The Degree of Irreversibility in Deterministic Finite Automata

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1 The Degree of Irreversiility in Deterministic Finite Automt Holger Bock Axelsen 1, Mrkus Holzer 2(B), nd Mrtin Kutri 2 1 Deprtment of Computer Science, University of Copenhgen, Universitetsprken 5, Copenhgen, Denmrk funkstr@di.ku.dk 2 Institut für Informtik, Universität Giessen, Arndtstr. 2, Giessen, Germny {holzer,kutri}@informtik.uni-giessen.de Astrct. Recently, Holzer et l. gve method to decide whether the lnguge ccepted y given deterministic finite utomton (DFA) cn lso e ccepted y some reversile deterministic finite utomton (REV-DFA), nd eventully proved NL-completeness. Here, we show tht the corresponding prolem for nondeterministic finite stte utomt (NFA) is PSPACE-complete. The recent DFA method essentilly works y minimizing the DFA nd inspecting it for foridden pttern. Weherestudythedegree of irreversiility for regulr lnguge, the miniml numer of such foridden ptterns necessry in ny DFA ccepting the lnguge, nd show tht the degree induces strict infinite hierrchy of lnguges. We exmine how the degree of irreversiility ehves under the usul lnguge opertions union, intersection, complement, conctention, nd Kleene str, showing tight ounds (some symptoticlly) on the degree. 1 Introduction In computtion theory, reversiility is the property tht computtions re oth forwrd nd ckwrd deterministic. In the context of finite stte models, reversiility cn usully e verified y simple inspection of the trnsition function, ensuring tht the induced computtion step reltion is n injective function on configurtions. Despite the pprent simplicity of reversile computtions, reversiility is n interesting property tht hs een studied in wide rry of contexts, including the thermodynmics of computtion [7], cross wide rry of utomt models [9], nd even in rootics [10]. It is well-known tht the reversily regulr lnguges, i.e., the lnguges ccepted y reversile deterministic finite utomt (REV-DFA), form strict suclss of the regulr lnguges, see, e.g., [6]. However, the exct cost of reversiility is still not well-understood: for exmple, chnging from one-wy The uthors cknowledge prtil support from COST Action IC1405 Reversile Computtion. H.B. Axelsen ws supported y the Dnish Council for Independent Reserch Nturl Sciences under the Foundtions of Reversile Computing project, nd y COST IC1405 STSM (short-term scientific mission) grnt. c Springer Interntionl Pulishing Switzerlnd 2016 Y.-S. Hn nd K. Slom (Eds.): CIAA 2016, LNCS 9705, pp , DOI: /

2 16 H.B. Axelsen et l. to two-wy tpes is sufficient to collpse the clsses [5]. Likewise, dding reversile trnsducer in front of the REV-DFA lso collpses to the regulr lnguges [1]. This motivtes further study into the reltionship etween the regulr nd reversily regulr lnguges, nd in prticulr towrds developing methods to understnd nd ridge the gp in terms of the internl structure of deterministic finite utomt (DFA). In this pper, we tke steps in this direction. Recently, Holzer et l. showed method for deciding if the lnguge ccepted y given DFA cn lso e recognized y some REV-DFA [4]. It ws lso shown tht this is n NL-complete prolem, nd decision method ws given, which essentilly works y minimizing the DFA nd inspecting it for the presence of foridden pttern. If this pttern is present in the miniml DFA, then there is no REV-DFA tht cn ccept the sme lnguge, nd if not, then there is. Wht mkes this prticulrly interesting is tht the pttern is structurlly more complex thn the simplest violtion of reversiility (see Sect. 2 for detils). This suggests tht the foridden pttern cptures n essentil spect of irreversiility, nd offers n pproch to studying the gp etween the reversily regulr nd regulr lnguges sed on the sence, presence, nd count of occurrences, of this pttern. Our contriutions re s follows. We show tht the generliztion of the prolem studied in [4] to nondeterministic finite utomt (NFA), i.e., the regulr reversiility prolem of whether the lnguge ccepted y given NFA is reversily regulr, is PSPACE-complete. Turning to DFAs, we introduce the notion of degree of irreversiility for DFAs, essentilly the numer of occurrences of the foridden pttern in given DFA, nd extend this to (regulr) lnguges y minimizing over ll DFAs ccepting the lnguge. Finlly, we show tht the degree of irreversiility induces strict, infinite hierrchy of lnguges. We then proceed to show exct ounds on the degree of irreversiility under the common lnguge opertions union, intersection, nd complement, nd symptoticlly tight ounds for conctention nd Kleene str. The pper is orgnized s follows. Section 2 covers the necessry preliminries. In Sect. 3 we show tht the regulr reversiility prolem is PSPACE-complete. Section 4 defines the degree of irreversiility, nd shows the relted hierrchy. We present the degree complexity results for common lnguge opertions in Sect. 5. Most proofs re omitted due to spce constrints, nd will e given in the full version of the pper. 2 Preliminries An lphet Σ is non-empty finite set, its elements re clled letters or symols. We write Σ for the set of ll words over the finite lphet Σ. We recll some definitions on finite utomt s contined, for exmple, in [3]. A deterministic finite utomton (DFA) is 5-tuple A =(Q, Σ, δ, q 0,F), where Q is the finite set of internl sttes, Σ is the lphet of input symols, q 0 S is the initil stte, F Q is the set of ccepting sttes, ndδ : Q Σ Q is the

3 The Degree of Irreversiility in Deterministic Finite Automt 17 prtil trnsition function. Note tht here the trnsition function is not required to e totl. Thelnguge ccepted y A is L(A) ={ w Σ δ(q 0,w) F }, where the trnsition function is recursively extended to δ : Q Σ Q. By δ R : Q Σ 2 Q, with δ R (q, ) ={ p S δ(p, ) =q }, we denote the reverse trnsition function of δ. Similrly, lso δ R cn e extended to words insted of symols. Two devices A nd A re sid to e equivlent if they ccept the sme lnguge, tht is, L(A) =L(A ). Let A = (Q, Σ, δ, q 0,F) e DFA ccepting the lnguge L. Thesetof words R A,q = { w Σ δ(q, w) F } refers to the right lnguge of the stte q in A. IncseR A,p = R A,q, for some sttes p, q Q, we sy tht p nd q re equivlent nd write p A q. The equivlence reltion A prtitions the stte set Q of A into equivlence clsses, nd we denote the equivlence clss of q S y [q] ={ p S p A q }. Equivlence cn lso e defined etween sttes of different utomt: stte p of DFA A nd stte q of DFA A re equivlent, denoted y p q, ifr A,p = R A,q. A stte p Q is ccessile in A if there is word w Σ such tht δ(q 0,w)= p, nd it is productive if there is word w Σ such tht δ(p, w) F.Ifp is oth ccessile nd productive then we sy tht p is useful. In this pper we only consider utomt with ll sttes useful. Let A nd A e two equivlent DFAs. Oserve tht if p is useful stte in A, then there exists useful stte p in A, with p p.adfaisminiml (mong ll DFAs) if there does not exist n equivlent DFA with fewer sttes. It is well known tht DFA is miniml if nd only if ll its sttes re useful nd inequivlent. Next we define reversile DFAs. Let A = (Q, Σ, δ, q 0,F) e DFA. A stte r Q is sid to e irreversile if there re two distinct sttes p nd q in Q nd letter Σ such tht δ(p, ) =r = δ(q, ). Then DFA is reversile if it does not contin ny irreversile stte. In this cse the utomton is sid to e reversile DFA (REV-DFA). Equivlently the DFA A is reversile, if every letter Σ induces n injective prtil mpping from Q to itself vi the mpping δ : Q Q with p δ(p, ). In this cse, the reverse trnsition function δ R cn e seen s (prtil) injective function δ R : Q Σ Q. Notice tht if p nd q re two distinct sttes in REV-DFA, then δ(p, w) δ(q, w), for ll words w Σ. Finlly, REV-DFA is miniml (mong ll REV-DFAs) if there is no equivlent REV-DFA with smller numer of sttes. In [4] the following structurl chrcteriztion of regulr lnguges tht cn e ccepted y REV-DFAs in terms of their miniml DFAs is given. The conditions of the chrcteriztion re illustrted in Fig. 1. Theorem 1. Let A =(Q, Σ, δ, q 0,F) e miniml deterministic finite utomton. The lnguge L(A) cn e ccepted y reversile deterministic finite utomton if nd only if there do not exist useful sttes p, q Q, letter Σ, nd word w Σ such tht p q, δ(p, ) =δ(q, ), ndδ(q, w) =q. Finlly we need some nottions on computtionl complexity theory. We clssify prolems on DFAs with respect to their computtionl complexity. Consider the complexity clss NL (PSPACE, respectively) which refers to the set

4 18 H.B. Axelsen et l. p q r p r=q q r=p w w w Fig. 1. The foridden pttern of Theorem 1: the lnguge ccepted y miniml DFA A cn e ccepted y REV-DFA if nd only if A does not contin the structure depicted on the left. Here the sttes p nd q must e distinct, ut stte r could e equl to stte p or stte q. The situtions where r = q or r = p re shown in the middle nd on the right, respectively here the word w nd its corresponding pth re gryed out ecuse they re not relevnt: in the middle, the word w tht leds from r to q is not relevnt since it cn e identified with the -loop on stte r = q. Also on the right hnd side, word w is not importnt ecuse we cn simply interchnge the roles of the sttes q nd r = p. of prolems ccepted y nondeterministic logspce ounded (polynomil spce, respectively) Turing mchines. Further, hrdness nd completeness is lwys ment with respect to deterministic logspce ounded reduciility, unless otherwise stted. 3 Complexity of the Regulr Reversiility Prolem In [4] it ws shown tht the regulr lnguge reversiility prolem given DFA A, decide whether L(A) is ccepted y ny REV-DFA is NL-complete. If the regulr lnguge is given y n NFA or regulr expression, the prolem ecomes intrctle. Theorem 2. The regulr lnguge reversiility prolem is PSPACE-complete, if the lnguge is given s nondeterministic finite utomton or regulr expression. Before we cn prove this result we need technicl lemm, which will e used in the PSPACE-hrdness rgument lter. Lemm 3. Let A =(Q, Σ, δ, q 0,F) e miniml DFA. If there is stte q Q, other thn the initil stte, such tht R A,q = Σ, then L(A) is irreversile. Let L Σ. Then the left derivtive of L with respect to the letter in Σ is the set 1 L = { w w L }. This nottion generlizes to words. By this definition, there is n ovious reltion etween these left derivtive set nd the sttes of the miniml finite utomton A ccepting L. To e more precise, the set u 1 L, foru Σ, is description of the stte q u = δ(q 0,u), where A =(Q, Σ, δ, q 0,F), nd vice vers. Now we re redy to proof Theorem 2 in convenient wy.

5 The Degree of Irreversiility in Deterministic Finite Automt 19 Proof (of Theorem 2). The continment within PSPACE is esily seen. For the hrdness we reduce the PSPACE-complete universlity prolem for regulr expressions [8] to the reversiility prolem for NFAs or regulr expressions. Let the regulr expression r e n instnce of the universlity prolem. We my ssume tht r is n expression over the lphet Σ = {, }. Then we construct the expression s = r + Σ + λ or equivlently the NFA depicted in Fig. 2 in deterministic logspce. Now ssume tht L(r) =Σ. Then it is esy to see tht L(s) =Σ, too, nd therefore reversile lnguge. On the other hnd, if L(r) Σ, then there is word u L(r). From this it follows tht u L(s) ut u L(s). Thus we conclude tht the sttes 1 L(s) nd 1 L(s) re not equivlent in the DFA ccepting L(s). Moreover, in tht DFA sttes L(s) nd 1 L(s) re not equivlent, too. Note tht the former stte is the initil stte of the DFA tht ccepts L(s). Since the right lnguge of the stte 1 L(s) is equl to Σ nd it is not equl to the initil stte, Lemm 3 pplies, nd the lnguge L(s) isnot reversile. This proves PSPACE-hrdness. q 1 A r q 0 q 2, Fig. 2. Finite utomton tht ccepts the lnguge L(s). It is uilt from the regulr expression r, wherea r is n NFA with initil stte q 1 tht ccepts the lnguge L(r). 4 On the Degree of Irreversiility For n utomton A we define its degree of irreversiility d(a) s the numer of irreversile sttes tht re prt of one of the foridden ptterns shown in Fig. 1. Oserve, tht since our DFAs need not to e complete nd only contin useful sttes, the non-ccepting sink stte does not count for the degree of irreversiility. This nottion is generlized to lnguges in the usul wy. This mens, for regulr lnguge L Σ we define its degree of irreversiility d(l) sthe minimum degree of irreversiility mong ll equivlent DFAs A, tht is, d(l) = min{ d(a) A is DFA with L(A) =L }. The next exmple explins our nottion.

6 20 H.B. Axelsen et l Fig. 3. DFA which ccepts + tht hs irreversiility degree one. Exmple 4. Consider the following DFA depicted in Fig. 3, which ccepts the union of nd. This utomton hs irreversiility degree one y stte 3. Note tht lthough stte 4 hs two ingoing -trnsitions, this stte does not yield foridden pttern s shown in Fig. 1. There is no word tht leds from stte 4 to either stte 1 or 3. Moreover, the lnguge ccepted y this utomton, which is + is lso of irreversiility degree one, since it is not reversile y Theorem 1. Next we consider the hierrchy on regulr lnguges tht is induced y the irreversiility degree. To this end let IREV k -DFA = { A A is DFA nd d(a) k }. We hve IREV 0 -DFA = { A A is reversile DFA } nd thus the equlity L (IREV 0 -DFA) = L (REV-DFA) holds, where the fmily of ll lnguges ccepted y n utomton of some type X is denoted y L (X). Moreover, y definition the inclusion IREV k -DFA IREV k+1 -DFA follows nd therefore the corresponding lnguge clsses stisfy L (IREV k -DFA) L (IREV k+1 -DFA), for k 0. By the exmple ove we hve L (REV-DFA) = L (IREV 0 -DFA) L (IREV 1 -DFA). Before we show tht the degree of irreversiility induces n infinite strict hierrchy we need some tool tht llows us to determine the irreversiility degree for n ritrry regulr lnguge. Since for the degree of irreversiility of lnguge L we quntify over ll equivlent DFAs we hve to show tht we cnnot trde more sttes for less irreversiility. The following exmple shows tht this is in fct not the cse in generl. Exmple 5. Consider the sustructure of DFA s depicted in Fig. 4. Itisnot hrd to see tht this pttern my pper in miniml DFA. Both sttes r 1 nd r 2 in the sustructure re irreversile. By splitting oth of these sttes, we otin connecting structure s shown in Fig. 5. The structure hs one irreversile stte only. Thus, the irreversiility degree of miniml DFA is not necessrily the irreversiility degree of the lnguge under considertion.

7 The Degree of Irreversiility in Deterministic Finite Automt 21 The Degree of Irreversiility in Deterministic Finite Automt d r 1 s r 2 c Fig. 4. Sustructure of DFA contining two irreversile sttes r 1 nd r 2. d d r 1 r 1 s r 2 r 2 c c Fig. 5. Sustructure of DFA with just one irreversile stte s otined fter splitting oth irreversile sttes. For specil clss of finite utomt, we cn show tht the miniml DFA lredy gives the degree of irreversiility. A DFA is simply-irreversile if ll irreversile sttes re of the form depicted in the middle nd right drwing shown in Fig. 1. Tht is, the irreversiility stte is entered y n -trnsition nd hs n -self-loop, which is the simplest form of irreversiility. For the lnguges ccepted y these utomt we cn prove the next result. Theorem 6. Let L e regulr lnguge nd A e its miniml deterministic finite utomton. If A is simply-irreversile, then the degree of irreversiility of A is equl to the irreversiility degree for L. Tht is d(l) =d(a). Now we re redy to show tht the strict hierrchy on regulr lnguges induced y the irreversiility degree is tight nd infinite. Theorem 7. For ll k 0, L (IREV k -DFA) L (IREV k+1 -DFA). Proof. Consider the lnguges L k over the lphet {, } defined s follows: for k 0set L 2k =( ) k nd L 2k+1 =( ) k. The lnguge L k,fork 0, is ccepted y the DFA A k =(Q k, {, },δ k,q 0,F k ) with Q k = {1, 2,...,k+1}, q 0 =1,F k = {k +1}, nd { i +1 ifi is odd nd 1 i<k+1 δ(i, ) = i if i is even nd 1 <i k +1 nd δ(i, ) = { i +1 ifi is even nd 1 i<k+1 i if i is odd nd 1 <i k +1.

8 22 H.B. Axelsen et l. By construction the DFA A k is miniml nd simply-irreversile. Thus, y the previous theorem the degree of irreversiility of A k is equl to the irreversiility degree of the lnguge L k. Since A k contins exctly k irreversile sttes, we hve d(l k )=k. This shows tht L k L (IREV k -DFA) \ L (IREV k 1 -DFA), for k 1. Finlly, we consider unry regulr lnguges nd their irreversiility degree. It is not difficult to see tht unry complete DFA consists of pth, which strts from the initil stte, followed y cycle of one or more sttes. Thus the irreversiility degree of ny unry DFA is t most one. Thus, the hierrchy on the irreversiility degree collpses to its second level nd L (IREV 1 -DFA) 2 {} is lredy equl to the clss of ll unry regulr lnguges. Moreover, we conclude tht L (IREV 0 -DFA) 2 {} is the clss of lnguges tht contins only finite or cyclic unry regulr lnguges. Here unry regulr lnguge is cyclic if it is ccepted y unry DFA which is cycle of one or more sttes. 5 Opertions on Lnguges nd Degree of Irreversiility In this section we study the descriptionl complexity of the opertion prolem for reversile lnguges. We strt with the Boolen opertions nd continue with the conctention nd Kleene str opertion. First we consider the union opertion. For the union of two reversile lnguge, the increse of the degree of irreversiility is liner in the sum of the numer of sttes of the involved utomt. This cn e seen in the next theorem. Theorem 8. Let m, n 1 e two integers, A e n m-stte nd B e n n-stte reversile deterministic finite utomton. Then the degree m+n of irreversiility for the lnguge L(A) L(B) is sufficient nd necessry in the worst cse. Proof. Let A =(Q A,Σ,δ A,q 0,A,F A )ndb =(Q B,Σ,δ B,q 0,B,F B ). In order to ccept the union of L(A) nd L(B) we pply the stndrd cross-product construction. To this end define C =(Q C,Σ,δ C,q 0,C,F C ), where Q C =(Q A Q B ) (Q A { }) ({ } Q B ), q 0,C =(q 0,A,Q 0,B ), nd F C =(Q A F B ) (F A Q B ) (F A { }) ({ } F B ). The trnsition function δ C is set to δ C ((p, q),)= (δ A (p, ),δ B (q, )) if oth δ A (p, ) ndδ B (q, ) re defined (δ A (p, ), ) if δ A (p, ) is defined nd δ B (q, ) is undefined (,δ B (q, )) if δ A (p, ) is undefined nd δ B (q, ) is defined nd furthermore δ C ((p, ),)=(δ A (p, ), ), if δ A (p, ) is defined, s well s δ C ((,q),) = (,δ B (q, )), if δ B (q, ) is defined, for Σ. So we hve

9 The Degree of Irreversiility in Deterministic Finite Automt 23 L(C) = L(A) L(B). From the m n + m + n sttes of C t most m + n re irreversile. To e more precise, none of the sttes from Q A Q B re irreversile. This is seen s follows: consider stte (r, r ) Q A Q B. Assume to the contrry tht (r, r ) is irreversile. Then there re different sttes (p, p )nd (q, q ) with δ C ((p, p ),)=(r, r )=δ C ((q, q ),), for some Σ. Since (p, p )is not equl to (q, q )wehvep q or p q. We only consider the cse p q y symmetric resons. But then we find tht r is n irreversile stte, ecuse δ A (p, ) =r = δ A (q, ), for the letter from ove. This is contrdiction, ecuse utomton A is reversile DFA. It is worth mentioning tht similr rgumenttion does not pply to sttes of the form (r, ) or(,r). This is seen y the counterexmple δ C ((r, p),)=(r, ) =δ C ((r, ),), for some Σ, which induces only δ A (r, ) =r nd δ B (p, ) is undefined n nlogous exmple cn e given for stte of the form (,r). Hence, this does not contrdict the irreversiility of either A or B. It remins to e shown tht the ound m + n is tight. Define the reversile DFA A =(Q A, {, },δ A,q 0,F) with Q A = {1, 2,...,m}, q 0 =1,F = {m}, nd the trnsition function is given y δ(i, ) =i +1,for1 i<m,ndδ(i, ) =i, for 1 i m. The utomton B is the sme s A, ut with n sttes, nd where the letters nd re interchnged. The utomton C constructed ove is esily seen to e miniml. Finlly we show tht ll sttes of the form (i, ) nd (,j), for 1 i m nd 1 j n, re irreversile nd yield foridden pttern s shown in Fig. 1. The elow given rgument shows even more, nmely tht the utomton C is simply-irreversile. We hve lredy rgued tht the stte (i, n) is ccessile. Then it is esy to see tht from stte (i, n) reding the utomton C enters stte (i, ), which hs loop. Therefore stte (i, ) is simply-irreversile. A similr rgument shows tht stte (,j) is simply-irreversile s well. By Theorem 6 the stted clim follows. A creful inspection of the previous proof revels tht we cn use prts of it for the intersection of two reversile lnguges. For two reversile utomt A nd B we construct n utomton C y the cross-product construction descried in the proof of Theorem 8 ut only using sttes of the form Q A Q B nd y ltering the set of ccepting sttes to e F = F A F B. Then L(C) =L(A) L(B). It ws shown tht none of the sttes from Q A Q B re irreversile. Hence C does not contin ny irreversile stte. Thus, we hve shown the following result. Theorem 9. Let m, n 1 e two integers, A e n m-stte nd B e n n-stte reversile deterministic finite utomton. Then the lnguge L(A) L(B) is ccepted y reversile deterministic finite utomton. Next we del with the complementtion opertion, nd show tht the degree of irreversiility cn e incresed y one. Theorem 10. Let n 1 e n integers nd A e n n-stte reversile deterministic finite utomton. Then the degree 1 of irreversiility for the complement of L(A) is sufficient nd necessry in the worst cse.

10 24 H.B. Axelsen et l. In the reminder of this section we investigte the effect of the conctention nd the Kleene str opertion on the degree of irreversiility. First we recll the construction of DFAs for the conctention [11]. Let A =(Q A,Σ,δ A,q 0,A,F A ) nd B =(Q B,Σ,δ B,q 0,B,F B ) e two DFAs. As in [11] we construct the DFA C =(Q C,Σ,δ C,q 0,C,F C ), where Q C =(Q A 2 QB ) \ (F A 2 QB\{q0,B} ), the initil stte is q 0,C = { (q 0,A, ) if q 0,A F A (q 0,A, {q 0,B }) otherwise, the finl sttes re F C = { (p, P ) (p, P ) Q C nd P F B }, nd the trnsition function is defined y δ C ((p, P ),)=(q, Q), for Σ, where q = δ A (p, ) nd { δ B (P, ) {q 0,B } if q F A Q = δ B (P, ) otherwise. Clerly, utomton C ccepts L(A) L(B) nd hs t most m 2 n 2 n 1 sttes. Thus, the construction gives rise to n exponentil upper ound on the numer of irreversile sttes. Theorem 11. Let m, n 2 e two integers, A e n m-stte nd B e n n-stte reversile deterministic finite utomton. Then the degree m 2 n 2 n 1 of irreversiility is sufficient for deterministic finite utomton to ccept the lnguge L(A) L(B). The next theorem gives n exponentil lower ound on the degree of irreversiility for the conctention opertion. Theorem 12. Let m, n 2 e two integers. There re reversile m-stte deterministic finite utomton A nd reversile n-stte deterministic finite utomton B such tht ny deterministic finite utomton ccepting L(A) L(B) hs t lest the degree (3m 2) 2 n 2 of irreversiility. Proof. Define the left utomton to e A =(Q A, {,, c, d},δ A,q 0,A,F A ) with Q A = {0, 1,...,m 1}, initil stte q 0,A = 0, finl sttes F A = {m 1}, nd the trnsition function { i +1 if0 i<m 1 δ A (i, ) = nd δ A (i, ) =δ A (i, c) =δ A (i, d) =i 0 otherwise for 0 i m 1. The right utomton is B =(Q B, {,, c, d},δ B,q 0,B,F B ) with Q B = {0, 1,...,n 1}, initil stte q 0,B = 0, finl sttes F A = {0}, nd the trnsition function

11 The Degree of Irreversiility in Deterministic Finite Automt 25 δ B (i, ) =i, for 0 i n 1, nd δ B (i, ) = { i +1 if0 i<n 1 0 otherwise, nd δ B (i, c) =i, for 0 <i n 1, nd δ B (i, d) =i, for i =0or2 i n 1. Both reversile utomt re depicted in Fig. 6. m 1 n 1, c, d, c, d 0, c, d 0, d 1, c, d 1, c, c, d 2 Fig. 6. The reversile utomt A (left) nd B (right) with m nd n sttes, respectively, tht witness the irreversiility degree lower ound for the conctention opertion. We construct the DFA C for the conctention L(A) L(B) s descried ove. In order to pply Theorem 6 we need to show tht C is miniml. Thus, one hs to verify tht every stte in C is useful nd defines distinct equivlence clss. Finlly, it remins to determine the lower ound on the irreversiility degree of C. We show tht ll sttes of C whose second component does not contin 0 nd 1 t the sme time re simply-irreversile. We hve lredy seen tht ll sttes of the form (p, P {0}) nd (p, P {1}) re rechle in C. We distinguish two cses: 1. Assume p = m 1. Then 0 P, ut then y ssumption 1 P.Wehve δ C ((p, P {1}),d)=(p, P )ndδ C ((p, P ),d)=(p, P ). Thus (p, P ) is simplyirreversile. 2. Let p = i with 0 i<m 1. If 0 P, then δ C ((p, P {0}),c)=(p, P )nd δ C ((p, P ),c)=(p, P ). Also in the cse 1 P, the two trnsitions δ C ((p, P {1}),d)=(p, P )ndδ C ((p, P ),d)=(p, P ) follow. In oth cses the stte (p, P ) is simply-irreversile. Next we count the numer of simply-irreversile sttes. The first item ove induces 2 n 2 possiilities, nd the second item 3(m 1) 2 n 2. There re (m 1) choices for p nd the numer of different sets P tht do not contin 0 or 1 is 3 2 n 2. For ech of the cses (i) oth 0 nd 1 re not in P, (ii) element 0 is

12 26 H.B. Axelsen et l. in P ut 1 is not, nd (iii) element 0 is not in P ut 1 is memer of P, there re 2 n 2 possiilities. This results in 3(m 1) 2 n 2 sets for the second item ove. Putting things together results in in t lest (3m 2) 2 n 2 simply-irreversile sttes in C. By Theorem 6 the stted clim follows. Finlly, we consider the Kleene str opertion. From [11] the tight worst cse ound for DFA to ccept the Kleene closure of n n-stte DFA lnguge is 2 n 1 +2 n 2. Thus, the upper ound for the irreversiility degree for the Kleene closure is exponentil. Theorem 13. Let n 2 e n integers nd A e n n-stte reversile deterministic finite utomton. Then the degree 2 n 1 +2 n 2 of irreversiility is sufficient for deterministic finite utomton to ccept the lnguge L(A). As in the cse of the conctention opertion we cn provide n exponentil lower ound. Theorem 14. Let n 3 e n integer. There is reversile n-stte deterministic finite utomton A such tht ny deterministic finite utomton ccepting L(A) hs t lest the degree 3 2 n 3 1 of irreversiility. References 1. Axelsen, H.B., Kutri, M., Mlcher, A., Wendlndt, M.: Boosting reversile pushdown mchines y preprocessing. In: RC 2016, LNCS. Springer (2016) 2. Glushkov, V.M.: The strct theory of utomt. Russ. Mth. Surv. 16, 1 53 (1961) 3. Hrrison, M.A.: Introduction to Forml Lnguge Theory. Addison-Wesley, Boston (1978) 4. Holzer, M., Jkoi, S., Kutri, M.: Miniml reversile deterministic finite utomt. In: Potpov, I. (ed.) DLT LNCS, vol. 9168, pp Springer, Heidelerg (2015) 5. Kondcs, A., Wtrous, J.: On the power of quntum finite stte utomt. In: FOCS 1997, pp IEEE (1997) 6. Kutri, M.: Aspects of reversiility for clssicl utomt. In: Clude, C.S., Freivlds, R., Kzuo, I. (eds.) Grusk Festschrift. LNCS, vol. 8808, pp Springer, Heidelerg (2014) 7. Lnduer, R.: Irreversiility nd het genertion in the computing process. IBM J. Res. Dev. 3, (1961) 8. Meyer, A.R., Stockmeyer, L.J.: The equivlence prolem for regulr expressions with squring requires exponentil time. In: SWAT 1972, pp IEEE (1972) 9. Morit, K.: Reversile computing nd cellulr utomt survey. Theoret. Comput. Sci. 395(1), (2008) 10. Schultz, U.P., Lursen, J.S., Ellekilde, L., Axelsen, H.B.: Towrds domin-specific lnguge for reversile ssemly sequences. In: Krivine, J., Stefni, J.-B. (eds.) RC LNCS, vol. 9138, pp Springer, Switzerlnd (2015) 11. Yu, S., Zhung, Q., Slom, K.: The stte complexity of some sic opertions on regulr lnguges. Theoret. Comput. Sci. 125, (1994)

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