In the following there are presented four different kinds of simulation games for a given Büchi automaton A = :

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1 Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile to mimick the ehviour of nother utomton nd lso to estlish lnguge continment mong nondeterministic utomt. Secondly, simultion reltions cn e used to reduce the stte spce of n utomton y otining its quotient with respect to the equivlence reltion underlying the simultion preorder. 4 Different kinds of simultions In the following there re presented four different kinds of simultion gmes for given Büchi utomton A = : 1) The ordinry simultion gme, denoted, 2) The direct (strong) simultion gme, denoted, 3) The delyed simultion gme, denoted, 4) The fir simultion gme, denoted. Ech of the gmes is plyed y two plyers, Spoiler nd Duplictor, in rounds s follows. At the strt, round 0, two peles, Red nd Blue, re plced on the two vertices q 0 nd q 0. Then Spoiler chooses trnsition nd moves Red to q i+1. Duplictor, responding, must choose trnsition nd moves Blue to q i+1. Here it cn e the cse tht no -trnsition strting from q i exists. In this cse the gme hlts nd Spoiler wins. Either the gme hlts, in which cse Spoiler wins, or the gmes produces two infinite runs: nd, uilt from the trnsitions tken y the two peles. For ech of the 4 simultion gmes there exist different rules to determine the winner: 1.) Ordinry simultion: Duplictor wins in ny cse s long s the gme does not hlt. (Firness conditions re ignored) 2.) Direct simultion: Duplictor wins, iff, for ll i, if, then lso 3.) Delyed simultion: Duplictor wins, iff, for ll i, if, then there exists j i such tht 4.) Fir simultion: Duplictor wins iff there re infinitely mny j such tht or there re only finitely mny i such tht A strtegy for Duplictor in one of the 4 simultion gmes from ove is function: which, given the history of the gme (ctully the choices of Spoiler) up to certin point, determines the next move of Duplictor. A strtegy f for Duplictor is winning strtegy if, no mtter how Spoiler plys, Duplictor lwys wins.

2 Simultion Reltion A stte q ordinry, direct, delyed, fir simultes stte q if there is winning strtegy for Duplictor. The simultion reltion, reflexive, trnsitive reltion (preorder or qusi-order), is denoted y, where * stnds for one of the 4 simultions ordinry(o), direct(di), delyed(de) nd fir(f). The reltions re ordered y continment: For direct, delyed nd fir holds: if then. Bisimultion Reltions For ll the mentioned simultions there re corresponding notions of isimultion, defined vi modifiction of the simultion gmes. The isimultion gme differs from the simultion gme in tht Spoiler gets to choose in ech round which of the two peles, Red or Blue, to move nd Duplictor hs to respond with the move of the other pele. Bisimultions define the following equivlence reltion continment: For isimultions there re lso winning rules for the isimultion gmes tht re similr to the rules for the 4 different simultion gmes: 1.) Ordinry: the sme rules s for the ordinry simultion gme 2.) Fir: If n ccept stte ppers infinitely often on one of the two runs π nd π, then n ccept stte must pper infinitely often on the other s well. 3.) Delyed: If n ccept stte is seen t position i of either run, then n ccept stte must e seen therefter t some position j i of the other run. 4.) Direct: If n ccept stte is seen t position i of either run, it must e seen t position I of oth runs. Quotienting A mjor motivtion for studying reltions is stte-spce reduction, the sic ide eing tht sttes tht simulte ech other re merged leding to reduction of the numer of sttes. This process is usully clled quotienting. Quotienting with respect to delyed simultion preserves the recognized lnguge, ut this is not true for with fir simultion s we will see lter on. For Büchi utomton A, nd n equivlence reltion on the sttes of A, let [q] denote the equivlence clss of with respect to. The quotient of A with respect to is the utomton, where For pplying the simultion reltions, corresponding to ech simultion preorder, n equivlence reltion o, di, de, f is defined, where, iff nd. The quotient with respect to di, de preserves the lnguge of ny utomton.

3 An exmple shows quotienting for delyed simultion. In this exmple oth utomt, the originl one nd the quotient, ccept the sme lnguge. Quotienting y delyed The otined quotient is smller thn the originl utomton nd ccepts the sme lnguge s the first one. The sttes round the middle stte re in one equivlence clss, ecuse one cn sty there with infinitely mny s. With one comes to nother stte nd there, if reding gin nother, one comes to the first stte. Whenever Spoilers mkes n -move, Duplictor cn mke one s well. If Spoiler then mkes -move to the outside, Duplictor nswers with move to the upper ccepting stte. In this cse Duplictor visits infinitely often n ccepting stte nd everything is fine. Thus the quotient utomton hs only two sttes. The quotient with respect to o, f does not preserve the lnguge of ny utomton, which is shown in the exmple elow:, Quotienting y fir, Quotienting is done in this exmple vi the equivlence reltion q f q tht holds for ll sttes q, q from the set Q. The left utomton is ccepting words with nd infinitely lternting. The right utomton ccepts words like ω or ω, ut the originl utomton doesn t.

4 So here it cn e seen, tht the quotient of the originl utomton with respect to f does not preserve the lnguge of the originl utomton. Prity Gmes A prity gme grph hs two disjoint sets of vertices V 0 nd V 1, whose union is V. It lso hs n edge set nd priority function tht ssigns priority to ech vertex. Such prity gme is plyed y two plyers, Zero nd One. The ply strts y plcing pele on vertex v 0. Therefter, the pele is moved ccording to the following rule: With the pele currently on vertex v i nd, Zero (One, respectively) plys nd moves the pele to neighour v i+1, such tht. If ever the rule ove cnnot e pplied, i.e., someone cn t move ecuse there re no outgoing edges, the gme ends nd the plyer who cnnot move loses. Otherwise, the gme goes on forever, nd defines pth, clled ply of the gme. The winner of the gme is determined s follows. Let k π e the minimum priority tht occurs infinitely often in the ply π; Zero wins if k π is even, wheres One wins if k π is odd. Now the gme grphs G * A re uilt, following the sme generl pttern, with some minor ltertions. The first gme grph will e with only three priorities (0,1,2). Formlly G f A is defined y The set of vertices for Zero:, This set of vertices contins ll vertices, where Spoiler hs moved the turn efore nd it is now Duplictor s turn to imitte the lel on the 3 rd position of the tupel, ecuse Spoiler hs tken this lel the move efore. So this lel rememers the move of Spoiler, q is the position of the Spoiler-pele nd q is the position of the Duplictor-pele. The set of vertices for One:, This set of vertices contins ll vertices, where Duplictor hs moved the turn efore nd it is now Spoiler s turn. Here is no 3 rd position in the tupel, ecuse Spoiler does not hve to imitte run move from Duplictor, tht hs to nswer to the moves of Spoiler. The set of the edges for Zero nd One:, The moves of the first set re the moves of Duplictor. Duplictor is in stte v(q 1,q 1,) nd hs to tke trnsition with lel to move from stte q 1 to stte q 2 if there exists the trnsition (q 1,, q 2 ). Only the stte of Duplictor chnges here, the Spoiler s stte q 1 remins the sme. The second set contins ll the moves of Spoiler. Spoiler s in stte v(q 1,q 1 ) nd cn choose ny trnsition (q 1,,q 2 ) form the trnsition set to move his pele to nother stte. In this cse only the Spoiler s stte is chnged.

5 The priority function:. This function mens, tht if neither Spoiler nor Duplictor hve infinitely often n ccepting stte in their runs tht the minimum prity numer tht occurs infinitely often is the 2. In this cse Duplictor wins. If Spoiler hs seen infinitely often n ccept stte ut not Duplictor, the minimum prity numer tht occurs infinitely often is the 1 nd so in this cse Spoiler wins. Lst ut not lest, if Duplictor sees infinitely often n ccepting stte no mtter wht Spoiler does, the minimum prity numer occurring is the 0 nd here lso Duplictor wins. The grph G f A cn e modified to otin G o A nd G di A, oth of which require only trivil modifiction to G f A. The prity gme grph G o A is exctly the sme s G f A, except tht ll nodes will receive priority 0. This mens, tht Duplictor wins s long s he cn mimick the moves of Spoiler nd he only loses, if he comes to ded-end nd cn t move nymore. The prity gme grph G di A is just like G o A, mening every vertex hs priority 0, ut some edges re eliminted: Finlly to define G de A the gme grph hs to e modified somewht more. For ech vertex of G f A there will e t most two corresponding vertices in G de A: The extr it encodes whether or not, thus fr in the simultion gme, the Red pele ws witnessed n ccept stte without Blue hving witnessed one since then. The edges of G de A re s follows: Lst ut not lest the priority function of G de A is descried s:.. Here, priority 1 will e ssigned to only those vertices in V 1 tht signify tht n unmtched ccept hs een encountered y Red.

6 References Crsten Fritz, Thoms Wilke: Simultion Reltions for Alternting Prity Automt nd Prity Gmes. DLT 2006, LNCS 4036, pp , Springer-Verlg (2006) Koush Etessmi, Thoms Wilke, Reecc A. Schuller: Fir Simultion Reltions, Prity Gmes nd Stte Spce Reduction for Büchi Automt. ICALP 2001, LNCS 2076, pp , Springer-Verlg (2001) Crsten Fritz: Simultion-Bsed Simplifiction of omeg-automt. PhD thesis, Technische Fkultät der Christin Alrecht Universität zu Kiel (2005) ville t

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