Asynchronous cellular automata for pomsets. Institut fur Algebra, Technische Universitat Dresden, D Dresden.

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1 Aynchronou cellular aomata for omet Manfred Drote 1, Paul Gatin 2, and Dietrich Kuke 1y 1 Intit fur Algebra, Techniche Univeritat Dreden, D-0102 Dreden fdrote,kukeg@math.tu-dreden.de 2 LIAFA, ERS 58, Univerite Pari 7, 2, lace Juieu, F Pari Cedex 05 Paul.Gatin@liafa.juieu.fr Abtract Thi aer extend to omet witho ao-concurrency the fundamental notion of aynchronou cellular aomata (ACA) which wa originally introduced for trace by Zielonka. We generalize to omet the notion of aynchronou maing introduced by Cori, Metivier and Zielonka and we how how to contruct a determinitic ACA from an aynchronou maing. Then we invetigate the relation between the exreivene of monadic econd order logic, nondeterminitic ACA and determinitic ACA. We can generalize Buchi' theorem for nite word to a cla of omet witho ao-concurrency which atify a natural axiom. Thi axiom enure that an aynchronou cellular aomaton work on the omet a a concurrent read and excluive owner write machine. More reciely, in thi cla non-determinitic ACA, determinitic ACA and monadic econd order logic have the ame exreive ower. Then we conider a cla where determinitic ACA are trictly weaker than nondeterminitic one. B in thi cla nondeterminitic ACA till cature monadic econd order logic. Finally it i hown that even thi equivalence doe not hold in the cla of all omet ince there the cla of recognizable omet language i not cloed under comlementation. 1 Introduction In a ditribed ytem, ome event may occur concurrently, meaning that they may occur in any order or imultaneouly or even that their execion may overla. Thi i the cae in articular when two event ue indeendent reource. On the other hand, ome event may caually deend on each other. For intance, the receiving of a meage m follow it ending. Therefore, a ditribed behavior may be abtracted a a omet, that i a et of event together with a artial order which decribe caual deendencie of event and with a labeling function. In thi aer, we mainly deal with Reearch uorted by PROCOPE y Suorted by the German Reearch Foundation (DFG). 1

2 omet witho ao-concurrency: concurrent event m have dierent label. Thee omet are called emi-word in [Sta81, Die94]. For tudie how general omet can be ued to rereent arallel rocee and how they can be comoed, we refer the reader e.g. to [Pra8, Gi88]. There are everal way to decribe the behavior of a ytem. For intance, logic formula are uited for ecication uroe. Deending on the roertie we have to exre, we can ue variou logic uch a temoral logic, rt order logic or (monadic) econd order logic. On the other hand, tranition ytem are often ued to give more oerational decrition. In thi aer, we will concentrate on thee two kind of decrition of ytem. When dealing with ditribed ytem, it i natural to look for tranition ytem which faithfully reect the concurrency. For intance, Petri net are a widely tudied cla of uch tranition ytem. Aynchronou cellular aomata (ACA) form another fundamental cla of tranition ytem with built-in concurrency. They were introduced for trace by Zielonka [Zie87, Zie89]. Mazurkiewicz introduced trace in order to decribe the behavior of one-afe Petri net [Maz77, Maz8]. A trace i a omet where the artial order i dictated by a tatic deendence relation over the action of the ytem. The rimary aim of thi work i to generalize the notion of ACA o that they can work on omet witho ao-concurrency. In Section 3, we dene our notion of ACA. There are two oible denition of run of an ACA on a omet and for each of thee denition there are two oible criteria for accetance. Thu, an ACA may work in four dierent mode. Section 4 tart with the roof that two of them are equivalent for nondeterminitic ACA. Aynchronou maing have roven to be a baic tool to contruct ACA for trace [CMZ93]. In Section 4.2, we give a denition of aynchronou maing for general omet. We how that a omet language recognized by an aynchronou maing can be acceted by a determinitic ACA in any mode. The ret of thi aer i devoted to the relation between ACA and monadic econd order (MSO) logic for omet. We rove in Section 4.3 that from a (non determinitic) ACA one can contruct a MSO formula which dene reciely the omet language acceted by the aomaton in a given mode. In Section 5, we rove the convere for the ecial ubcla of omet for which the ACA work a a concurrent read and excluive owner write (CROW) machine. Thee omet are called CROW-omet. More reciely, from a MSO formula we contruct a determinitic ACA which in a given mode accet reciely the CROW-omet dened by the formula. Therefore, for CROW-omet, we have the equivalence between (exitential) MSO logic, determinitic ACA and non determinitic ACA (for any of the alternative mode). Thi reult i crucial ince it oen the way of model checking for ditribed ytem whoe behavior are decribed a CROW-omet. In Section, we retrict our attention to k-omet and ACA. We can how that the exreive ower of non-determinitic ACA in any mode, MSO logic and exitential MSO logic coincide. Thu, in articular, any ACA running in a given mode can be imulated by a non-determinitic ACA that run in any other mode. B thi im- 2

3 ulating aomaton ha to be non-determinitic ince, a we how, the four mode give rie to incomarable concet of determinitic recognizability. Thu, in the cla of k- omet, non-determinitic ACA are trictly more exreive than determinitic one and determinitic ACA are more exreive than aynchronou maing. Furthermore, emtine, univerality and equivalence for ACA are decidable within thi cla. In Section 7, we return to the cla of all omet. We how that in thi context the mode give rie to two incomarable concet of nondeterminitic recognizability. Thi already imlie that for omet MSO logic i trictly more exreive than ACA in any mode. Thi negative reult i harened in the nal art where we reent an examle of a rt-order denable omet roerty that cannot be recognized by an ACA. Since the negation of thi roerty i recognizable, we obtain that the cla of recognizable omet language i not cloed under comlementation. The roof of the oitive reult are baed on the claical reult by Zielonka [Zie87], on a cloe analyi of the omet and aomata in conideration, and on a technique develoed by Thoma [Tho90] for aynchronou aomata for trace. The negative reult are hown by earating examle. Preliminary verion of thee reult have aeared in the extended abtract [DG9] and [Ku98]. 2 Preliminarie 2.1 Pomet Let be a nite et, called alhabet. A omet over i (an iomorhim cla of) a nite labeled artial order t = (V; ; ) where V i a nite et of vertice, i the artial order on V and : V! i the labeling function. The emty omet (;; ;; ;) will be denoted by 1. Througho the aer we will mainly deal with omet witho ao-concurrency, that i omet t = (V; ; ) uch that?1 (a) i totally ordered for all a 2. Let = (V ; ; ) and t = (V t ; t ; t ) be two omet. We ay that i a rex of t, if i (iomorhic to) a downward cloed ubomet of t, that i, if V i a downward cloed ubet of V t (i.e. V V t and for all u; v 2 V t, u t v and v 2 V imly u 2 V ), i the retriction of t to V (i.e. = t \V V ) and i the retriction of t to V. The rex order relation i a artial order on the et of all omet (even if we allow ao-concurrency). Since we have aumed non ao-concurrency, there i a unique way to embed a rex of t a a downward cloed ubomet of t. Hence, we will identify a downward cloed ubet of vertice with the correonding rex of the omet. Let 1 = (V 1 ; 1 ; 1 ) and 2 = (V 2 ; 2 ; 2 ) be two rexe of a omet t = (V t ; t ; t ). Then, V 1 [ V 2 i a downward cloed ubet of V t and the correonding rex of t i 1 [ 2 = (V 1 [ V 2 ; 1 [ 2 ; 1 [ 2 ) where 1 [ 2 i the labeling which coincide with 1 on V 1 and with 2 on V 2 (note that 1 and 2 agree on V 1 \ V 2 ). Let t = (V; ; ) be a omet. The downward cloure of a vertex v i denoted by #v = fu 2 V j u vg. The trict downward cloure of a vertex v i denoted by 3

4 +v = #v n fvg. Since #v and +v are downward cloed ubet of V, we will identify thee et with the correonding rexe of t. Let 1 ; : : : ; n be airwie dijoint alhabet and let = 1 _[ _[ n. Intuitively, we can view [n] = f1; : : : ; ng a a et of label of equential rocee and 1 ; : : : ; n a the et of action of thee equential rocee. Let :! [n] be the maing which aociate with each letter a 2 the roce (a) 2 [n] which exece the letter a, i.e. a 2 (a). Let t = (V; ; ) be a omet. We ay that a vertex v cover a vertex u, denoted by u?< v, if u < v and there i no vertex w uch that u < w < v. We ay that two vertice u; v 2 V are incomarable or concurrent, denoted by u k v, if neither u v nor u v. We may ee the covering relation a the decrition of the interaction between the rocee. More reciely, we conider that an event v 2 V read the tate of the rocee (fu j u?< vg) and write in the roce (v), which, by abue of notation, will be abbreviated by (v). We will not allow concurrent write, therefore two concurrent event u k v m write in dierent rocee (u) = (v). Thi lead to the following Denition 2.1 A ( 1 ; 2 ; : : : ; n )-omet or -omet ~ i a omet t = (V; ; ) for which?1 ( i ) i totally ordered for all 1 i n. The et of all ~-omet will be denoted by P(~). Note that with thi notation the et P() i the et of word over. It i eaily een that -omet ~ are ecial omet witho ao-concurrency. If the et 1 ; : : : ; n are all ingleton P(~) i the et of all omet witho ao-concurrency. For A, we denote A (t) = #?1 (A) the leat rex of a S omet t which contain all element labeled with letter from A. Note A (t) = vj(v)2a #v. For a 2 and i 2 [n], we will ue the following imlied a (t) fag (t) i (t) i (t). Note that if?1 (A) i totally ordered A (t) i either emty or ha exactly one maximal vertex. In articular, if t i a -omet ~ i (t) i either emty or ha exactly one maximal vertex. Occaionally, we will i (t) with it maximal vertex. 2.2 Trace We recall now baic denition for Mazurkiewicz trace which will be needed in thi aer. The reader i referred to [DR95] for a general reentation of trace theory. A deendence alhabet i a air (; D) where i a nite alhabet and D i a reexive and ymmetric relation over called the deendence relation. Intuitively, two deendent action (a; b) 2 D m be execed equentially while two indeendent action (a; b) =2 D may occur concurrently. More formally, one conider the congruence relation over the free monoid? generated by the relation f(ab; ba) j (a; b) =2 Dg. A trace i imly an equivalence cla of word for the congruence. The trace monoid i then the quotient M (; D) =? =. 4

5 We give now an equivalent denition of trace which i more adequate in our context. Baically, a trace can be een a a omet which atie additional requirement. More reciely, we will ee that a trace over the deendence alhabet (; D) i a omet t = (V; ; ) uch that for all vertice u; v 2 V, ((u); (v)) 2 D =) u v or v u (1) u?< v =) ((u); (v)) 2 D (2) Note that a linearization of a omet t may be identied with a word of?. Now, let t = (V; ; ) be a omet atifying condition (1) and (2). Then, the et of linearization of t i reciely a trace, that i, an equivalence cla for. Hence, with each omet t atifying (1) and (2), one can aociate a trace '(t). Converely, a word u 2? dene a labeled linear order (V u ; u ; u ) over the occurrence of action of u: V u = f(a; i) j 1 i juj a g (juj a denote the number of occurrence of a in u); (a; i) u (b; j) if the i-th a occur before the j-th b in u; and u ((a; i)) = a. Since two equivalent word u v have the ame et of occurrence T of action (V u = V v ), we can aociate with a trace [u] the omet ([u]) = (V u ; vu v; u ). One can check that ([u]) atie condition (1) and (2) and that and ' are invere bijection. Thi exlain why the two denition are equivalent. We will now dene recognizable trace language. A trace aomaton i a quadrule A = (Q; T; I; F ) where Q i a nite et of tate, I Q i the et of initial tate, F Q i the et of nal tate and T Q Q i the et of tranition which atie the diamond roerty: for all (a; b) 2 ( ) n D and q; q 0 ; q 00 2 Q, if (q; a; q 0 ) 2 T and (q 0 ; b; q 00 ) 2 T then there exit ome q 0 2 Q uch that (q; b; q 0 ) 2 T and (q 0 ; a; q 00 ) 2 T. A word w = a 1 a n 2? i acceted by A if there i a run q 0 ; a 1 ; q 1 ; : : : ; a n ; q n uch that q 0 2 I, q n 2 F and (q i?1 ; a i ; q i ) 2 T for all 1 i n. A trace t 2 M (; D) i acceted by A if ome linear extenion of t i acceted by A. Note that, thank to the diamond roerty of a trace aomaton, if ome linear extenion of a trace i acceted by A then all linear extenion of t are acceted by A. A trace language L M (; D) i recognizable if it i the et of trace acceted by ome trace aomaton. Equivalently, it i a recognizable language in the monoid M (; D), a uual, in the ene of [Eil74]. 3 Aynchronou cellular aomata Denition 3.1 A ( 1 ; 2 ; : : : ; n )-aynchronou cellular aomaton (or ~ -ACA) i a tule A = ((Q i ) i2[n] ; ( a;j ) a2;j[n] ; F ) where 1. for all i 2 [n], Q i i a nite et of local tate for roce i, Q 2. for all a 2 and J [n], a;j : Q i2j i! P(Q (a) ) i a (nondeterminitic) tranition function (where P denote the ower et oerator) and 3. F S J[n] Q i2j Q i i a et of acceting tate. 5

6 The aomaton i determinitic if all the tranition function are determinitic, i.e. if j a;j ((q i ) i2j )j 1 for all a 2, J [n] and q i 2 Q i for i 2 J. We now exlain how a ~ -ACA can accet a ~ -omet t = (V; ; ). The idea i that the ~-ACA conit of n local rocee whoe local tate are Q i. Then, any event x 2 V change the tate of it roce (x), only. Thi change deend on the local tate of the rocee in the read domain of thi event. There are (at leat) two reaonable read domain of an event x: The rt one i that it read only the local tate reached at the event covered by x. In articular, in thi mode it may haen that x doe not read the lat tate of it own roce (x). In the econd reading mode, x read for each roce that acted in the at of x the tate at the lat event below x on thi roce. Since the rt reading mode i a retriction of the econd one, we refer to it a the R? -mode. The econd i called the R + -mode. To dene thee two kind of run uniformly, let t = (V; ; ) be a ~-omet and x 2 V. Then R? (x) := fy 2 V j y?< xg = (max(+x)) and R + (x) := fy 2 V j y < xg = (+x): Note that R? (x) R + (x). For 2 f+;?g, an R -run of A on t i a function r : V! S i2[n] Q i uch that r(x) 2 (x);r (x)(r(@ i (+ x)) i2r (x)) for any x 2 V. Note that an R? -run can be een a a run on the Hae-diagram of the ~ -omet. To comare it with an R + -run, let t = (V; ; ) be a ~ -omet and dene the binary relation E on V by E = f(@ i (+x); x) j x 2 V; i 2 R + (x)g: Then?< E. Hence i the tranitive and reexive cloure of E. An R + -run can be een a a run on the directed acyclic grah (V; E; ). Let u conider two examle that make the dierence between the two reading mode clear. In both examle, we reent a determinitic ~ -ACA and then decribe the et of all ~-omet that admit an R -run of thi aomaton. Later, we will ee that thee et cannot be acceted by a ~-ACA in the other reading mode. Examle 3.2 Let n = 3, 1 = fag, 2 = fbg and 3 = fcg. We conider the aomaton given by Q 1 = Q 2 = Q 3 = fqg and ( ; x = c and 1 2 J x;j ((q i ) i2j ) = fqg otherwie for any x 2 fa; b; cg. Next we decribe the et of -omet ~ that admit an R? -run of thi aomaton: Since there i only one tate, for any omet t = (V; ; ) there i only one maing r : V! fqg. Thi maing i an R? -run i no c-labeled event

7 cover an a-labeled one. Indeed, if we have x?< y with (x) = a and (y) = c, then (y);r? (y)((q i ) i2r? (y)) = ; and therefore r doe not atify the condition for an R? -run. Note that a ~ -omet ha an R + -run of thi aomaton i it doe not contain an a-labeled event below ome c-labeled one. Examle 3.3 Let n = 4, 1 = fag, 2 = fbg, 3 = fcg and 4 = fdg. The aomaton i given by Q 1 = f0; 1g, Q 2 = Q 3 = Q 4 = f0g, and ( f(q 1 + 1) mod 2g if 1 2 J a;j ((q i ) i2j ) = f1g otherwie b;j ((q i ) i2j ) = f0g c;j ((q i ) i2j ) = f0g ( f0g if q 1 = 0 or 1 =2 J d;j ((q i ) i2j ) = ; otherwie. Thi aomaton i meant to run in the R + -mode. Note that in thi cae the rt roce imly count it event modulo 2. The econd and third roce do \nothing". Now conider the fourth roce. It allow a tranition a long a no event from the rt roce occurred. Once uch an a-labeled event occurred, it only allow a tranition if the tate on the lat uch event i 0. Thu, the aomaton cannot roceed if ome d-labeled event dominate an odd number of a-labeled event. Since thi i the only cae where it cannot roceed, thi ~-ACA allow an R + -run on a ~-omet i any d-labeled event dominate an even number of a-labeled one. In the R? -mode, the rt roce doe not count the occurrence of a-labeled event modulo 2 becaue it retart with 1 whenever ome a-labeled event doe not directly cover another uch a-labeled element. Similarly, there are two oibilitie to dene when a run i ucceful. The rt one i to conider all local nal tate. Alternatively, we may retrict our attention to the tate that correond to the maximal element of the omet in conideration. More formally, let A be a ~ -ACA with nal tate F and let t be a ~ -omet. Then F + (t) := (t) denote the et of local rocee that erform at leat one te when t i execed. The et F? (t) := (max(t)) comrie thoe local rocee that erform a maximal event. For ; 2 f+;?g, an R -run r i F -ucceful i r(@ i (t)) i2f (t) 2 F: Examle 3.4 Our next examle i a determinitic -ACA ~ which accet reciely the et of ~-omet atifying condition (2) of Section 2.2. More reciely, let (; D) be a deendence alhabet with = 1 _[ _[ n where each i i a clique of (; D). We dene the -ACA ~ A = ((Q i ) i2[n] ; ( a;j ) a2;j[n] ; F ) where Q i = i [ f?g for all i 2 [n] and ( fag if q j =? and (a; q j ) 2 D for all j 2 J a;j ((q j ) j2j ) = f?g otherwie 7

8 S for all a 2 and J [n]. Finally, the et of acceting tate i F = J[n] i2j i. In an R? -run of thi aomaton, each roce remember the lat action erformed and therefore i able to check that an event cover only deendent event. One can eaily check that R? F (A) i the et of ~-omet (V; ; ) uch that for all u; v 2 V, if u?< v then ((u); (v)) 2 D. For intance, if (; D) = a b c d with 1 = fa; bg, 2 = fcg and 3 = fdg, we give below a rejecting run and an acceting run of A. In thi icture, each vertex v i labeled by the air ((v); r(v)). Note that, in order to obtain the tate of minimal vertice, we aly tranition function of the form (v);;. Q 1 2 b; b a;? b;? c; c c;? 3 d; d d; d d; d 1 2 a; a b; b b; b b; b c; c c; c 3 d; d d; d d; d Examle 3.5 Later, we will ee that for any deendence alhabet (; D), it i oible to contruct a nondeterminitic ACA A that accet reciely the trace over (; D). Here, we give uch an aomaton for the imle deendence alhabet (; D) = a b c. We conider the rocee 1 = fag, 2 = fbg and 3 = fcg. The et of tate of the ( 1 ; 2 ; 3 )-ACA A are Q 1 = Q 2 = Q 3 = fa; bg fb; cg and all oible combination of tate are acceting. Intuitively, a tate (x; y) claim that the next event labeled by a or by b i actually labeled by x (and imilarly the next event labeled by c or by b i actually labeled by y). The tranition function are the following (we only give the non emty tranition): b;; = b;f1g ((b; b)) = b;f2g ((b; b)) = b;f3g ((b; b)) = b;f1;3g ((b; c); (a; b)) = f(b; b); (a; b); (b; c); (a; c)g a;f1g ((a; b)) = a;f2g ((a; b)) = f(a; b); (b; b)g a;f1g ((a; c)) = a;f2g ((a; c)) = f(a; c); (b; c)g c;f2g ((b; c)) = c;f3g ((b; c)) = f(b; c); (b; b)g c;f2g ((a; c)) = c;f3g ((a; c)) = f(a; c); (a; b)g Here i an R? -run of thi aomaton: 8

9 a; (a; c) a; (b; c) b; (a; c) b; (b; b) b; (b; c) b; (a; b) c; (a; b) c; (b; b) It i eay to ee that all trace tarting with b admit an F? -ucceful R? -run of A. Although le trivial, the convere i alo true. Therefore, in the mode R? F?, thi aomaton accet the et of trace tarting with b. By changing the initial condition of the aomaton, we can recognize all trace tarting with a or with c or with a and c. For intance, if we et b;; = c;; = ; and a;; = f(a; b); (b; b)g we accet all trace tarting with a. Now let P P(~) be ome et of ~-omet and let ; 2 f+;?g. For a ~-ACA A, the language of omet from P acceted by A in the mode R F, denoted by R F (A; P), i the et of all ~-omet t 2 P that admit an F -ucceful R -run. Then R F (P) denote the et of all language R F (A; P) for ome -ACA ~ A. The et dr F (P) contain all language R F (A; P) for ome determinitic -ACA ~ A. Often, we will abbreviate (d)r F (A; P(~)) by (d)r F (A). We conclude thi ection with a few remark. Firt, the covering of omet by the chain formed by the xed equential rocee i crucial in the denition of aynchronou cellular aomata. It allow u to ue a xed number of local tate and to determine the read and write domain of the action uing the labeling and the order relation. The weaket covering i when each i i a ingleton. In thi cae we have a et of local tate er letter a in the aynchronou cellular aomata for trace [Zie89, CMZ93]. Note that, even with thi trivial covering, our denition i not the ame a that of Zielonka for trace. Mainly, in our denition, a run of the ACA i over the Hae diagram (over the directed acyclic grah (V; E; ), reectively) of the omet wherea with Zielonka' ACA for trace, a run i in fact over the deendence grah of the trace. A deendence grah i an intermediary rereentation of a trace between it Hae diagram and it directed acyclic grah (V; E; ). Thi intermediary rereentation i oible thank to the exitence of a tatic deendence relation over action. More reciely, our denition of ACA for omet and that of Zielonka for trace dier in three reect. Firt, Zielonka' denition ue a global initial tate which in our cae i coded in the tranition function of the form a;;. Second, the read domain in our denition deend on the actual omet wherea in Zielonka' denition a xed et of rocee i read even if the lat execion of ome of thee rocee are far below the current action. Third (eecially in the accetance mode F? ), we do not necearily read the nal tate of all local rocee to determine whether a run i ucceful wherea in Zielonka' denition the tate of all rocee are collected globally to decide accetance. 9

10 4 ACA on general omet - oitive reult 4.1 The accetance mode Firt we how that changing the accetance mode between? and + reerve the exreive ower of nondeterminitic aynchronou cellular aomata. Our contruction yield a nondeterminitic aomaton even if we tart with a determinitic one. Later, in Prooition.4, we will ee that in general for a determinitic ACA there i no determinitic aomaton that accet the ame language in the other accetance mode, i.e. that the following theorem doe not hold for determinitic ACA. Theorem 4.1 Let 2 f+;?g. Then R F? (P(~)) = R F + (P(~)), i.e. for any ~-ACA A there exit nondeterminitic ~ -ACA A 1 and A 2 uch that R F? (A; P( ~ )) = R F + (A 1 ; P( ~ )) and R F + (A; P( ~ )) = R F? (A 2 ; P( ~ )): Proof. Firt, we contruct the ~-ACA A 1. Thi ~-ACA A 1 will imulate the run of A on ome omet t and additionally will gue the maximal node of t for each roce. To do thi, along a ucceful R -run of A 1 on t, any local roce i nondeterminitically ick one and only one node x with (x) = i. Thi node end a ignal End i uward. To check thi gue, whenever a node y with (y) = i receive the ignal End i, the aomaton to, i.e. no ucceor tate i dened and therefore the aomaton i forced to reject. Note that i 2 F + (t) i actually in F? (t) i there i no node z with (z) = i that received the ignal End i. Hence, the aomaton A 1 can come the tule of nal maximal tate of the run of A from it tule of nal tate and accet or reject the omet accordingly. Now we contruct the ~-ACA A 2. The idea i a follow: Similarly to the contruction above, A 2 imulate A. Additionally, any local roce i 2 [n] guee it maximal node and end a ignal (End i ; i ) uward where i i the current tate of roce i. Thi ignal i forwarded uward by all tranition. The ACA A 2 to if a node of roce i receive the ignal End i ince thi mean that the gueed node wa not maximal in it chain. By reading the maximal tate of the run of A 2 on t we can now recover the nal tate of each roce in F + (t) and accet or reject the omet according to the accetance condition of A. 4.2 Aynchronou maing Aynchronou maing were introduced in [CMZ93] in order to imlify the contruction of ACA for trace. Here we generalize thi notion to ~-omet. The domain of an aynchronou maing m be a rex cloed ubet of the et of ~ -omet, that i a ubet Q of ~ -omet uch that if ome ~ -omet i a rex of t 2 Q then 2 Q. For intance, P( ~ ) and M (; D) are rex cloed et of ~ -omet. Denition 4.2 Let Q be a rex cloed et of ~ -omet and let S be a nite et. A maing : Q! S i aynchronou if for all t = (V; ; ) 2 Q, 10

11 1. for all vertice x 2 V, the value (#x) i uniquely determined by (+x) and (x). 2. for all A; B, the value (@ A[B (t)) i uniquely determined by (@ A (t)) and (@ B (t)). A language L Q i recognized by an aynchronou maing : Q! S whenever L =?1 ((L)). Prooition 4.3 Let Q P(~) be a rex cloed et of ~-omet and ; 2 f+;?g. Let L Q be a language of ~ -omet recognized by ome aynchronou maing : Q! S. Then there exit a determinitic ~ -aynchronou cellular aomaton A uch that R F (A; Q) = L. Proof. In thi roof, we conider only the cae = =?. For the other mode, the roof remain eentially the ame. One only ha to change accordingly the denition of the tranition function and of the nal et. Thi i left to the reader. Our roof follow the ame idea a the correonding one for trace. Aume that : Q! S recognize the language L Q. We dene a determinitic -ACA ~ A a follow: 1. For all i 2 [n], let Q i = S, 2. For all a 2, J [n] and (q i ) i2j 2 S J, let a;j ((q i ) i2j ) = f(t) j t 2 Q; t = #x for ome x uch that 3. F = f(@ i (t)) i2f? (t) j t 2 Lg. (x) = a; R? (x) = J and (@ i (+x)) = q i for all i 2 Jg Claim 1. A i determinitic. Indeed, let a 2, J [n] and (q i ) i2j 2 S J. Chooe t = #x and t 0 = #x 0 in Q with (x) = (x 0 ), R? (x) = J = R? (x 0 ) and (@ i (+x)) = q i = (@ i (+x 0 )) for all i 2 J. We have +x ([i2j i )(+x) and +x 0 ([i2j i )(+x 0 ). Hence, uing the denition of aynchronou maing, we deduce (+x) = (+x 0 ) and ince (x) = (x 0 ) it follow that (t) = (#x) = (#x 0 ) = (t 0 ) which rove the claim. Claim 2. Let t = (V; ; ) 2 Q be a ~-omet. S Then, the maing r : V! Q i2[n] i dened by r(x) = (#x) i the R? -run of A on t. One only ha to check that r(x) 2 (x);r? (x) (r(@ i (+ x)) i2r? (x) ) for any x 2 V. B thi follow directly from the denition of the tranition function of A. 11

12 Claim 3. R? F? (A; Q) = L. Note rt that t ([i2f? (t) i )(t) for all t 2 Q. Now aume that t 2 R? F? (A). Then for the unique run r of A on t, we have r(@ i (t)) i2f? (t) 2 F and there exit t 0 2 L uch that F? (t) = F? (t 0 ) and (@ i (t)) = (@ i (t 0 )) for all i 2 F? (t) = F? (t 0 ). Since i aynchronou, it follow that (t) = (t 0 ). Therefore, t 2?1 ((L)) = L which rove one incluion. The convere i trivial. Note that, for trace language, the convere of Prooition 4.3 i alo true imlying that all alternative mode of ACA are equivalent for trace. Indeed, it i eay to how that a trace language acceted by an ACA i a recognizable trace language, whatever mode i choen for acceting run. Moreover, if L i a recognizable trace language, the exitence of an aynchronou maing which recognize L wa roven in [CMZ93]. Finally, by Prooition 4.3, from thi aynchronou maing one can eaily get for each mode an ACA which accet L. The equivalence between alternative denition of acceting run will be extended to a more general cla of omet in Section 5. Section will deal with a cla of omet where the equivalence for nondeterminitic ACA remain true while it will not hold for determinitic ACA. In Section 7 we will how that even the equivalence for nondeterminitic ACA doe not hold for the cla of all -omet. ~ In addition, in the general etting of P( ), ~ the convere of Prooition 4.3 i fale (cf. dicuion after Prooition.4). 4.3 From ACA to MSO In thi ection, we will dene monadic econd order (MSO) formula and their interretation over omet. We will then rove that for all ACA A (determinitic or not), there exit an MSO formula which dene the language acceted by A. Let be a nite alhabet. Formula of the MSO language over that we conider involve rt order variable x; y; z : : : for vertice and monadic econd order variable X; Y; Z; : : : for et of vertice. They are built u from the atomic formula (x) = a for a 2, x y, and x 2 X by mean of the boolean connective :; _; ^;!; $ and quantier 9; 8 (both for rt order and for econd order variable). If we ue the quantier 9 and 8 only for rt-order variable, we obtain formula of rt-order logic. Formula witho free variable are called entence. For intance, the following formula are rt order and monadic econd order entence reectively. ' 1 ::= 9x((x) = a ^ 8y(x y! :(y) = b)) ' 2 ::= 9X9Y (8x(x 2 X _ x 2 Y ) ^ 9x (x 2 X) ^ 9y (y 2 Y ) ^ 8x8y(x 2 X ^ y 2 Y! :x y ^ :y x)) The atifaction relation j= between omet t = (V; ; ) and a entence ' of the monadic econd order logic i dened canonically with the undertanding that rt order variable range over the vertice of V and econd order variable over ubet of V. The et of omet which atify a entence ' i denoted by L('). For intance, L(' 1 ) i the 12

13 et of omet which have a vertex labeled by a with no vertex labeled by b above and L(' 2 ) i the et of non connected omet. In order to make the formula more readable, we will ue everal abbreviation which can be eaily tranlated in our MSO language. For intance, we will write x < y for x y ^ :y x x?< y for x < y ^ :9z(x < z ^ z < y) (x) 2 A (x) = (y) for for a2a 1in (x) = a ((x) 2 i ^ (y) 2 i ) X \ Y = ; for :9x(x 2 X ^ x 2 Y ) Note that the language dened by a formula can contain omet with ao-concurrency (concurrent vertice with the ame label). We do not need to retriction on the omet dened by a formula becaue all retriction we need can be exreed by MSO (or even rt-order) formula. For intance the et P(~) of ~-omet i dened by the formula ' ~ ::= 8x8y( (x) = (y)! (x y _ y x)) and the et M (; D) of trace over a deendence alhabet i dened by the formula 8x8y ([((x); (y)) 2 D! (x y _ y x)] ^ [x?< y! ((x); (y)) 2 D]) W where ((x); (y)) 2 D tand for the formula ((x) = a ^ (y) = b). (a;b)2d We are now ready to tate Theorem 4.4 Let A be a oibly nondeterminitic ~ -ACA and let ; 2 f+;?g. There exit an exitential monadic econd order entence ' over ~ uch that L(') = R F (A; P( ~ )): Proof. Let A = ((Q i ) i2[n] ; ( a;j ) a2;j[n] ; F ) be a -ACA. ~ We will contruct an MSO entence which will be atied exactly by thoe ~-omet acceted by A in the mode R? F?. For the other mode, the roof i eentially the ame: one only ha to change the formula tranition and S S acceted accordingly. Let k be the number of tate in Q i2[n] i. We may aume that Q i2[n] i = [k] = f1; : : : ; kg. The following formula claim the exitence of an F? -ucceful R? -run of the aomaton. ::= 9X 1 : : : 9X k? artition(x1 ; : : : ; X k ) ^?8x tranition(x) ^ acceted We will now exlain thi formula and give the ub-formula artition, tranition and acceted. An R? -run over a ~ -omet t = (V; ; ) i coded by the MSO variable 13

14 X 1 ; : : : ; X k. More reciely, X i tand for the et of vertice maed on the tate i by the R? -run. The formula artition(x 1 ; : : : ; X k ) make ure that the MSO variable X 1 ; : : : ; X k decribe a maing from V to S i2[n] Q i. artition(x 1 ; : : : ; X k ) ::= 8x _ i2[k] x 2 X i 1 A ^ ^ 1i<jk X i \ X j = ; Then, we have to claim that thi labeling of vertice by tate agree with the tranition function of the aomaton. _ tranition(x) ::= (x) = a ^ x ^ 2 X q ^ 8y (y?< x! (y) 2 J) q2 a;j ((q i ) i2j ) ^ 9y (y?< x ^ (y) = i ^ y 2 X qi ) where the dijunction Q range over all letter a 2, tate q 2 Q (a), ubet J [n] and tule (q i ) i2j 2 Q i2j i uch that q 2 a;j ((q i ) i2j ). It remain to tate that the R? -run reache a nal tate of the aomaton. acceted ::= _ (f i ) i2j 2F i2j 8x? (:9y x < y)! (x) 2 J ^ ^ i2j 9x? (:9y x < y) ^ (x) = i ^ x 2 X fi : In fact, the formula decribe an F? -acceting R? -run of the aomaton only for ~-omet. Therefore, we need in addition the formula ' ~ decribed above. Finally, the theorem follow from L(' ~ ^ ) = R? F? (A): The convere of the theorem above doe not hold a we will ee by Theorem 7.3 and Prooition CROW-omet In thi ection, we rove that the convere of Theorem 4.4 hold for the ecial ubcla of ~ -omet which atify the CROW axiom dened below. Denition 5.1 A ~ -omet t = (V; ; ) atie the Concurrent Read and Excluive Owner Write (CROW) axiom if for all x; y; z 2 V, x?< y; x < z and y k z =) (x) = (z): The et of ~ -omet which atify the CROW axiom i denoted by C ROW ( ~ ).! 14

15 A oible interretation of thi axiom i to think of the ACA a a Concurrent Read and Excluive Owner Write (CROW) machine. More reciely, we conider n rocee whoe et of action are 1 ; : : : ; n reectively. Each roce ha a memory which can be read by all action b can be written by it own action only (Owner Write). We allow concurrent read of memorie b no concurrent write. A mentioned in Section 2.1, thi retriction i already enforced by the very denition of -omet. ~ Witho further retriction, two concurrent event may reectively read from and write to the ame location. Thi i the cae when there exit two concurrent event y k z uch that z write in the memory of ome roce i ( (z) = i) and y read the memory of thi roce i ((x) = i for ome x?< y). Thi i reciely the ituation which i forbidden by the CROW axiom. Theorem 5.2 Let ' be an MSO entence over ~ and let 2 f+;?g. There exit a determinitic ~ -ACA A uch that L(') \ C ROW ( ~ ) = R? F (A; C ROW ( ~ )): In order to rove thi theorem, one can ue an induction on the tructure of the formula. Dijunction and exitential quantication are eaily dealt with when nondeterminitic ACA are allowed. On the other hand, comlement i eay for determinitic ACA. Whence the core of uch an aroach i the determinization of ACA. For thi roblem, tarting from a nondeterminitic ACA A, one can directly contruct an aynchronou maing which accet the language R? F (A; C ROW (~)) and then ue Prooition 4.3. Thi contruction i imilar to that of [Mu9] and ue the aynchronou time taming of Cori, Metivier & Zielonka [CMZ93] b the roof are more involved. In articular, it i known that for trace the maing i aynchronou by itelf [CMZ93, DM95] b thi i not the cae for CROW-omet. Here we give a imler roof which ue Zielonka' theorem. For thi, we rt ma CROW-omet into trace by imly changing the labeling. Let 0 = P([n]) be a new et of label and for all i 2 [n], let 0 i = i P([n]) be the aociated new rocee. By a light abue of notation, let again be the maing that aociate with any (a; M) 2 0 the roce i with (a; M) 2 0 i. Thi i jied ince (a; M) 2 0 i a 2 i i, i.e. (a; M) = (a). Intuitively, the econd comonent of a label in 0 tand for the read domain of the action. We dene an embedding g from P( ) ~ into P( ~ 0 ) by g(v; ; ) = (V; ; 0 ) where for all x 2 V, 0 (x) = ((x); R? (x)). Note that g i well dened, ince for all i 2 [n], 0?1 ( 0 i) =?1 ( i ) i totally ordered. Let D 0 be the deendence relation dened on 0 by D 0 = f((a; A); (b; B)) j (a) = (b) _ (a) 2 B _ (b) 2 Ag: Hence, two action are deendent if either they both write in the ame roce, or one read the roce written by the other. Prooition 5.3 C ROW (~) = g?1 (M ( 0 ; D 0 )) 15

16 Proof. We rt rove that C ROW ( ) ~ g?1 (M ( 0 ; D 0 )). Let t = (V; ; ) be a CROW-omet from C ROW (~) and let g(t) = (V; ; 0 ). Let x; y 2 V and aume that x?< y. Then, (x) 2 R? (y) and it follow that ( 0 (x); 0 (y)) 2 D 0. Now, let y; z 2 V and aume that ( 0 (y); 0 (z)) 2 D 0. If (y) = (z) then y k z ince t i a -omet. ~ If (y) = (z), we have for intance (z) 2 R? (y). Hence, there exit x 2 V uch that x?< y and (x) = (z). Therefore, x and z m be ordered. Since x < z would contradict the CROW-axiom, it follow z x, whence z < y. Therefore, g(t) 2 M ( 0 ; D 0 ). Converely, let t = (V; ; ) 2 g?1 (M ( 0 ; D 0 )) and let g(t) = (V; ; 0 ). Let x; y; z 2 V be uch that x?< y, x < z and y k z. By denition, (x) 2 R? (y) and ( 0 (y); 0 (z)) =2 D 0. Therefore, (z) =2 R? (y) and it follow that (x) = (z). Prooition 5.4 Let ' be an MSO entence over. There exit an MSO entence ' 0 over 0 uch that L(') \ C ROW ( ~ ) = g?1 (L(' 0 ) \ M ( 0 ; D 0 )) Proof. Let ' 0 be the MSO entence over 0 obtained W from ' by ubtiting for atomic formula of the form (x) = a the dijunction J[n] 0 (x) = (a; J): ' 0 = ' 2 4 _ J[n] 0 (x) = (a; J), 3 (x) = a5 : Let t = (V; ; ) 2 L(') \ C ROW (~). We have g(t) = (V; ; 0 ) 2 M ( 0 ; D 0 ) by Prooition 5.3 and it remain to how that g(t) j= ' 0. Thi i clear ince (x) = a if and only if 0 (x) = (a; J) for ome J [n]. The convere can be hown imilarly. Prooition 5.5 Let A 0 be a (determinitic) ~ 0 -ACA and 2 f+;?g. There exit a (determinitic) ~-ACA A uch that R? F (A; P(~)) = g?1 (R? F (A 0 ; P( ~ 0 ))). Proof. Let A 0 = ((Q i ) i2[n] ; ( 0 ) a 0 ;J a 0 2 ;J[n]; F ) be a ~ 0 0 -ACA. For all a 2 and J [n], let a;j = 0. We claim that the aomaton A = ((Q (a;j);j i) i2[n] ; ( a;j ) a2;j[n] ; F ) i the required -ACA. ~ Note that if A 0 i determinitic then o i A. We rt how that in order to accet a omet in g(p(~)) the ACA A 0 only ue tranition function of the form 0. Indeed, let t = (V; ; ) 2 P(~ ) and let g(t) = (a;j);j (V; ; 0 ). Then R? (x) = 0 (fy 2 V j y?< xg) = (fy 2 V j y?< xg) for all x 2 V. Therefore, in a run of A 0 on g(t) the tranition function ued are of the form 0 = 0 (x);r? (x) 0 = ((x);r? (x));r? (x) (x);r? S (x). It follow that a maing r : V! Q i2[n] i i an F -ucceful R? -run of A 0 on g(t) if and only if it i an F -ucceful R? -run of A on t, that i, t 2 R? F (A; P( ~ )) () g(t) 2 R? F (A 0 ; P( ~ 0 )) () t 2 g?1 (R? F (A 0 ; P( ~ 0 ))): 1

17 The rooition follow. Proof of Theorem 5.2 Let ' be an MSO entence over. By Prooition 5.4, there exit an MSO entence ' 0 over 0 uch that L(') \ C ROW ( ~ ) = g?1 (L(' 0 ) \ M ( 0 ; D 0 )): The language L(' 0 ) \ M ( 0 ; D 0 ) i a recognizable trace language [Tho90]. Hence by [CMZ93], there exit an aynchronou maing from M ( 0 ; D 0 ) into a nite et which recognize L(' 0 )\M ( 0 ; D 0 ). By Prooition 4.3, there exit a determinitic ~ 0 -ACA A 0 uch that R? F (A 0 ; M ( 0 ; D 0 )) = L(' 0 ) \ M ( 0 ; D 0 ). It follow by Prooition 5.5 that there exit a determinitic ~-ACA A uch that R? F (A; P(~)) = g?1 (R? F (A 0 ; P( ~ 0 ))). Finally, alying Prooition 5.3 we obtain R? F (A; C ROW ( ~ )) = R? F (A; P( ~ )) \ C ROW ( ~ ) = g?1 (R? F (A 0 ; P( ~ 0 ))) \ g?1 (M ( 0 ; D 0 )) = g?1 (R? F (A 0 ; M ( 0 ; D 0 ))) = g?1 (L(' 0 ) \ M ( 0 ; D 0 )) = L(') \ C ROW (~): A a corollary of Theorem 4.4 and 5.2 we obtain that (exitential) MSO entence, nondeterminitic -ACA ~ in the reading mode R? and determinitic -ACA ~ in the reading mode R? have the ame exreive ower for C ROW (~)-omet. Theorem 5. Let L C ROW ( ~ ) and 2 f+;?g. The following are equivalent: 1. L i denable by a monadic econd order entence, 2. L i denable by an exitential monadic econd order entence, 3. there exit a nondeterminitic ~ -ACA A uch that L = R? F (A; C ROW ( ~ )); 4. there exit a determinitic ~-ACA A uch that L = R? F (A; C ROW (~)): In the remainder of thi ection, we rove an analogou reult for the mode R + F. We do thi by contructing a determinitic ACA A 0 from a determinitic ACA A uch that R? F (A; C ROW ( ~ )) = R + F (A 0 ; C ROW ( ~ )). Then Theorem 4.4 together with Theorem 5. give the deired reult. The main tak of thi contruction i to rovide an ACA with the ability to ditinguih immediate redeceor among all redeceor in an R + -run. We how that thi i oible in the next rooition uing the following lemma. 17

18 Lemma 5.7 ([CMZ93]) For any trace monoid M, there exit a nite et S and an aynchronou maing : M! S uch that (t) uniquely determine the label of the maximal element of t, i.e. the et (max(t)), for each trace t 2 M. Note that the mere maing t 7! (max(t)) i not aynchronou b it i eay to obtain an aynchronou maing atifying the condition of the lemma above by uing the aynchronou time taming introduced in [CMZ93]. Notation. Let t = (V; ; ) be a ~-omet. Then t 0 := (V; ; R? ) i a trace in M ( 0 ; D 0 ) with g(t) = t 0. We write (t; R? ) a an abbreviation for g(t) = (V; ; R? ). In addition, let (#x; R? ) denote (#x; \(#x #x); ( R? ) #x ) and imilarly for (+x; R? ) whenever x 2 V. Prooition 5.8 There exit a determinitic ~ -ACA A 00 = ((Q 00 ) i2[n] ; ( 00 a;j ) a2;j[n]; F 00 ) (note that all rocee have the ame et Q 00 of local tate) and a maing : Q 00! 2 [n] uch that (i) R + F (A 00 ; C ROW ( ~ )) = C ROW ( ~ ). (ii) Let t = (V; ; ) be a CROW-omet and let r be the R? -run of A 00 r = R?, that i, R? (x) = (r(x)) for all x 2 V. on t. Then Proof. Let : M ( 0 ; D 0 )! S be the aynchronou maing given by Lemma 5.7. The common local tate ace of the ACA A 00 i given by Q 00 = 2 [n] S. For a 2 i, let 00 = f(;; (a; ;))g. Now let ; = J [n], (M a;; j; j ) 2 Q 00 for j 2 J and a 2 i. Then a;j((m 00 j ; j ) j2j ) conit of all air (M; ) 2 Q 00 uch that there exit a trace t 2 M ( 0 ; D 0 ) with (1) J = j (t) = j for j 2 J, (2) M = (max(t)), and = (t (a; M)). Finally, all tule of local tate are acceting. Firt we how that the tranition function are indeed determinitic: So let (M j ; j ) 2 Q 00 for j 2 J [n], and a 2 i. Let t; t 0 2 M ( 0 ; D 0 ) be trace W uch that J = (t) = (t 0 ) and j j (t) j (t 0 ) for any j 2 (t). Clearly, t j2(t) j(t) and imilarly for t 0. Since i an aynchronou maing, (t) = (t 0 ) follow j (t) j (t 0 ) for j 2 (t) = (t 0 ). Then max(t) = max(t 0 ) =: M by the choice of the aynchronou maing. Now let y 2 max(t). Then (y) 2 M imlying that 0 (y) and (a; M) are deendent. Since thi hold for all y 2 max(t), the trace t (a; M) i rime. Similarly, the trace t 0 (a; M) i rime. Since, a we aw above, (t) = (t 0 ), the aynchronicity of imlie (t (a; M)) = (t 0 (a; M)) =:. Thu, we howed that (M; ) i the only element of a;j((m 00 j ; j ) j2j ), i.e. the aomaton A 00 i determinitic. To how the rt tatement of Prooition 5.8, it i ucient to rove that any CROW-omet allow a run of the ACA A 00. Therefore, let t = (V; ; ) 2 C ROW ( ). ~ 18

19 Dene r : V! 2 [n] S by r(x) := (R? (x); (#x; R? )). Let x 2 V, r(x) = (M; ) and r@ j (+x) = (M j ; j ) for j 2 R + (x). We have to how that (M; ) 2 00 (x);r + (x) ((M j; j ) j2r + (x)): Then t := (+x; R? ) i a trace from M ( 0 ; D 0 ) with R + (x) = (+x) = (t) and j j (+x; R? ) = (@ j (+x); R? ) for all j 2 R + (x). Thu, (1) hold. Clearly, M = R? (x) = max(+x) = max(t) and = (#x; R? ) = ((+x; R? ) ((x); M)) which rove (2). Thu, r i indeed an R + -run of A 00 on t. Since A 00 i determinitic, r i the only oible run of A 00 on t. Dening to be the rt rojection from Q 00 to 2 [n], the econd tatement i obviou. Now we can eaily contruct an ACA that imulate an R? -run of a given ACA in the mode R + a follow: Let A = ((Q i ) i2[n] ; ( a;j ) a2;j[n] ; F ) be a ~-ACA and let A 00 = ((Q 00 ) i2[n] ; ( 00 ) a;j a2;j[n]; F 00 ) be the ~-ACA from Prooition 5.8. Then dene Q 0 i := Q i Q 00 and F 0 := f(q j ; q 00 j ) j2j j (q j ) j2j 2 F g. For a 2, J [n] and (q j ; q 00 j ) 2 Q 0 j for j 2 J, let a;j((q 0 j ; q 00 j ) j2j ) conit of all tule (q; q 00 ) with q 00 2 a;j((q j ) j2j ) and q 2 a;(q 00 ) ((q j ) j2(q 00 ) ). Note that, ince A 00 i determinitic, A 0 i determinitic whenever A i. Now let t = (V; ; ) be a CROW-omet and let r 0 be an F -ucceful R + -run of A 0 on t. Then, by Prooition 5.8 (ii), 1 r 0 i an R? -run of A on t. By the denition of the acceting tate of A 0, it i F -ucceful. Hence R + F (A 0 ) R? F (A). To how the other incluion, let r be an F -ucceful R? -run of A on t. By Prooition 5.8 (i), there i an R + -run r 00 of A 00 on t. Let r 0 = r r 00. We how that thi i an R + -run of A 0 on t. Let x 2 V. By Prooition 5.8 (ii), R? (x) = (r 00 (x)). Since r i an R? -run of A, we get that r(x) 2 a;(r 00 (x))(r@ j (+x) j2(r (x))). Hence 00 r r 00 i an R + -run on t that i F -ucceful ince the econd comonent doe not inuence the accetance. Thu R + F (A 0 ) = R? F (A). Hence, uing Theorem 4.4 and 5. we get Theorem 5.9 Let L C ROW ( ~ ) and ; 2 f+;?g. The following are equivalent: 1. L i denable by a monadic econd order entence, 2. L i denable by an exitential monadic econd order entence, 3. there exit a nondeterminitic ~ -ACA A uch that L = R F (A; C ROW ( ~ )); 4. there exit a determinitic ~ -ACA A uch that L = R F (A; C ROW ( ~ )): k-omet Let t = (V; ; ) be a ~ -omet. Furthermore, let k be a oitive integer and C` V for 1 ` k. We call the tule (C 1 ; C 2 ; : : : ; C k ) a k-chain covering of t if 1. C` i a chain for ` = 1; 2; : : : ; k, 19

20 2. V = S`2[k] C` and 3. for any x; y 2 V with x?< y there exit ` 2 [k] with x; y 2 C`. The ~ -omet t i a k-omet if it ha a k-chain covering. Let P k denote the et of all k-omet over ~. Remark.1 Let t = (V; ; ) 2 C ROW ( ~ ). For i; j 2 [n] and i = j dene C i;j :=?1 ( i ) [ fy 2?1 ( j ) j 9x 2?1 ( i ) : x?< yg. Then the et fc i;j j i; j 2 [n]; i = jg atie roertie 2 and 3 given above. Furthermore, C i;j i a chain ince t i a CROWomet. Thu, C ROW (~) P n(n?1)..1 Searating the determinitic clae for k-omet Examle.2 Let n = 3, k = 2, 1 = fag, 2 = fbg and 3 = fcg. Furthermore, let L be the et of all k-omet (V; ; ) over ( 1 ; 2 ; 3 ) uch that?1 (a) i even,?1 (b) and?1 (c) are nonemty and no a-labeled element dominate ome b- or c-labeled one. Then L i in dr F + (P k ), b not in dr F? (P k ) for 2 f+;?g. Proof. Let Q 1 = f0; 1g and Q 2 = Q 3 = f0g. Furthermore, we dene tranition function a follow: a;j ((q i ) i2j ) = 8 >< >: ; 2 2 J or 3 2 J f(q 1 + 1) mod 2g J = f1g f1g J = ; and b;j ((q i ) i2j ) = c;j ((q i ) i2j ) = f0g for any J and q i. With F = f(0; 0; 0)g, we get a determinitic ~-ACA A = ((Q i ) i2[3] ; ( d;j ); F ) uch that R? F + (A; P k ) = R + F + (A; P k ) = L witneing L 2 dr F + (P k ). Suoe, A i a determinitic -ACA ~ uch that L = R F? (A; P( )) ~ for = + or =?. Furthermore, let ` = jq 2 Q 3 j + 1. To derive a contradiction, let t m (for m 2 N) denote the -omet ~ deicted in Figure 1. The labeling i dened canonically by (a i ) = a, (b 1 ) = b and (c 1 ) = c. Since t m 2 L, there i an F? -ucceful R -run r m of A on t m. Since ` i larger than the number of tule from Q 2 Q 3, there are i < j ` uch that r i (b 1 ) = r j (b 1 ) and r i (c 1 ) = r j (c 1 ). Now conider the -omet ~ t = (V; ; ) on Figure 1 that doe not belong to L. The labeling on t i dened canonically. Let r : V! Q 1 [ Q 2 [ Q 3 be the retriction of r j, i.e. r = r j V. Since A i determinitic, we get r fa 1 ; a 2 ; : : : ; a 2i+2 g = r i fa 1 ; a 2 ; : : : ; a 2i+2 g, in articular r(a 2i+2 ) = r i (a 2i+2 ). Thu, r i an R -run of A on t. Since F? (t) = f2; 3g = F? (t j ), the run r i F? -ucceful contradicting L = R F? (A; P( )). ~ Examle.3 Let n = k = 2, 1 = fag and 2 = fbg. Furthermore, let L conit of all k-omet (V; ; ) over ( 1 ; 2 ) that have a larget element x uch that (x) = b. Then L i in dr F? (P k ), b not in dr F + (P k ) for any 2 f+;?g. 20

21 b 1 a 1 a 2 a 2m a 2m+1 a 2m+2 The k-omet t m c 1 b 1 a 1 a 2 a 2i a 2i+1 a 2i+2 a 2j+1 The k-omet t c 1 Figure 1: cf. Proof of Examle.2 Proof. Clearly, there i a determinitic ~-ACA that allow an R -run on any ~-omet. Let F conit of all tule (q i ) i2j of local tate with J = f2g. Then thi ~-ACA A accet L in the mode R F?, i.e. R F? (A; P k ) = L witneing L 2 dr F? (P k ). We want to how that there i no determinitic ~-ACA A uch that R F + (A; P k ) = L. By contradiction, aume A i uch a -ACA. ~ Let ` = jq 1 j+2 and conider the k-omet t = (V; ; ) with V = fa i j i = 1; 2; : : : ; `g [ fb 1 g, a 1 < a 2 < a` < b 1 and with the canonical labeling. Then t 2 L. Hence there i an F + -ucceful R -run r of A on t. Since ` > jq 1 j + 1, there are i < j < ` uch that r(a i ) = r(a j ). Now conider the k-omet t 1 and t 2 with V 1 = V 2 = fa` j ` = 1; 2; : : : ; jg [ fb 1 g and the canonical labeling. The order relation are dened by a 1 < 1 a 2 < 1 a 3 < 1 a j < 1 b 1 (i.e. t 1 i a linear ordering with maximal element b 1 ) and a 1 < 2 a 2 < 2 a 3 < 2 a j and a i < 2 b 1 (i.e. in t 2, the a-labeled element are linearly ordered, b the maximal element b 1 cover a i ). Since t 1 2 L, there i an F + -ucceful R -run r 1 of A on t 1. Since A i determinitic, we have r 1 (a`) = r(a`) for ` j. Thi imlie r 1 (a i ) = r 1 (a j ) ince the equality hold for the run r. Hence r 1 i an R -run on t 2, too. Thi imlie that r 1 i an F + -ucceful R -run on t 2, contradicting L = R F + (A; P k ). The two examle above can be generalized to rove the following Prooition.4 Let n; k 3. Then the clae dr + F + (P k ), dr? F + (P k ), dr + F? (P k ), and dr? F? (P k ) are airwie incomarable. Hence, alo the clae dr + F + (P( ~ )), dr? F + (P( ~ )), dr + F? (P( ~ )), and dr? F? (P( ~ )) are airwie incomarable. 21

22 Proof. The incomarability of the clae from fdr + F + (P k ); dr? F + (P k )g and thoe from fdr + F? (P k ); dr? F? (P k )g i witneed by the two examle above. A language that i determinitically accetable in the mode R + F, b not in the mode R? F i eaily obtained from the language in Examle.2 a follow: L conit of all k-omet uch that any b-labeled element dominate an even number of a-labeled element. Thi language i in dr + F (P k ) for 2 f+;?g. To how that it i not in dr? F (P k ), one add an additional maximal b-labeled element b 2 to the k-omet t m and t (Figure 2). Then, the reading domain of thi additional element in mode R? i reciely the accetance domain of the original omet on acceting mode F?. Therefore, the roof goe through a before. Similarly, one can adot the idea from Examle.3 to obtain a language that i in dr? F (P k ) b not in dr + F (P k ). To how that the clae dr F (P( )) ~ are mually incomarable, let ; ; 0 ; 0 2 f+;?g with dr F (P(~)) dr 0 F 0 (P(~)). Now let A be a determinitic ~-ACA. By our aumtion dr F (P( )) ~ dr 0 F 0 (P( )), ~ there i a determinitic -ACA ~ A 0 uch that R F (A; P( )) ~ = R 0 F 0 (A 0 ; P( )). ~ Hence in articular R F (A; P k ) = R F (A; P(~)) \ P k = R 0 F 0 (A 0 ; P(~)) \ P k = R 0 F 0 (A 0 ; P k ) and therefore = 0 and = 0 by what we howed above. b 1 b 2 a 1 a 2 a 2m a 2m+1 a 2m+2 The k-omet t m c 1 b 1 b 2 a 1 a 2 a 2i a 2i+1 a 2i+2 a 2j+1 The k-omet t c 1 Figure 2: cf. Proof of Prooition.4 Note that the et of k-omet form a rex cloed cla of ~-omet. Hence Prooition 4.3 can be alied. Therefore, any language of k-omet recognizable by an aynchronou maing i in the interection of all clae dr F (P k ). Since thee clae are incomarable, the language recognizable by an aynchronou maing form 22