Buffered Simulation Games for Büchi Automata

Transcription

1 Buffered Simultion Gmes for Büchi Automt Milk Hutglung nd Mrtin Lnge nd Etienne Lozes School of Electr. Eng. nd Computer Science, University of Kssel, Germny Abstrct. We introduce buffered simultions, fmily of simultion reltions tht better pproximte lnguge inclusion thn stndrd simultions. Buffered simultions re bsed on two-plyer gme in which the second plyer, clled here Duplictor, my skip his turn nd respond to given move in lter round of the gme. We consider vrious vrints of these buffered simultions nd estblish their decidbility nd exct complexity. 1 Introduction Nondeterministic Büchi utomt (NBA) re n importnt formlism for the specifiction nd verifiction of rective systems. While they hve originlly been introduced s n uxiliry device in the quest for decision procedure for Mondic Second-Order Logic [3] they re by now commonly used in such pplictions s LTL softwre modelchecking [9, 15], or size-chnge termintion nlysis for recursive progrms [18, 11]. A lrge mount of work in the re of utomt theory for verifiction is devoted to combting the stte spce explosion problem. One mjor issue of utomt mnipultion is to keep the number of sttes s smll s possible. In prticulr it is crucil to void explicit complementtion, becuse ll of the existing complementtion procedures [19, 20, 17] often drsticlly increse the sizes of the mnipulted utomt. Quite often, complementtion is not needed in itself, but rther is used for deciding whether the lnguge of n utomton is included in nother one. Since the erly works of Dill et l [7], simultions hve been intensively used in utomt-bsed verifiction. While problems like lnguge inclusion or utomt minimistion re typiclly PSPACE-hrd, simultion reltions re chep to compute. Simultions re interesting with respect to severl spects. On the one hnd, they offer sound, but incomplete, pproximtion of lnguge inclusion tht my be sufficient in mny prcticl cses. On the other hnd, simultions cn be used for quotienting utomt [4, 12, 10], for pruning trnsitions [1, 2], or for improving existing lgorithms like the Rmsey-bsed [11] or the ntichin lgorithm [8]. In this pper, we introduce new fmily of simultions for Büchi utomt, clled buffered simultions. In buffered simultion, two plyers clled Spoiler nd Duplictor move two pebbles long utomt trnsitions, but unlike in stndrd simultions, Spoiler nd Duplictor s moves do not lwys lternte. Indeed, Duplictor cn skip The Europen Reserch Council hs provided finncil support under the Europen Community s Seventh Frmework Progrmme (FP7/ ) / ERC grnt greement no

2 his turn nd wit to see Spoiler s next moves before responding. Spoiler nd Duplictor shre first-in first-out buffer: everytime Spoiler moves long n -lbelled trnsition, he dds n into the buffer, wheres everytime Duplictor mkes step long b-lbelled trnsition, he removes b from the buffer. Since Duplictor hs more chnces to defet Spoiler thn in stndrd simultions, buffered simultions better pproximte lnguge inclusion. Finitry versions of buffered simultions were recently studied for tht purpose [16] nd for utomt minimiztion [6]. In this pper, we study their infinitry counterprt, or in other words we consider the cse of the shred FIFO buffer being unbounded. We study two notions of buffered simultion gmes, clled continuous nd look-hed simultion gmes, respectively. Their rules only differ in the wy tht Duplictor must use the buffer: in look-hed simultions, Duplictor is forced to flush the buffer, so tht he ctches up with Spoiler everytime he decides to mke move. Thus, the buffer is flushed completely with ech of Duplictor s moves. In the continuous cse, Duplictor cn choose to only consume prt of the buffer with every move, nd it need not ever be flushed. Decidbility of these simultions is not obvious. In the finitry cses, it is provided by rther stright-forwrd reduction to prity gmes but gmes with unbounded buffers would yield prity gmes of infinite size. Moreover, questions bout systems with unbounded FIFO buffers re often undecidble; for instnce, liner-time properties of system of two mchines nd one buffer re known to be undecidble [5]. Our min contribution is the decidbility of buffered simultions together with exct mesurements of their complexities: we estblish PSPACE completeness of look-hed simultions nd EXPTIME-completeness of continuous simultions. The lower bounds provide limits to the bility to efficiently pproximte lnguge inclusion through simultion reltions: while simultion bsed on finite buffers is efficiently decidble [6, 16], one hs to py for the incresed preciseness through infinite buffers. We lso study Duplictor s degree of look-hed [14], i.e. how lrge the buffer cn grow in these gmes, nd provide n upper bound when it exists. Outline. The first section collects stndrd notions on Büchi utomt nd simultions. Continuous simultion is introduced in section 2, nd bsed on some topologicl considertions, series of bsic results re derived. Section 3 defines look-hed simultions nd discuss some differences with continuous simultion. In section 4, we introduce quotient gmes nd prove the decidbility of the continuous nd look-hed simultions. Section 5 estblishes the corresponding complexity lower bounds. Section 6 contins some concluding remrks. Due to spce constrints, we often only give some proof sketches. 2 Bckground A non-deterministic Büchi utomton (NBA) is tuple A = (Q, Σ, δ, q 0, F ) where Q is finite set of sttes with q 0 being designted strting stte, δ Q Σ Q is trnsition reltion, nd F Q is set of ccepting sttes. The stte q Q is clled ded end, if there is no Σ nd q Q such tht (q,, q ) δ. If w = 1... n, w sequence q 0 1 q 1... q n is clled w-pth from q 0 to q n, q 0 qn, if (q i, i+1, q i+1 )

3 δ for ll i {0,..., n 1}. It is n ccepting w-pth, q 0 qn, if there is some i {1,..., n} such tht q i F. A run of A on word w = 1 2 Σ ω is n infinite sequence ρ = q 0 1 q such tht (q i, i+1, q i+1 ) δ for ll i 0. The run is ccepting if there is some q F such tht q = q i for infinitely mny i. We write Runs(A) to denote the set of runs of A, nd ARuns(A) the set of ccepting runs. The lnguge of A is the set L(A) of infinite words for which there exists n ccepting run. Remrk 1. The set (Q Σ) ω is equipped with stndrd structure of metric spce. The distnce d(x, y) between two infinite sequences x 0 x 1 x 2... nd y 0 y 1 y 2... is the rel 1 2 i, where i is the first index for which x i y i. Intuitively, two words re significntly close if they shre significntly long prefix. The sets Runs(A) nd ARuns(A) re subsets of (Q Σ) ω ; Runs(A) hs the prticulrity of being closed subset, nd it is thus compct spce, wheres ARuns(A) is not. Fir simultion [13] is n extension of stndrd simultion to Büchi utomt. The esiest wy of defining fir simultion is by mens of gme between two plyers clled Spoiler nd Duplictor. Let us fix two NBA A = (Q, Σ, δ, q I, F ) nd B = (Q, Σ, δ, q I, F ). Spoiler nd Duplictor re given ech pebble initilly plced on owns pebble, nd this pebble is plced on q 0 := q I nd q 0 := q I respectively. Then, on ech round i 1, 1. spoiler chooses letter i Σ, trnsition (q i 1, i, q i ) δ, nd moves his pebble to q i ; 2. duplictor, responding, chooses trnsition (q i 1, i, q i ) δ nd moves his pebble to q i. Either the ply termintes becuse one plyer reches ded end, nd then the opponent wins the ply. Or the gme produces two infinite runs ρ = q 0 1 q 1,... nd ρ = q 0 1 q Given these two infinite runs, Duplictor is declred the winner of the ply if either Spoiler s run ρ is not ccepting, or Duplictor s run ρ is ccepting. We sy tht A firly simulted by B, A f B, if Duplictor hs winning strtegy for this gme. Clerly, A f B implies L(A) L(B), but the converse does not hold in generl. Remrk 2. Notice tht stndrd simultion, s defined for lbelled trnsition systems, is specil cse of fir simultion. Indeed, for given lbelled trnsition system (Q, Σ, δ), nd given stte q, we cn define the NBA A(q) with q I := q s the initil stte, nd F := Q s the set of ccepting sttes. Then q simultes q in the stndrd sense (without tking cre of firness) if nd only if A(q) f A(q ). We write q q when q simultes q in the stndrd sense. 3 Continuous Simultion Continuous simultions re defined by gmes in which Duplictor is llowed to see in dvnce some finite, unbounded number of Spoiler s moves. For exmple, Duplictor cn decide his first move depending on the three first moves of Spoiler. We first formlly define this notion nd then explin its reltion with continuity. w

4 Definition. Let A = (Q, Σ, δ, q I, F ) nd B = (Q, Σ, δ, q I, F ) be two NBA. In the continuous fir simultion gme, Spoiler nd Duplictor now shre FIFO buffer b. Initilly, Spoiler s pebble is on q 0 := q I, Duplictor s pebble is on q 0 := q I, nd the buffer b is empty. On ech round i 1: 1. Spoiler chooses letter i Σ, trnsition (q i 1, i, q i ) δ, moves the pebble to q i, nd dds i to the buffer b. 2. Duplictor either skips his turn, or picks some r 1 smller thn the buffer size b, removes the letters i b +1, i b +2,..., i b +r out of b, nd moves his pebble to some sttes q i b, q i b +1,... until q i b +r, following some trnsitions (q j, j+1, q j+1 ) δ for ll j vrying from i b to i b + r 1. A ply of the gme defines finite or infinite trce ρ for Spoiler (finite if Spoiler reches ded end), nd finite or infinite trce ρ for Duplictor (finite if Duplictor eventully lwys skips his turn). Duplictor is declred the winner of the ply if either ρ is finite, or ρ is not ccepting, or ρ is infinite nd ccepting. We sy tht B continuously firly simultes A, A f co B, if Duplictor hs winning strtegy. We lso consider the (unfir) continuous simultion co for pirs of LTS sttes by considering LTS with distinguished stte s NBA where ll sttes re ccepting. Exmple 1. Consider the following two NBA A (left) nd B (right) over the lphbet Σ = {, b, c}. b c Σ Σ Then Duplictor hs winning strtegy for the continuous fir simultion gme: he skips his turn until Spoiler follows either b or c. However, if we ignore the ccepting sttes nd consider these utomt s trnsition system, then Spoiler hs winning strtegy for the continuous simultion: he itertes the loop, nd then either Duplictor wits forever nd loses the ply, or he mkes move nd it is then esy for Spoiler to defet him. b c Σ Σ Bounded Buffers. Note tht in the continuous simultion gmes, Duplictor my remove s mny letters hs he wishes during his turn. However, if Duplictor hs winning strtegy, he lso hs lzy winning strtegy, in which he lwys removes t most one letter from the buffer in round. The bility to remove more thn one letter in round becomes n importnt spect of the gme if we get interested in the mount of spce resources needed by Duplictor in the ppliction of his strtegy. We sy tht winning strtegy for Duplictor is universlly bounded if there is n integer B such tht Duplictor keeps the buffer size b smller thn B in ll plys following the strtegy; it is sid to be existentilly bounded if there is bound B for ech ply, but the

5 bound my depend on the ply. For instnce, for the continuous fir simultion gme of Exmple 1, Duplictor hs n existentilly bounded winning strtegy, but not universlly bounded one, becuse the bound on the buffer depend on the time tht Spoiler will spend looping on the second stte. Similrly, one cn find pirs of Büchi utomt such tht Duplictor hs winning strtegy, but not n existentilly bounded one. We write co nd f co when Duplictor hs n existentilly bounded strtegy, nd co nd f co, when Duplictor hs uniformly bounded strtegy. Topologicl Chrcteriztion. We cll function f : ARuns(A) ARuns(B) word preserving if for ll ρ ARuns(A), f(ρ) nd ρ re lbelled with the sme word. It cn be seen tht L(A) L(B) holds if nd only if there is word preserving function f : ARuns(A) ARuns(B). Proposition 1. Let A, B be two NBA. The following holds: A f co B if nd only if there is continuous word preserving function f : ARuns(A) ARuns(B). Existentilly bounded nd uniformly bounded gmes lso hve such topologicl chrcteriztion, nmely f co corresponds to the Lipschitz word preserving functions, nd f co to the loclly Lipschitz ones. This result hs some interesting consequences. First, it shows tht co nd f co re trnsitive reltions, since the composition of two continuous functions is continuous; using similr rgument, it cn be seen tht co, f co, co, nd f co re trnsitive s well. Second, since continuous functions re the sme s Lipschitz functions on compct spce, Proposition 1 shows tht, in the bsence of firness conditions, ny winning strtegy for Duplictor cn be turned into uniformly bounded strtegy. Corollry 1. The preorders co, co, nd co re equl. A third ppliction of Proposition 1 is tht we could show tht co (but not f co) is decidble in 2-EXPTIME using result of Holtmnn et l. [14]. We will however see in Section 5 tht this complexity is not optiml. 4 Look-Ahed Simultions We now consider vrint of the continuous simultion gmes clled look-hed simultion gmes (the terminology follows [6]). Look-hed simultion gmes proceed exctly like the continuous ones, except tht now Duplictor hs only two possibilities: either he skips his turn, or he flushes the entire buffer. Formlly, the definition of the gme only differs from the one of Section 3 in tht the number r of letters removed by Duplictor in round is either 0 or the size b of the buffer b, wheres continuous simultion llowed ny r {0,..., b }. We write A f l B if Duplictor hs winning strtegy for the look-hed fir simultion gme ssocited with the two utomt A, B. Similrly, we define the look-hed fir simultion for LTS, l, nd the ccording existentilly bounded l nd uniformly bounded l vrints.

6 Exmple 2. Consider gin A nd B s in Exmple 1. It holds tht A f l B, becuse Duplictor cn flush the buffer once he hs seen the first b or c. Clerly, ech look-hed simultion implies its continuous counterprt, but the converse does not hold. Exmple 3. Consider the following two NBA A (left) nd B (right) over the lphbet Σ = {, b, c}. b, c b c b c Duplictor wins the continuous fir simultion: uniformly bounded winning strtegy for Duplictor is to skip his first turn, nd then to remove one letter in turn during ll the rest of the ply. On the other hnd, Spoiler wins the look-hed simultion, becuse the first time Duplictor flushes the buffer, he hs to commit to choice between the two right sttes, nd thus mkes prediction bout the next letter tht Spoiler will ply. Unlike continuous simultions, look-hed simultions do not enjoy nturl topologicl chrcteriztion. A remrkble consequence of tht is tht look-hed simultions cese to be trnsitive (see Myr nd Clemente [6] for counter-exmple). With this respect, it is not cler whether ll winning strtegies cn be turned into uniformly bounded ones. We show in the next section tht this property of continuous simultions extends to look-hed simultions despite the bsence of topologicl chrcteriztion. 5 Quotient Gmes We now estblish the decidbility of buffered simultions. For this, we define quotient gme tht hs finite stte spce, nd show tht it is equivlent to the buffered simultion gme. Continuous Quotient Gme. The quotient gme is bsed on the congruence reltion ssocited with the Rmsey-bsed lgorithm for complementtion. We briefly recll its definition. Let us fix two Büchi utomt A = (Q, Σ, δ, q I, F ) nd B = (Q, Σ, δ, q I, F ) for simplicity we ssume they shre the sme stte spce nd only differ in their initil stte. We sy tht two finite words w 1, w 2 Σ re equivlent, w 1 w 2, if for ll q, q (1) q w1 q if nd only if q w2 q, nd (2) q w1 q if nd only if q w2 q. We write [w] to denote the equivlence clss of w with respect to the congruence. We sy clss [w] is idempotent if [ww] = [w]. Definition 1. The continuous quotient gme is plyed between plyers Refuter nd Prover s follows. Initilly, Refuter s pebble is on q 0 := q I, Prover s pebble is on q 0 := q I, nd the bstrct buffer [b] contins [ε]. On ech round i 1:

7 [w 1] 1. Refuter chooses two equivlence clsses [w 1 ], [w 2 ] nd stte q i, such tht q i 1 [w 2] q i q i nd [w 2 ] is idempotent [b] [w1] 2. Prover chooses q i such tht q i 1 q i q i, nd [w 2] is idempotent. The vlue [b] of the bstrct buffer is set to [w 2 ] for the next turn. Prover wins the ply if Refuter gets stuck or if the ply is infinitely long. Observe tht the quotient gme cn be seen s rechbility gme from Refuter s point of view. Moreover, since hs t most 3 Q 2 equivlence clsses, the stte spce of the quotient gme is finite nd bounded by Q 2.(3 Q Q 2 ). Proposition 2. Whether Prover hs winning strtegy for the continuous quotient gme is decidble in EXPTIME. [w 2] Lifting Quotient Strtegies. We show tht quotient gmes chrcterise the reltion f co. Lemm 1. A f co B only if Prover hs winning strtegy for the continuous quotient gme. Proof. Let us ssume tht Refuter hs winning strtegy for the continuous quotient gme. We wnt to show tht then Spoiler hs winning strtegy for the continuous fir simultion gme. We ctully consider vrint of the continuous fir simultion gme in which Spoiler my dd more thn one letter in round, nd Duplictor only removes one letter in round. Clerly, Spoiler hs winning strtegy for this vrint if nd only if he hs winning strtegy for the continuous fir simultion gme s defined in Section 3. Spoiler s strtegy bsiclly follows the one of Refuter. In the first round, Spoiler dds into the buffer some representtives w 1, w 2 of the equivlence clsses plyed by Refuter. Spoiler then dds w 2 into the buffer on every round for while. At some point, if Duplictor does not get stuck, he hs finished to remove w 1 w 2 from the buffer. Then Spoiler considers the stte q 1 in which Duplictor is, nd looks t wht Refuter would ply if Prover would hve picked q 1. Iterting this principle, Spoiler mimicks Refuter s strtegy. A key rgument in the proof of the converse direction is the following lemm which is esily proved using Rmsey s theorem. Lemm 2. Let q 0 1 q be n infinite ccepting run. Then there re i, j, k, i < j < k, such tht q i = q j = q k is ccepting, nd i+1... j j+1... k i+1... k. Lemm 3. A f co B if Prover hs winning strtegy for the continuous quotient gme. Proof. When the continuous simultion gme strts, Duplictor just skips his turn for while. Then Spoiler strts providing n infinite ccepting run q 0 0 q if he does not, Duplictor wits forever nd wins the ply. At some point, Lemm 2 pplies: the buffer contins w 1 w 2 w 2, [w 2 ] = [w 2] is idempotent, nd Spoiler is in stte q tht dmits [w 2 ]-loop. Then Duplictor considers the stte q in which Prover would move

8 if Refuter plyed [w 1 ], [w 2 ], q in the first round; Duplictor then removes w 1 w 2 from the buffer, nd moves to this stte q. Duplictor proceeds identiclly in the next rounds, nd either Spoiler eventully gets stuck, or he follows non-ccepting run, or the ply is infinite. Look-Ahed Quotient Gme. In order to estblish the decidbility of look-hed simultions, we introduce look-hed quotient gme. The gme essentilly differs from the continuous quotient gme in tht it does not use buffer. Definition 2. We cll look-hed quotient gme the following gme plyed by Refuter nd Prover; initilly, Refuter s pebble is on q 0 := q I, Prover s pebble is on q 0 := q I. On ech round i 1: [w 1] 1. Refuter chooses two equivlence clsses [w 1 ], [w 2 ] nd stte q i, such tht q i 1 [w 2] q i q i nd [w 2 ] is idempotent. [w 1] [w 2] 2. Prover chooses q i such tht there is q i 1 q i q i. Prover wins the ply if Refuter gets stuck or if the ply is infinitely long. Following the sme kind of rguments we used for the continuous quotient gme, the result below cn be estblished. Proposition 3. A f l B if nd only if Prover hs winning strtegy for the look-hed quotient gme. The size of the ren of look-hed quotient gme is gin exponentil in the size of the utomt; but there re only Q 2 positions for Refuter, so look-hed quotient gmes cn be solved slighty better thn continous ones. Proposition 4. Whether Prover hs winning strtegy for the look-hed quotient gme cn be decided in PSPACE. Proof. We consider the following non-deterministic lgorithm tht guesses the set W of ll pirs (q 0, q 0) of initil configurtions of the gme such tht Duplictor hs winning strtegy. For ll (q 0, q 0) in W, the following cn then be checked in polynomil spce: for ll [w 1 ], [w 2 ], nd q 1 tht could be plyed by Spoiler, there is q 1 tht cn be plyed by Duplictor such tht (q 1, q 1) is in W. Degrees of Look-Ahed. The quotient gmes for co nd l only differ from the ones of Definitions 1 nd 2 in tht they do not require the loop lbelled with w 2 to be ccepting (or equivlently, ll sttes re ssumed to be ccepting). We clim tht for such quotient gmes, it is possible to lift winning strtegy for Prover to uniformly bounded winning strtegy for Duplictor. Agin, the key of the proof is n ppliction of Rmsey s theorem. Lemm 4. Let A be NBA. Then there is number n(a) such tht for ll trces q 0 1 q q l with l n(a), there re i, j, k such tht i < j < k, q i = q j = q k nd nd i+1... j j+1... k i+1... k.

9 When we lift Prover s strtegy to Duplictor s strtegy, we thus only need buffer bounded by n(a). Theorem 1. Let x {l, co}. Then x, x nd x coincide. 6 The Complexity of Buffered Simultions We provide lower bounds tht mtch the upper ones on deciding continuous nd lookhed simultions of the previous section. These re estblished by encoding vrious tiling problems. Definition 3. A tiling system is tuple T = (T, H, V, t I, t F ), where T is set of tiles, H, V T T re the horizontl nd verticl comptibility reltions, t I T is n initil tile, nd t F is n ccepting tile. Let n, m be two nturl numbers. A vlid n m tiling for T is function t : {1,..., n} {1,..., m} T such tht (1) t 1,1 = t I, (2) for ll i = 1,..., n, for ll j = 1,..., m 1, (t i,j, t i,j+1 ) H, (3) for ll i = 1,..., n 1, for ll j = 1,..., m, (t i,j, t i+1,j ) V, nd (4) t i,j = t F for some i, j. Theorem 2. Deciding l nd f l is PSPACE-complete. Proof. PSPACE-hrdness of f l follows from the one of l. To prove the ltter we reduce the problem of the existence, for given n in unry nd given tiling system T, of m > 0 nd m n vlid tiling. We define two utomt A, B over the lphbet T, such tht ll sttes re ccepting, nd the sizes of A, B re polynomil in T + n, nd moreover A f l B holds if nd only if vlid tiling exists. A ccepts n infinite word if nd only if it is of the form w 1 w 2... w i, where ech w i T n is such tht every two consecutive tiles t, t ppering in w i re such tht (t, t ) H. Similrly, B is defined such tht it ccepts n infinite sequence of tiles t 0 t 1... if nd only if it either t 0 t I or there is some i n such tht (t i n, t i ) V, nd for ll j i, t j t F. Theorem 3. Deciding f co, nd co is EXPTIME-complete. Proof. We consider n EXPTIME-hrd gme-theoretic vrint of the tiling problem. Let us fix some tiling system T. The gme is plyed by two plyers: Strter nd Completer. The tsk for Completer is to produce vlid tiling, wheres Strter s gol is to mke it impossible. On every round i 1, 1. Strter selects the tile t i,1 strting the i-th row; if i = 1, the tile must be t I, otherwise it must be verticlly comptible with t i 1,1 ; 2. Completer selects the tiles t i,2,... t i,n completing the i-th row; if this does not define vlid i n-tiling, Completer loses. If it does, nd if t i,j = t F for some j, then Completer wins. Otherwise, the ply proceeds to the next round. If Strter gets stuck by choosing verticlly comptible t i,1, Duplictor wins the ply. In ll other cses, Completer wins the ply. Given tiling system T = (T, H, V, t 0, t F ), we construct two NBA A, B of polynomil size, tht only contin ccepting sttes, such tht there is winning strtegy for

10 Strter in the tiling gme if nd only if there is winning strtegy for Duplictor in the continuous simultion gme (A co B). Without loss of generlity, we ssume tht Strter cn lwys move. We consider the lphbet T {0, 1}. Spoiler s utomton A is defined such tht n infinite word w is ccepted by A if nd only if it is of the form b 0 w 0 b 1 w 1 b 2 w 2..., where for ll i 0, b i {0, 1}, w i T n, nd two consecutive tiles in w i re in the horizontl reltion. Duplictor s utomton does severl things. He forces Spoiler to repet the previous row when bit 1 occurs, i.e. if Spoiler wins plying w i 1w i+1, then w i = w i+1. Duplictor lso forces Spoiler to provide verticlly mtching row when bit 0 occurs, i.e. if Spoiler wins plying w i 0w i+1, then w i nd w i+1 must be verticlly comptible consecutive rows. However, Duplictor does more: he lwys forces Spoiler to strt the row with given tile t; this tile is determined by the stte q t (or neighbour ) in which Duplictor currently is. Informlly, the sttes q t of Duplictor s utomton B re such tht (1) q ti is the initil stte of B, nd (2) if one strts reding from q t, the following holds: (P1) for n infinite word strting with 0t..., with t t, one cn pick n ccepting run tht does not depend on the infinite suffix; (P2) for n infinite word strting with bv1v..., b {0, 1}, v, v T n, nd v v, one cn pick n ccepting run tht does not depend on the infinite suffix; (P3) for n infinite word strting with bv0v..., b {0, 1}, v, v T n, if there is i {1,..., n} such tht the i-th letters of v nd v re not verticlly comptible, then one cn pick n ccepting run tht does not depend on the infinite suffix; (P4) if v T n 1v does not contin t F, then q t q t ; (P5) if v T n 0v does not contin t F, then q t q t for ll t such tht (t, t ) V. Let us first show tht if Completer hs winning strtegy, then Spoiler hs winning strtegy. Spoiler plys s follows: first, he moves long 0v 1, where v 1 is the first row of the tiling. Then he itertes 1v 1 for while. This forces Duplictor to eventully remove 0v 1 from the buffer, nd commit to choosing some q t, due to (P4) nd (P5). Spoiler then considers the second row v 2 tht Completer would nswer if Strter would put t t the beginning of the second row. Spoiler picks this row v 2, nd ply 0v 2, followed by itertions of 1v 2, nd repets the sme principle. Let us now show tht if Strter hs winning strtegy, then Duplictor hs winning strtegy. Duplictor first wits for the 2n + 2 first letters of Spoiler. Becuse of (P1..3), Spoiler hs nothing better to do thn to ply 0v1v for some v encoding vlid first row of tiling. Duplictor considers the tile t tht would be plyed by Strter in the second row if Completer plyed v on the first row. Duplictor then removes 0v nd ends in the stte q t. From there, he wits gin for n + 1 letters, so tht the buffer now contins 1vbv for some b {0, 1}. Repeting the sme process if b = 1, he cn force Spoiler to eventully ply 0v where v codes row verticlly comptible with v nd strting with t. Iterting this principle results in n infinite ply won by Duplictor. 7 Conclusion We introduced severl buffered simultions for trnsitions systems nd Büchi utomt tht differ from stndrd simultions in tht Duplictor is llowed to skip his turn nd

11 only respond to given Spoiler s move with n rbirrily long dely. We estblished the decidbility of these simultions by mens of quotient gme, nd we precisely chrcterized their complexity, EXPTIME-complete for the continuous simultion, nd PSPACE-complete for the look-hed simultion. It seems very likely tht these complexity results extend to vrints of these simultions. Let us briefly mention few of these vrints. First, it is possible to define continuous nd look-hed bisimultions. For this, we my llow Spoiler to swp the pebbles t the beginning of his turn, but only on the turns where the buffer is empty. Continuous bisimultion is bit degenerted, nd ctully collpses to continuous simultion equivlence becuse Duplictor my lwys keep the buffer non-empty nd prevent Spoiler from swpping the pebbles. Look-hed bisimultion is strictly included in look-hed simultion equivlence, but it is not n equivlence reltion for the sme reson tht look-hed simultion is not trnsitive. It is questionble whether there would be better forms of buffered bisimultions. Finitry versions of the look-hed nd continuous simultions hve been introduced indepently by Clemente nd Myr [6] for utomt minimiztion nd by the uthors of this pper [16] for lnguge inclusion testing. For the prticulr purpose of utomt minimiztion, it is interesting to consider delyed nd direct simultions. It is not difficult to check tht the infinitry version of delyed/direct buffered simultions cn be used for quotienting/pruning Büchi utomt exctly like their finitry counterprt, nd tht they hve the sme complexities s the buffered fir simultions. Finitry buffered simultions pper thus s computtionlly chep pproximtions of the simultions we introduced in this work. References 1. P. A. Abdull, Y.-F. Chen, L. Clemente, L. Holík, C.-D. Hong, R. Myr, nd T. Vojnr. Simultion subsumption in rmsey-bsed Büchi utomt universlity nd inclusion testing. In Proc. 22nd Int. Conf. on Computer-Aided Verifiction, CAV 10, volume 6174 of LNCS, pges Springer, P. Aziz Abdull, Y.-F. Chen, L. Clemente, L. Holík, C.-D. Hong, R. Myr, nd T. Vojnr. Advnced rmsey-bsed Büchi utomt inclusion testing. In Proc. 22nd Int. Conf. on Concurrency Theory, CONCUR 11, volume 6901 of LNCS, pges Springer, J. R. Büchi. On decision method in restricted second order rithmetic. In Proc. Congress on Logic, Method, nd Philosophy of Science, pges 1 12, Stnford, CA, USA, Stnford University Press. 4. D. Bustn nd O. Grumberg. Simultion-bsed minimiztion. ACM Trns. Comput. Logic, 4(2): , April G. Cécé nd A. Finkel. Verifiction of progrms with hlf-duplex communiction. Inf. Comput., 202(2): , Lorenzo Clemente nd Richrd Myr. Advnced utomt minimiztion. In Proc. 40th Symp. on Principles of Progrmming Lnguges, POPL 13, pges ACM, D. L. Dill, A. J. Hu, nd H. Wong-Toi. Checking for lnguge inclusion using simultion preorders. In Proc. 3rd Int. Workshop on Computer-Aided Verifiction, CAV 91, volume 575 of LNCS, pges Springer, L. Doyen nd J.-F. Rskin. Antichins for the utomt-bsed pproch to model-checking. Logicl Methods in Computer Science, 5(1), 2009.

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