction note tht (q ) F if nd only if q F R, nd if A R hs the I-property, then q F R if nd only if q F R, nd the ltter gin holds if nd only if (q ) F. Thus, the second sttement is proven. The finl sttement now follows from the fct tht (A R ) R = A. If we hve iutomton A with oth the - nd the F-property, then Lemm 15 implies tht the dul of the iutomton A hs the - nd the I- property. Further, Lemm 10 implies, tht the lnguge ccepted y the dul is cyclic. In fct, we cn show the following result. Corollry 16. Let A = (Q, Σ,,, I, F) e nondeterministic iutomton tht hs oth the - nd the F-property, nd let A R = (Q, Σ, R, R, F, I) e the dul of A. Then L(A R ) = (L(A) R ), i.e., utomton A R ccepts the cyclic shift of the reversl of L(A). 12

14 Proof. Let w L(A), then y Corollry 6, we hve (I w) F, i.e., there is stte p I, nd stte q F, such tht q p w. By Lemm 14, this is equivlent to p q R w R, which mens tht w R is ccepted y A R. Since y Corollry 11, the lnguge L(A R ) is cyclic, we know tht (w R ) L(A R ), for every w L(A), so (L(A) R ) L(A R ). For the other inclusion let w L(A R ), then we know tht there re words u, v Σ, with uv = w, such tht q 0 (q f R u) R v, for some q f F, nd q 0 I. Lemm 10 implies tht lso q 0 q f R vu, which is equivlent to q f q 0 R (vu) R, y Lemm 14. This mens tht (vu) R is ccepted y A, so vu L(A) R. Since w = uv (vu), we hve w (L(A) R ). Wht cn e sid out L(A R ), if A is iutomton with oth the - nd the I-property? Unfortuntely, the lnguge cnnot e identified y some opertion on the lnguge L(A), ecuse regulr cyclic lnguge cn e ccepted y structurlly different iutomt tht hve the - nd the I-property. From these structurl differences, lso different dul utomt, tht ccept different lnguges, cn result, s the following exmple shows. Exmple 17. Consider the cyclic lnguge L = {, }, which is ccepted y ll the three iutomt A 1, A 2, nd A 3, tht re depicted in Figure 5. Note tht Fig. 5. Three different nondeterministic iutomt A 1 (left), A 2 (middle), nd A 3 (right), ll ccepting the lnguge {, }. these three utomt hve oth the - nd the I-property. The corresponding dul utomt A R 1, AR 2, nd AR 3 re depicted in Figure 6. Note tht these three utomt hve oth the - nd the F-property. Further note, tht the lnguges L(A R 1 ) = {, }, L(AR 2 ) = {}, nd L(A 3) R = {} ccepted y the dul utomt re pirwise distinct. 5 Brzozowski-Like Minimiztion for Biutomt An interesting lgorithm for minimizing deterministic finite utomt is tht of Brzozowski [2]: given (deterministic or nondeterministic) finite utomton A, it computes P([P(A R )] R ), which turns out to e the miniml deterministic finite utomton for L(A). While the minimlity of the constructed utomton needs some rgumenttion, the fcts tht it ccepts the correct lnguge, nd tht it is deterministic finite utomton re esy to see, since the dul B R 13

15 Fig. 6. The dul utomt A R 1 (left), A R 2 (middle), nd A R 3 (right) of the corresponding iutomt from Figure 5 ccepting pirwise different lnguges. of finite utomton B ccepts the reverse of the lnguge ccepted y B, i.e., L(B R ) = L(B) R. We hve seen in the Section 4, tht the reltion etween the lnguges ccepted y iutomton A nd its dul A R is not s simple s for clssicl finite utomt. Nevertheless we cn show tht Brzozowski s minimiztion lgorithm cn still e used for minimiztion of iutomt. More precisely, we prove tht for every (deterministic or nondeterministic) iutomton A with oth the - nd the F-property, the utomton P([P(A R )] R ) is the unique miniml deterministic iutomton, tht hs oth the - nd the F-property. Note tht the middle utomton P(A R ) is deterministic iutomton, tht hs the I-property. The following lemm strts with this middle utomton. Lemm 18. Let A e deterministic iutomton with oth the - nd the I-property, nd with no unrechle sttes. Then P(A R ) is miniml iutomton with oth the - nd the F-property. Proof. Let A = (Q, Σ,,, q 0, F) e deterministic iutomton with oth the - nd the I-property, where ll sttes q Q re rechle. Further, let A R = (Q, Σ, R, R, F, {q 0 }) e its dul iutomton, nd let B = P(A R ) e the powerset iutomton of A R. Assume tht B = (Q B, Σ, B, B, q B 0, F B). Lemms 2 nd 15 imply tht B hs oth the - nd the F-property. In the following, we prove tht B does not hve pir of distinct, ut equivlent sttes. Since y definition of P ll sttes in B = P(A R ) re rechle, the minimlity of B then follows from [5]. Let P 1, P 2 Q B e two distinct sttes of B, then we my ssume tht there is n element q Q, with q P 1 \ P 2. Since q is rechle in the deterministic iutomton A, there re words u, v Σ, such tht q = (q 0 u) v. Since A hs oth the - nd the I-property, Lemm 10 implies q = q 0 vu. This mens tht in the dul utomton A R, we hve q 0 q R(vu) R, y Lemm 14. Furthermore, this mens tht in the powerset iutomton B we hve q 0 (P 1 B (vu) R ), ecuse q P 1. Since the ccepting sttes of B re the sets P Q B with q 0 P, it follows tht the word (vu) R is ccepted y B, when strting from stte P 1. Now ssume, for the ske of contrdiction, tht (vu) R is lso ccepted y B, when strting from stte P 2. Then, since B hs the - nd the F-property, we know from Corollry 6, tht q 0 P 2 B (vu) R, i.e., tht there is stte p P 2, with q 0 p R (vu) R. From Lemm 14, nd the fct tht A is deterministic, 14

16 we otin p = q 0 vu = q, which is contrdiction to q / P 2. We hve shown tht (vu) R is ccepted y B when strting from stte P 1, ut not when strting from stte P 2, so P 1 nd P 2 cnnot e equivlent. Now we re le to prove the min result of this section. Theorem 19. Let A e (deterministic or nondeterministic) iutomton with oth the - nd the F-property. Then P([P(A R )] R ) is the unique miniml iutomton with oth the - nd F-property, for the lnguge L(A). Proof. Let A = (Q, Σ,,, I, F) e iutomton with oth the - nd the F-property. Then let B = P(A R ) nd C = P(B R ), where we ssume tht B = (Q B, Σ, B, B, q0 B, F B), nd C = (Q C, Σ, C, C, q0 C, F C). Then B is deterministic iutomton with the - nd the I-property, with no unrechle sttes. So y Lemm 18, the utomton C is miniml iutomton with the - nd the F-property. It follows from [5], tht C is the unique miniml utomton, mong ll iutomt with these two properties, ccepting L(C). It remins to prove L(A) = L(C), which, due to Corollry 6, cn e done y only resoning out forwrd trnsitions: For ll words w Σ, we hve w L(A) if nd only if (I w) F. By Lemm 14, this holds if nd only if (F Rw R ) I, which is the sme s q0 B Bw R F B, y definition of B. We gin use Lemm 14, to see tht q0 B B w R F B holds if nd only if q0 B F B RB w, which y definition of C is equivlent to q0 C C w F C. Thus, we hve w L(A) if nd only if w L(C). We illustrte the lgorithm in the following exmple. Exmple 20. Let A = (Q, {, },,, I, F) e nondeterministic iutomton with Q = {0, 1, 2, 3, 4}, I = {0, 1}, F = {1, 4}, nd whose trnsition functions, nd re depicted on the left in Figure 7. Note tht A hs oth the - nd the F-property. The corresponding powerset iutomton B = P(A R ) of the dul of A is depicted in the middle in Figure 7. Finlly, the miniml deterministic iutomton C = P(B R ) is depicted on the right in Figure 7, where gin the sink stte, nd ll trnsitions leding to it re omitted. Let us follow the rgumenttion in the proof of Lemm 18, to show tht the sttes p nd s cnnot e equivlent. Note tht p = { {0}, {1, 3}, {1, 4} }, nd s = { {1, 3}, {1, 4} } differ in the element {0}, which is rechle in the iutomton B, for exmple y first reding with forwrd trnsition, nd then with ckwrd trnsition: ({1, 4} ) = {0}. One cn check tht y Lemm 10, this stte of B is lso reched y first reding forwrds, nd then forwrds: ({1, 4} ) = {0}. This in turn mens y Lemm 14 tht the word is ccepted from stte p, ut not from stte s, since {0} p \ s. Thus, sttes p, nd s cnnot e equivlent. References 1. Brzozowski, J.A.: Derivtives of regulr expressions. J. ACM 11, (1964) 2. Brzozowski, J.A.: Cnonicl regulr expressions nd miniml stte grphs for definite events. In: Mthemticl Theory of Automt, MRI Symposi Series, vol. 12, pp Polytechnic Press, NY (1962) 15

17 0 1 p r 2 3 1,4 2 1,3 0 q, s 4 t Fig. 7. Left: The nondeterministic iutomton A with the - nd the F-property. Middle: The deterministic iutomton B = P(A R ), with the - nd the I-property. Right: The miniml deterministic iutomton C = P(B R ) with the - nd F-property. The stte symols p, q, r, s, t re revitions for susets of the stte set of B. It is p = {{0}, {1, 3}, {1, 4}}, q = {{0}, {2}, {1, 3}, {1, 4}}, r = {{1, 3}}, s = {{1, 3}, {1, 4}}, nd t = {{1, 4}}. The sink stte in the iutomt B nd C nd trnsitions leding to it re not shown. 3. Chmprnud, J.-M., Duernrd, J.-P., Jenne, H., Mignot, L.: Two-sided derivtives for regulr expressions nd for hirpin expressions. rxiv: v1 [cs.fl] (2012) 4. Jirásková, G., Klím, O.: Descriptionl complexity of iutomt. In: Kutri, M., Moreir, N., Reis, R. (eds.) Proceedings of the 14th Interntionl Workshop Descriptionl Complexity of Forml Systems. LNCS, vol. 7386, pp Springer, Brg, Portugl (2012) 5. Klím, O., Polák, L.: On iutomt. RAIRO Informtique théorique et Applictions / Theoreticl Informtics nd Applictions 46(4), (2012) 6. Louknov, R.: Liner context free lnguges. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) Proceedings of the 4th Interntionl Colloquium Theoreticl Aspects of Computing. LNCS, vol. 4711, pp Springer, Mcu, Chin (2007) 7. Pin, J.-E.: Hndook of Forml Lnguges, Vol. 1, Word, Lnguge, Grmmr, vol. 1, chp. Syntctic Semigroups, pp Springer (1997) 8. Rosenerg, A.L.: A mchine reliztion of the liner context-free lnguges. Inform. Control 10, (1967) 16

18 Institut für Informtik Justus-Lieig-Universität Giessen Arndtstr. 2, Giessen, Germny Recent Reports (Further reports re ville t A. Mlcher, K. Meckel, C. Mereghetti, B. Plno, Descriptionl Complexity of Pushdown Store Lnguges, Report 1203, My M. Holzer, S. Jkoi, On the Complexity of Rolling Block nd Alice Mzes, Report 1202, Mrch M. Holzer, S. Jkoi, Grid Grphs with Digonl Edges nd the Complexity of Xms Mzes, Report 1201, Jnury H. Gruer, S. Guln, Simplifying Regulr Expressions: A Quntittive Perspective, Report 0904, August M. Kutri, A. Mlcher, Cellulr Automt with Sprse Communiction, Report 0903, My M. Holzer, A. Mletti, An n log n Algorithm for Hyper-Minimizing Sttes in (Minimized) Deterministic Automton, Report 0902, April H. Gruer, M. Holzer, Tight Bounds on the Descriptionl Complexity of Regulr Expressions, Report 0901, Ferury M. Holzer, M. Kutri, nd A. Mlcher (Eds.), 18. Theorietg Automten und Formle Sprchen, Report 0801, Septemer M. Holzer, M. Kutri, Flip-Pushdown Automt: Nondeterminism is Better thn Determinism, Report 0301, Ferury 2003 M. Holzer, M. Kutri, Flip-Pushdown Automt: k + 1 Pushdown Reversls re Better Thn k, Report 0206, Novemer 2002 M. Holzer, M. Kutri, Nondeterministic Descriptionl Complexity of Regulr Lnguges, Report 0205, Septemer 2002 H. Bordihn, M. Holzer, M. Kutri, Economy of Description for Bsic Constructions on Rtionl Trnsductions, Report 0204, July 2002 M. Kutri, J.-T. Löwe, String Trnsformtion for n-dimensionl Imge Compression, Report 0203, My 2002 A. Klein, M. Kutri, Grmmrs with Scttered Nonterminls, Report 0202, Ferury 2002 A. Klein, M. Kutri, Self-Assemling Finite Automt, Report 0201, Jnury 2002 M. Holzer, M. Kutri, Unry Lnguge Opertions nd its Nondeterministic Stte Complexity, Report 0107, Novemer 2001 A. Klein, M. Kutri, Fst One-Wy Cellulr Automt, Report 0106, Septemer 2001 M. Holzer, M. Kutri, Improving Rster Imge Run-Length Encoding Using Dt Order, Report 0105, July 2001 M. Kutri, Refining Nondeterminism Below Liner-Time, Report 0104, June 2001 M. Holzer, M. Kutri, Stte Complexity of Bsic Opertions on Nondeterministic Finite Automt, Report 0103, April 2001 M. Kutri, J.-T. Löwe, Mssively Prllel Fult Tolernt Computtions on Syntcticl Ptterns, Report 0102, Mrch 2001 A. Klein, M. Kutri, A Time Hierrchy for Bounded One-Wy Cellulr Automt, Report 0101, Jnury 2001 M. Kutri, Below Liner-Time: Dimensions versus Time, Report 0005, Novemer 2000 M. Kutri, Efficient Universl Pushdown Cellulr Automt nd their Appliction to Complexity, Report 0004, August 2000 M. Kutri, J.-T. Löwe, Mssively Prllel Pttern Recognition with Link Filures, Report 0003, June 2000