R-automata. 1 Introduction. Parosh Aziz Abdulla, Pavel Krcal, and Wang Yi

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1 R-utomt Proh Aziz Abdull, Pvel Krcl, nd Wng Yi Deprtment of Informtion Technology, Uppl Univerity, Sweden Emil: Abtrct. We introduce R-utomt nite tte mchine which operte on nite number of unbounded counter. The vlue of the counter cn be incremented, reet to zero, or left unchnged long the trnition. R-utomt cn be, for exmple, ued to model ytem with reource (modeled by the counter) which re conumed in mll prt but which cn be replenihed t once. We dene the lnguge ccepted by n R- utomton reltive to nturl number D the et of word llowing run long which no counter vlue exceed D. A the min reult, we how decidbility of the univerlity problem, i.e., the problem whether there i number D uch tht the correponding lnguge i univerl. We preent proof bed on nite monoid nd the fctoriztion foret theorem. Thi theorem w pplied for ditnce utomt in [12] pecil ce of R-utomt with one counter which i never reet. A econd technicl contribution, we extend the decidbility reult to R-utomt with Büchi cceptnce condition. 1 Introduction We conider ytem operting on reource which re conumed in mll prt nd which cn be (or hve to be) replenihed completely t once. To model uch ytem, we introduce R-utomt nite tte mchine extended by nite number of unbounded counter correponding to the reource. The counter cn be incremented, reet to zero, or left unchnged long the trnition. When the vlue of counter i equl to zero then the tock of thi reource i full. Incrementing counter men uing one unit of the reource nd reetting counter men the full replenihment of the tock. We dene the lnguge ccepted by n R-utomton reltive to nturl number D the et of word llowing n ccepting run of the utomton uch tht no counter vlue exceed D in ny tte long the run. We tudy the problem of whether there i number D uch tht the correponding lnguge i univerl. Thi problem correpond to the fct tht with tock ize D, the ytem cn exhibit ll the behvior without running out of reource. We how tht thi problem i decidble in 2-EXPSPACE. A econd technicl contribution, we extend the decidbility reult to R-utomt with Büchi cceptnce condition. Thi work h been prtilly upported by the EU CREDO project.

2 To prove decidbility of the univerlity problem, we dopt the technique from [12] nd extend it to our etting. We reformulte the problem in the lnguge of nite monoid nd olve it uing the fctoriztion foret theorem [11]. In [12], thi theorem i ued for olving the limitedne problem for ditnce utomt. Ditnce utomt re ubcl of R-utomt with only one counter which i never reet. In contrt to thi model, we hndle everl counter nd reet. Thi extenion cnnot be encoded into the ditnce utomt. The deciion lgorithm del with btrction of collection of run in order to nd nd nlyze the loop creted by thee collection. The min tep in the correctne proof i to how tht ech collection of run long the me word cn be plit (fctorized) into hort repeted loop, poibly neted. Hving uch fctoriztion, one cn nlyze ll the loop to check tht none of the counter i only increed without being reet long them. If none of the counter i increed without being reet then we cn bound the counter vlue by contnt derived from the length of the loop. Since the length of the loop i bounded by contnt derived from the utomton, ll word cn be ccepted by run with bounded counter. Otherwie, we how tht there i +-free regulr expreion uch tht for ny bound there i word obtined by pumping thi regulr expreion which doe not belong to the lnguge. Therefore, the lnguge cnnot be univerl for ny D. Relted work. The concept of ditnce utomt nd the limitedne problem were introduced by Hhiguchi [6]. The limitedne problem i to decide whether there i nturl number D uch tht ll the ccepted word cn lo be ccepted with the counter vlue mller thn D. Dierent proof of the decidbility of the limitedne problem re reported in [7, 10, 12]. The lt of thee reult [12] i bed on the fctoriztion foret theorem [11, 4]. The model of R- utomt, which we conider in thi pper, extend tht of ditnce utomt by introducing reet nd by llowing everl counter. Furthermore, ll the work mentioned bove only conider the limitedne problem on nite word, while we here extend the decidbility reult of the univerlity problem to the ce of innite word. Ditnce utomt were extended in [8] with dditionl counter which cn be reet following hierrchicl dicipline reembling prity cceptnce condition. R-utomt relx thi dicipline nd llow the counter to be reet rbitrrily. Univerlity of imilr type of utomt for tree lnguge i tudied in [5]. A model with counter which cn be incremented nd reet in the me wy in R-utomt, clled B-utomt, i preented in [3]. B-utomt ccept innite word uch tht the counter re bounded long n innite ccepting computtion. Decidbility of our problem cn be obtined uing the reult from [3]. However, thi would require complementtion of B-utomton which reult in non-elementry blowup of the utomton tte pce. The fct tht R-utomt cn hve everl counter which cn be reet llow, for intnce, to cpture the btrction of the mpled emntic of timed utomt [9, 1]. A mpled emntic given by mpling rte ɛ = 1/f for ome poitive integer f llow time to p only in tep equl to multiple of ɛ. The number of dierent clock vlution within one clock region ( bounded et of

3 vlution) correpond to reource. It i nite for ny ɛ while innite in the tndrd (dene time) emntic of timed utomt. Timed utomt cn generte run long which clock re forced to tke dierent vlue from the me clock region (n increment of counter), tke exctly the me vlue ( counter i left unchnged), or forget bout the previouly tken vlue ( counter reet). 2 Preliminrie Firt, we introduce the model of R-utomt nd it unprmeterized emntic. Then, we introduce the prmeterized emntic, the lnguge ccepted by the utomton, nd the univerlity problem. R-utomt. R-utomt re nite tte mchine extended with counter. A trnition my incree the vlue of counter, leve it unchnged, or reet it bck to zero. The utomton on it own doe not hve the cpbility of teting the vlue of the counter. However, the emntic of thee utomt i prmeterized by nturl number D which dene n upper bound on counter vlue which my pper long the computtion of the utomton. Let N denote the et of non-negtive integer. An R-utomton with n counter i 5-tuple A = S, Σ,, 0, F where S i nite et of tte, Σ i nite lphbet, S Σ {0, 1, r} n S i trnition reltion, 0 S i n initil tte, nd F S i et of nl tte. Trnition re lbeled (together with letter) by n eect on the counter. The ymbol 0 correpond to leving the counter vlue unchnged, the ymbol 1 repreent n increment, nd the ymbol r repreent reet. We ue t, t 1,... to denote element of {0, 1, r} n which we cll eect. A pth i equence of trnition ( 1, 1, t 1, 2 ),( 2, 2, t 2, 3 ),..., ( m, m, t m, m+1 ), uch tht 1 i m.( i, i, t i, i+1 ). An exmple of n R-utomton i given in Figure 1. b, (0, 1), (1, 0) 0 1, (0, 1) b, (r, r) 2, (0, r) Fig. 1. An R-utomton with two counter.

4 Unprmeterized emntic. We dene n opertion on the counter vlue follow: for ny k N, k 0 = k, k 1 = k + 1, nd k r = 0. We extend thi opertion to n-tuple by pplying it componentwie. The opertionl emntic of n R-utomton A = S, Σ,, 0, F i given by lbeled trnition ytem (LTS) A = Ŝ, Σ, T, ŝ 0, where the et of tte Ŝ contin pir, (c 1,..., c n ), S, c i N for ll 1 i n, with the initil tte ŝ 0 = 0, (0,..., 0). The trnition reltion i dened by (, (c 1,..., c n ),,, (c 1,..., c n) ) T if nd only if,, t, nd (c 1,..., c n) = (c 1,..., c n ) t. We hll cll the tte of the LTS congurtion. We write, (c 1,..., c n ), (c 1,..., c n) if (, (c 1,..., c n ),,, (c 1,..., c w n) ) T. We extend thi nottion lo for word,, (c 1,..., c n ), (c 1,..., c n), where w Σ +. Pth in n LTS re clled run to ditinguih them from pth in the underlying R-utomton. Oberve tht the LTS contin innitely mny tte, but the counter vlue do not inuence the computtion, ince they re not teted nywhere. In fct, for ny R-utomton A, A i biimilr to A conidered nite utomton (without counter nd eect). The LTS induced by the R-utomton from Figure 1 i in Figure 2. 0, (0, 0) 1, (1, 0) b 1, b 1, (1, 2) b 1, (1, 3) b b b b 2, (0, 1) 2, (0, 0) Fig. 2. The unprmeterized emntic of the R-utomton in Figure 1. Prmeterized Semntic. Now we dene the D-emntic of R-utomt. We ume tht the reource ocited to the counter re not innite nd we cn ue them only for bounded number of time before they re replenihed gin. If mchine trie to ue reource which i lredy completely ued up, it i blocked nd cnnot continue it computtion. For given D N, let ŜD be the et of congurtion retricted to the congurtion which do not contin counter exceeding D, i.e., ŜD = {, (c 1,..., c n ), (c 1,..., c n ) Ŝ nd (c 1,..., c n ) (D,..., D)} ( i pplied componentwie). For n R-utomton A, the D-emntic of A, denoted by A D, i A retricted to ŜD. We write, (c 1,..., c n ) D, (c 1,..., c n) to denote the trnition reltion of A D. We extend thi nottion for word,, (c 1,..., c n ) w D, (c 1,..., c n) where w Σ +. The 2-emntic of the R-utomton from Figure 1 i in Figure 3. We bue the nottion to void tting the counter vlue explicitly when it i not necery. We dene the rechbility reltion nd D over pir of tte nd word follow. For, S nd w Σ +, w if nd only

5 0, (0, 0) 1, (1, 0) b 1, b 1, (1, 2) b b b 2, (0, 1) 2, (0, 0) Fig. 3. The 2-emntic of the R-utomton in Figure 1. if there i pth (, 1, t 1, 1 ), ( 1, 2, t 2, 2 ),..., ( w 1, w, t w, ) uch tht w w = 1 2 w. For ech D N, D if lo for ll 1 i w, t 1 t 2 t i (D,..., D). It lo hold tht w D if nd only if there w i run, (0,..., 0) D, (c 1,..., c n ). Lnguge. The (unprmeterized or D-) lnguge of n R-utomton i the et of word which cn be red long the run in the correponding LTS ending in n ccepting tte (in congurtion whoe rt component i n ccepting tte). The unprmeterized lnguge ccepted by n R-utomton A w i L(A) = {w 0 f, f F }. For given D N, the D-lnguge ccepted by w n R-utomton A i L D (A) = {w 0 D f, f F }. The unprmeterized lnguge of the R-utomton from Figure 1 i b. The 2-lnguge of thi utomton i (ɛ + b + bb + bbb). Problem Denition. Now we cn k quetion bout lnguge univerlity of n R-utomton A prmeterized by D, i.e., i there nturl number D uch tht L D (A) = Σ. Figure 4 how n R-utomton A uch tht L 2 (A) = Σ. b, 0 b, 0, r 0 1, 1, 1 2 b, 0 Fig. 4. A 2-univerl R-utomton. The lnguge denition nd the univerlity quetion cn lo be formulted for innite word with Büchi cceptnce condition. The unprmeterized ω- lnguge of the utomton from Figure 1 i b ω + b ω. The 2-ω-lnguge of thi utomton i (ɛ + b + bb + bbb) ω.

6 3 Univerlity The min reult of the pper i the decidbility of the univerlity problem for R-utomt formulted in the following theorem. Theorem 1. For given R-utomton A, the quetion whether there i D N uch tht L D (A) = Σ i decidble in 2-EXPSPACE. Firt, we introduce nd lo formlly dene the necery concept (pttern, fctoriztion, nd reduction) together with n overview of the whole proof. Then we how the contruction of the reduced fctoriztion tree nd tte the correctne of thi contruction. Finlly, we preent n lgorithm for deciding univerlity. All proof cn be found in the full verion of thi pper [2]. 3.1 Concept nd Proof Overview When n R-utomton A i not univerl for ll D N then there i n innite et X of word uch tht for ech D N there i w D X nd w D / L D (A). We y then tht X i counterexmple. The min tep of the proof i to how tht there i n X which cn be chrcterized by +-free regulr expreion. In fct, we how tht X lo tie number of dditionl propertie which enble u to decide for every uch +-free regulr expreion, whether it correpond to counterexmple or not. Another tep of the proof i to how tht we need to check only nitely mny uch +-free regulr expreion in order to decide whether there i counterexmple t ll. Pttern. The tndrd procedure for checking univerlity in the ce of nite utomt i ubet contruction. Whenever there re non-determinitic trnition 1 nd 2 then we build ummry trnition {} { 1, 2 }. Thi ummry trnition y tht from the et of tte {} we get to the et of tte { 1, 2 } fter reding the letter. In the ce of R-utomt, ubet contruction i in generl not gurnteed to terminte ince the vlue of the counter might grow unboundedly. To del with thi problem, we exploit the fct tht the vlue of the counter do not inuence the computtion of the utomton. Therefore, we perform n btrction which hide the ctul vlue of the counter nd conider only the eect long the trnition inted. The btrction led to more complicted vrint of ummry trnition nmely o clled pttern. We dene commuttive, ocitive, nd idempotent opertion on the et {0, 1, r}: 0 0 = 0, 0 1 = 1, 0 r = r, 1 1 = 1, 1 r = r, nd r r = r. In fct, if we dene n order 0 < 1 < r then i the opertion of tking the mximum. We extend thi opertion to eect, i.e., n-tuple, by pplying it componentwie (thi preerve ll the propertie of ). An eect obtined by dding everl other eect through the ppliction of the opertor ummrize the mnner in which the counter re chnged. More preciely, it decribe whether counter i reet or whether it i increed but not reet or whether it i only left untouched. A pttern σ : (S S) 2 {0,1,r}n i function from pir of utomton tte to et of eect. Let u denote pttern by σ, σ 1, σ,.... A n exmple,

7 conider pttern σ involving tte nd nd two counter. Let σ(, ) = {(0, 0), }, σ(, ) = {, (1, 0)}, σ(, ) = {} nd σ(, ) = {}. Thi pttern i depicted in Figure 5. Clerly, for given R-utomton there re only nitely mny pttern; let u denote thi nite et of ll pttern by P. We dene n opertion on P follow. Let (σ 1 σ 2 )(, ) = {t, t 1, t 2. t 1 σ 1 (, ), t 2 σ 2 (, ), t = t 1 t 2 }. Note, tht i ocitive nd it h unit σ e, where σ e (, ) = {(0,..., 0)} if = nd σ e (, ) = otherwie. Therefore, (P, ) i nite monoid. For ech word we obtin pttern by running the R-utomton long thi word. Formlly, let Run : Σ + P be homomorphim dened by Run() = σ, where t σ(, ) if nd only if (,, t, ). Loop. In the ce of nite utomt, et of tte L nd word w contitute loop in the ubet contruction if L w L, i.e., trting from L nd reding w, we end up in L gin. The intuition behind the concept of loop i tht everl itertion of the loop hve the me eect ingle itertion. In our btrction uing pttern, loop re word w uch tht w yield the me pttern w 2, w 3,.... We cn kip the trting et of tte, becue the function Run trt implicitly from the whole et of tte S (if there re no run between ome tte then the correponding et of eect i empty). More preciely, word w i loop if Run(w) i n idempotent element of the pttern monoid. Two loop re identicl if they produce the me pttern. Oberve tht the pttern in Figure 5 i idempotent. Fctoriztion. We how tht ech word cn be plit into hort identicl loop repeted mny time. The loop cn poibly be neted, o tht thi plit (fctoriztion) dene fctoriztion tree. The ide i tht ince we hve uch fctoriztion for ech word, it i ucient to nlyze only the (hort) loop nd either nd run with bounded mximl vlue of the counter or ue the loop tructure to contruct counterexmple regulr expreion. On higher level we cn ee fctoriztion of word function which for every word w = 1 2 l return it fctoriztion tree, i.e., nite tree with brnching degree t let 2 (except for the leve) nd with node lbeled by ubword of w uch tht the lbeling function tie the following condition: if node lbeled by v h children lbeled by w 1, w 2,..., w m then v = w 1 w 2 w m, if m 3 then σ = Run(v) = Run(w i ) for ll 1 i m nd σ i idempotent, the leve re lbeled by 1, 2,..., l from left to right. An exmple of uch tree i in Figure 5b. It follow from the fctoriztion foret theorem [11, 4] tht there i uch (totl) function which return tree whoe height i bounded by 3 P where P i the ize of the monoid. We dene the length of loop the length of the word (or pttern equence) provided tht only the two longet itertion of the neted loop re counted. Thi concept i dened formlly in Subection 3.3. We y tht the loop re hort if there i bound given by the utomton o tht the length of ll the loop i horter thn thi bound. A conequence of the fctoriztion foret theorem i tht there i fctoriztion uch tht ll loop re hort.

8 (0, 0), (1, 0), c c cbbc bbc b b c b c () (b) Fig. 5. A pttern involving two tte nd two counter () nd fctoriztion tree (b). Run(bbc) = Run(b) = Run(b) = Run(c) nd it i idempotent. Reduction. We hve dened the loop o tht the itertion of loop hve the me eect the loop itelf. Therefore, it i enough to nlyze ingle itertion to tell how the computtion look when the loop i iterted n rbitrry number of time. By prt in n idempotent pttern σ, we men n element (n eect) in the et σ(, ) for ome tte nd. We will ditinguih between two type of prt, nmely bd nd good prt. A bd prt correpond only to run long which the incree of ome counter i t let big the number of the itertion of the loop. A prt i good if there i run with thi eect long which the incree i bounded by the mximl incree induced by two itertion of the loop. Formlly, we dene function reduce which for ech pttern return pttern contining ll good prt of the originl pttern, but no bd prt. Then we illutrte it on number of exmple. For pttern σ, core(σ) i dened follow: { core(σ)(, σ(, ) = ) {0, r} n if = otherwie Let reduce(σ) = σ core(σ) σ. For n utomton with one tte, one counter, nd loop w with pttern σ, if σ(, ) = {(1)} then the whole pttern i bd, i.e., reduce(σ)(, ) =. Notice tht ny run over w k incree the counter by k. On the other hnd, if σ(, ) = {(0)} or σ(, ) = {(r)} then the whole pttern i good, i.e., reduce(σ) = σ. With more complicted pttern we need more creful nlyi. Let u conider loop w with pttern σ where σ(, ) = {(0)}, σ(, ) = {(1)}, σ(, ) = {(1)}, nd σ(, ) = {(1)}. We will motivte why the prt (1) σ(, ) i good. For ny k, we cn tke the run over w k which trt from, move to fter the rt itertion, ty in for k 2 itertion, nd nlly move bck to fter the k th itertion. Then, the eect of the run i (1). Furthermore, the counter incree long the run i bounded by twice the mximl counter incree while reding w. In fct, uing imilr reoning, we cn how tht ll prt of σ re good (which i conitent with the fct tht reduce(σ) = σ). A the lt exmple, let u conider the pttern from Figure 5. Firt, we how tht the prt (1, 0) σ(, ) i bd. The only run over w k with eect

9 (1, 0) i the one which come bck to fter ech itertion. However, thi run incree the rt counter by k. On the other hnd, the prt σ(, ) i good by imilr reoning to the previou exmple. In fct, we cn how tht ll other prt of the pttern re good (which i conitent with the vlue of reduce(σ) in Figure 6). (0, 0), (0, 0) (0, 0), (0, 0), = (1, 0), (1, 0), Fig. 6. σ core(σ) σ = reduce(σ) where σ i the pttern from Figure 5 Reduced Fctoriztion Tree. For fctoriztion of word w, we need to check whether there i run which goe through good prt in every loop. In order to do tht, we enrich the tree tructure, o tht ech node will now be lbeled, in ddition to word, lo by pttern. The pttern re dded by the following function: given n input equence of pttern, the leve re lbeled by the element of the equence, node with brnching degree 2 re lbeled by the compoition of the children lbel, nd we lbel ech node with brnching degree t let 3 by σ, where σ i the idempotent lbel of ll it children. Now, bed on thi lbeling, we build reduced fctoriztion tree for w in everl tep (formlly decribed in Subection 3.2). We trt with the equence of pttern obtined by Run from the letter of the word. In ech tep, we tke the reulting equence from the previou tep, build fctoriztion tree from it, nd lbel it by pttern decribed bove. Then we tke the lowet node uch tht they hve t let 3 children nd they re lbeled by pttern σ uch tht reduce(σ) σ. We chnge the lbel of thee node to reduce(σ). We pck the ubtree of thee node into element of the new equence nd we leve other element of the equence unmodied. Thi procedure eventully terminte nd return one tree with the following propertie (the importnt invrint i hown in Lemm 1): if node lbeled by σ h two children lbeled by σ 1, σ 2 then σ = σ 1 σ 2, if node lbeled by σ h m children lbeled by σ 1,..., σ m, m 3, then σ i = σ j for ll 1 i, j m, σ 1 i idempotent, nd σ = reduce(σ 1 ). An exmple of reduced fctoriztion tree i in Figure 7. We how tht there i fctoriztion function uch tht the height of ll reduced fctoriztion tree produced by it i bounded by 3 P 2 (Lemm 2) uing the fctoriztion foret theorem nd property of the reduction function tht if reduce(σ) σ then reduce(σ) < J σ, where < J i the uul ordering of the J -cle on P, J i tndrd Green' reltion; σ J σ if nd only if there re σ 1, σ 2 uch tht σ = σ 1 σ σ 2 ; σ < J σ if nd only if σ J σ nd σ J σ (Lemm 2 in [2]).

10 σ 1, bcdecc σ 2, b reduce(σ 5), cdecc σ 3, σ 4, b σ 5, c σ 5, de σ 5, c σ 5, c σ 6, d σ 7, e Fig. 7. An exmple reduced fctoriztion tree. σ 1 = σ 2 reduce(σ 5), σ 2 = σ 3 σ 4, nd σ 5 = σ 6 σ 7. For ll leve lbeled by ˆσ, â, ˆσ = Run(â). Correctne. Let σ be the lbel of the root of reduced fctoriztion tree for word w nd let pump(r, k) for +-free regulr expreion r nd for k N be the word obtined by repeting ech r 1, where r 1 i ubexpreion of r, k-time. Then if σ( 0, f ) for ome f F then there i run from 0 to over w in 8 P 2 -emntic, otherwie, there i +-free regulr expreion r uch tht for ll D there i k uch tht there i counter which exceed D long ll run from 0 to f, f F, over pump(r, k). The previou item re formulted in Subection 3.3, Lemm 4 nd Lemm 5. Reltion to Simon' Approch. There re everl importnt dierence between the method preented in thi pper nd tht of Simon [12]. Our notion of pttern i function to et of eect, while in Simon' ce it i function to the et {0, 1, ω}. Becue of the reet nd the fct tht there re everl counter, it i not poible to linerly order the eect. Thu, collection of utomton run cn be btrcted into everl incomprble eect. The et re necery in order to remember ll of them. Furthermore, the dierent notion of pttern require new notion of reduction which doe not remove loop lbeled lo by reet. We need to how then tht ppliction of thi notion of reduction during the contruction of the reduced fctoriztion tree preerve the correctne. 3.2 Contruction of the Reduced Fctoriztion Tree We dene lbeled nite tree to cpture the looping tructure of pttern equence. Let Γ be et of nite tree with two lbeling function Pt nd Word, which for ech node return pttern nd word, repectively. We will bue the nottion nd, for tree T, we ue Pt(T ) or Word(T ) to denote Pt(N) or Word(N), repectively, where N i the root of T. We lo identify node with the ubtree in which they re root. We cn then y tht node T h children T 1,..., T m nd then ue T i ' tree. For tree T, we dene it height h(t )

11 h(t ) = 1 if T i lef, h(t ) = 1 + mx{h(t 1 ),..., h(t m )} if T 1,..., T m re children of the root of T. By Γ + we men the et of nonempty equence of element of Γ. By (Γ + ) + we men the et of nonempty equence of element of Γ +. Let u denote element of Γ + by γ, γ 1, γ,.... For γ Γ +, let γ denote the length of γ. Let f : Γ + P be homomorphim with repect to dened by f(t ) = Pt(T ). We cll function d : Γ + (Γ + ) + fctoriztion function if it tie the following condition. If d(γ) = (γ 1, γ 2,..., γ m ) then γ = γ 1 γ 2 γ m, if m = 1 then γ = 1, nd if m 3 then f(γ) = f(γ i ) for ll 1 i m nd f(γ) i n idempotent element. For fctoriztion function d we dene two function tree : Γ + Γ nd con : Γ + Γ + inductively follow. Let σ, w denote tree which conit of only the root lbeled by σ nd w. tree(γ) = γ if γ = 1, σ 1 σ 2, w 1 w 2 with children tree(γ 1 ), tree(γ 2 ), if d(γ) = (γ 1, γ 2 ), σ i = Pt(tree(γ i )), w i = Word(tree(γ i )) for i {1, 2}, reduce(σ), w 1 w 2 w m with children tree(γ 1 ),..., tree(γ m ), if m 3, d(γ) = (γ 1, γ 2,..., γ m ), σ = Pt(tree(γ 1 )), nd w i = Word(tree(γ i )) for ll 1 i m. The function tree build tree (reembling fctoriztion tree) from the equence of tree ccording to the function d. The only dierence from trightforwrdly following the function d i tht the lbeling function Pt might be chnged by the function reduce. Let u color the tree in the function con either green or red during the inductive contruction of new equence. γ con(γ) = if γ = 1. Mrk γ green. con(γ 1 ) con(γ 2 ) con(γ m ) if d(γ) = (γ 1, γ 2,..., γ m ) nd either m = 2 or there i 1 i m uch tht con(γ i ) contin red tree or reduce(f(γ 1 )) = f(γ 1 ). tree(γ) if d(γ) = (γ 1, γ 2,..., γ m ), m 3, no con(γ i ) contin red tree nd reduce(f(γ 1 )) f(γ 1 ). Mrk the tree red. The function con updte the equence of tree trying to leve much poible untouched, but whenever Pt would be chnged by the reduce function for the rt time (on the lowet level), it pck the whole equence into ingle tree with chnged Pt lbel of the root uing the function tree. The importnt property of the contruction i tht for ech tree in the new equence it hold tht whenever node h more thn two children, they re ll lbeled by identicl idempotent pttern. Let u cll tree blnced if whenever node T h children T 1, T 2,..., T m, where m 3, then Pt(T 1 ) = Pt(T 2 ) = = Pt(T m ), it i n idempotent element in P, nd Pt(T ) = reduce(pt(t 1 )).

12 T B T A T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T 11 T 12 T 13 T 14 T 15 Fig. 8. Appliction of con to T 1 T 15. The blck node repreent the node for which reduce(σ) σ. The reulting equence i T 1T 2T 3T 4T AT 8T 9T BT 15. Lemm 1. For γ Γ +, if ll tree in γ re blnced then ll tree in con(γ) re blnced. Now we how how to get equence of tree from run of the utomton. Let treerun : Σ + Γ + be homomorphim with repect to the word compoition dened by treerun() = Run(),. Aume tht there i fctoriztion function d xed. Let for word w Σ +, γ w be dened con n (treerun(w)), where n N i the let uch tht con n (treerun(w)) = con n+1 (treerun(w)). Note tht γ w i lwy dened, becue for ll γ Γ +, con(γ) γ nd if con(γ) = γ then con(γ) = γ. Let T w = tree(γ w ). We cll T w the reduced fctoriztion tree of w contructed by d. From Lemm 1 it follow tht T w i blnced (note tht if con n (γ) = con n+1 (γ) then con n (γ) contin only green tree). Remrk. Notice tht we do not explicitly mention the fctoriztion function d in the denition of reduced fctoriztion tree T w contructed by d from word w. It i lwy cler from the context which fctoriztion function we men. We how tht for ech R-utomton there i fctoriztion function uch tht for ny w the height of the tree T w i bounded by contnt computed from the prmeter of the utomton. Lemm 2. Given n R-utomton A, there i fctoriztion function d uch tht for ll word w Σ +, h(t w ) 3 P Correctne To formulte the rt correctne lemm, we dene the following concept of length function l : Γ N inductively by 1 if T i lef l(t ) = l(t 1 ) + l(t 2 ) if T h two children T 1, T 2 2 mx{l(t 1 ),..., l(t m )} if T h children T 1,..., T m, m 3 By induction on h(t w ) nd uing the bound derived in Lemm 2, one cn how the following clim.

13 Lemm 3. Given n R-utomton A, there i fctoriztion function d uch tht for ll word w Σ +, l(t w ) 8 P 2. w We y tht w or D relize t if there i witneing pth (, 1, t 1, 1 ), ( 1, 2, t 2, 2 ),..., ( w 1, w, t w, ) uch tht t = t 1 t 2 t w. Let u dene Run D (w) to be the pttern obtined by running the utomton over w in the D-emntic. Formlly, Run D (w)(, ) contin t if nd only if w D relize t. Note tht the function Run D i not homomorphim with repect to the word compoition. We lo dene reltion on pttern by σ σ if nd only if for ll,, σ(, ) σ (, ). From Lemm 3 we how tht there i fctoriztion function uch tht for every w, Pt(T w ) correpond to the run of the R-utomton which cn be performed in the D-emntic for ny big enough D. Thi i formulted in the following lemm. Lemm 4. Given n R-utomton, there i fctoriztion function uch tht for ll w Σ + nd for ll D N, D 8 P 2, Pt(T w ) Run D (w). Of prticulr interet re run trting in the initil tte. Corollry 1. Given n R-utomton A, there i fctoriztion function uch tht for ll word w, if Pt(T w )( 0, ) then there i run 0, (0,..., 0) w D, (c 1,..., c n ) where D = l(t w ). It remin to how tht if the reltion between the pttern in the previou lemm i trict then there i word for ech D which i witne for the trictne, i.e., the run over thi word in the D-emntic generte mller pttern thn over the originl word. Thee witne word re generted from +-free regulr expreion r by pumping r 1 for ll ubexpreion r 1 of r. Let u dene function re which for reduced fctoriztion tree return +-free regulr expreion inductively by Word(T ) if T i lef re(t ) = re(t 1 ) re(t 2 ) if T h two children T 1, T 2 (re(t 1 )) if T h children T 1, T 2,..., T m, m 3 For +-free regulr expreion r nd nturl number k > 0, let the function pump(r, k) be dened inductively follow: pump(, k) =, pump(r 1 r 2, k) = pump(r 1, k) pump(r 2, k), nd pump(r, k) = pump(r, k) k. For exmple, pump((bc d), 2) = bccdbccd. Lemm 5. Given n R-utomton nd fctoriztion function, for ll w Σ + nd ll D N there i k N uch tht Run D (pump(re(t w ), k)) Pt(T w ). A pecil ce re run trting from the initil tte. Corollry 2. Given n R-utomton, for ny w Σ +, if Pt(T w )( 0, ) = v then D k uch tht there i no run 0, (0,..., 0) D, (c 1,..., c n ) where v = pump(re(t w ), k).

14 3.4 Algorithm To check the univerlity of n R-utomton A, we hve to check ll pttern σ uch tht σ = Pt(T w ) for ome w Σ + nd ome fctoriztion function. If there i σ uch tht for ll f F, σ( 0, f ) = then for ll D N, L D (A) Σ. Thi give u the following lgorithm. Recll tht σ e denote the unit of (P, ). The lgorithm ue et of pttern P the dt tructure. Given n R- utomton A = S, Σ,, 0, F on the input, it nwer 'YES' or 'NO'. The et P i initilized by P = {σ σ = Run(), Σ} {σ e }. While P incree the lgorithm perform the following opertion: pick σ 1, σ 2 P nd dd σ 1 σ 2 bck to P. pick σ P uch tht σ i idempotent nd dd reduce(σ) bck to P. If there i σ P uch tht for ll f F, σ( 0, f ) =, nwer 'NO', otherwie, nwer 'YES'. The correctne i tted in the following theorem. See [2] for the full proof. Theorem 2. The lgorithm i correct nd run in 2-EXPSPACE. Proof. The lgorithm terminte becue P i nite. It correctne follow from the previou two corollrie. The lgorithm need pce P (the number of different pttern). The ize of P i 2 (3n ) S 2 ( S 2 dierent pir of tte, 2 (3n) different et of eect). Therefore, the lgorithm need double exponentil pce. 4 Büchi Univerlity The univerlity problem i lo decidble for R-utomt with Büchi cceptnce condition. Theorem 3. For given R-utomton A, the quetion whether there i D N uch tht L ω D (A) = Σω i decidble in 2-EXPSPACE. To how thi reult, we need to extend pttern by ccepting tte informtion. A pttern i now function σ : S S 2 {0,1} {0,1,r}n, where for, nd, t σ(, ), the vlue of encode whether there i pth from to relizing t which meet n ccepting tte. For intnce, σ(, ) = { 0, (0, r), 1, } men tht there re two dierent type of pth between nd : they either relize (0, r) but do not viit n ccepting tte, or relize nd viit n ccepting tte. We dene the compoition by dening the compoition on the ccepting tte: 0 0 = 0, 0 1 = 1 0 = 1 1 = 1. The et of pttern (denote gin P) with i nite monoid. We dene the function reduce in the me wy before, i.e., the ccepting tte informtion doe not ply ny role there. Clerly, either reduce(σ) = σ or reduce(σ) < J σ, o the reduced fctoriztion tree hve bounded height. Lemm 4 nd Lemm 5 lo hold, becue (non)viiting n ccepting tte doe not inuence the run in the D-emntic.

15 Thi llow u to ue the me lgorithm for the nite word univerlity problem, except for the condition for concluding non-univerlity. The condition i whether there re σ 1, σ 2 P uch tht σ 2 i idempotent nd for ll uch tht σ 1 ( 0, ) the following hold. If, t σ 2 (, ) then either = 0 or t / {0, r} n. For full proof of Theorem 3 ee [2]. 5 Concluion We hve dened R-utomt nite utomt extended with unbounded counter which cn be left unchnged, incremented, or reet long the trnition. A the min reult, we hve hown tht the following problem i decidble in 2-EXPSPACE. Given n R-utomton, i there bound uch tht ll word re ccepted by run long which the counter do not exceed thi bound? We hve lo extended thi reult to R-utomt with Büchi cceptnce condition. A future work, one cn conider the (bounded) univerlity or limitedne quetion to vector ddition ytem (VASS) or reet vector ddition ytem (R-VASS), where the ltter form upercl of R-utomt. The limitedne problem cn be hown undecidble for R-VASS for both nite word nd ω-word ce, while it i n open quetion for VASS. The univerlity problem cn be hown to be undecidble for R-VASS for ω-word ce, in other ce it i open. Reference 1. P. A. Abdull, P. Krcl, nd W. Yi. Smpled univerlity of timed utomt. In Proc. of FOSSACS'07, volume 4423 of LNCS, pge 216. Springer, P. A. Abdull, P. Krcl, nd W. Yi. R-utomt. Technicl Report , IT Deprtment, Uppl Univerity, Jun M. Boj«czyk nd T. Colcombet. Bound in omeg-regulrity. In Proc. of LICS'06, pge IEEE Computer Society Pre, T. Colcombet. Fctorition foret for innite word. In Proc. of FCT'07, volume 4639 of LNCS, pge Springer, T. Colcombet nd C. Löding. The non-determinitic Motowki hierrchy nd ditnce-prity utomt. In Proc. of ICALP'08, volume 5126 of LNCS, pge Springer, K. Hhiguchi. Limitedne theorem on nite utomt with ditnce function. Journl of Computer nd Sytem Science, 24(2):233244, K. Hhiguchi. Improved limitedne theorem on nite utomt with ditnce function. Theoreticl Computer Science, 72(1):2738, D. Kirten. Ditnce deert utomt nd the tr height problem. Informtique Theorique et Appliction, 39(3):455509, P. Krcl nd R. Pelnek. On mpled emntic of timed ytem. In Proc. of FSTTCS'05, volume 3821 of LNCS, pge Springer, H. Leung. Limitedne theorem on nite utomt with ditnce function: n lgebric proof. Theoreticl Computer Science, 81(1):137145, I. Simon. Fctoriztion foret of nite height. Theoreticl Computer Science, 72(1):6594, I. Simon. On emigroup of mtrice over the tropicl emiring. Informtique Theorique et Appliction, 28(3-4):277294, 1994.