Bisimilarity of one-counter processes is PSPACE-complete

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1 Bisimilrity of one-counter processes is PSPACE-complete Stnislv Böhm 1, Stefn Göller 2, nd Petr Jnčr 1 1 Techn. Univ. Ostrv (FEI VŠB-TUO), Dept of Computer Science, Czech Republic 2 Universität Bremen, Institut für Informtik, Germny Abstrct. A one-counter utomton is pushdown utomton over singleton stck lphbet. We prove tht the bisimilrity of processes generted by nondeterministic one-counter utomt (with no ε-steps) is in PSPACE. This improves the previously known decidbility result (Jnčr 2000), nd mtches the known PSPACE lower bound (Srb 2009). We dd the PTIME-completeness result for deciding regulrity (i.e. finiteness up to bisimilrity) of one-counter processes. 1 Introduction Among the vrious notions of behviorl equivlences of (rective) systems, (strong) bisimilrity plys n importnt rôle (cf, e.g., [16]). For instnce, vrious logics cn be chrcterized s the bisimultion-invrint frgment of richer logics. A fmous theorem due to vn Benthem sttes tht the properties expressible in modl logic coincide with the bisimultion-invrint properties expressible in first-order logic [28]. Similr such chrcteriztions hve been obtined for the modl µ-clculus [8] nd for CTL [17]. Another importnt notion is wek bisimilrity tht generlizes (strong) bisimilrity by distinguishing ε-moves corresponding to internl behvior. There re numerous further notions of equivlences. For more detiled tretment of the different behviorl equivlences in the context of concurrency theory, the reder is referred to [4]. The (wek/strong) bisimilrity problem consists in deciding if two given sttes of given trnsition system re wekly/strongly bisimilr. On finite trnsition systems both wek nd strong bisimilrity is well-known to be complete for deterministic polynomil time [1]. Moreover, on finite trnsition systems wek bisimilrity cn be reduced to strong bisimilrity in polynomil time by computing the trnsitive closure. In the lst twenty yers lot of reserch hs been devoted to checking behviorl equivlence of infinite-stte systems, see [23] for n up-to-dte record. In the setting of infinite-stte systems, see lso [14] for Myr s clssifiction of infinite-stte systems, the sitution is less cler. There re numerous clsses of infinite-stte systems for which decidbility of bisimilrity is not known. Three such intricte open problems re (i) wek bisimilrity on bsic prllel processes (BPP, subclss of Petri nets), (ii) strong bisimilrity of process lgebrs (PA), nd (iii) wek bisimilrity of bsic process lgebrs (BPA). On the negtive side, we mention undecidbility of wek bisimilrity of PA by Srb [22]. On the positive side we mention n importnt result by Sénizergues who S. Böhm nd P. Jnčr re supported by the Czech Ministry of Eduction, project No. 1M0567.

2 shows tht bisimilrity on equtionl grphs of finite out degree [19] ( slight generliztion of pushdown grphs) is decidble. See lso Stirling s unpublished pper [25] for shorter proof of this, using ides from concurrency theory. For normed PA processes Hirshfeld nd Jerrum prove decidbility of strong bisimilrity [7]. When focussing on the computtionl complexity of bisimilrity checking of infinite-stte systems for which this problem is decidble, the sitution becomes even worse. There re only very few clsses of infinite-stte systemss for which the precise computtionl complexity is known. For instnce, when coming bck to one of the bove-mentioned positive results by Sénizergues/Stirling concerning (slight extensions of) pushdown grphs, primitive recursive upper bound is not yet known. However, EXPTIME hrdness of this problem ws proven by Kučer nd Myr [13]. As one of the few results on infinite systems where the upper nd lower complexity bounds mtch, we cn mention [10] where it is shown tht bisimilrity on bsic prllel processes is PSPACE-complete. In this pper we study the computtionl complexity of deciding strong bisimilrity over processes generted by one-counter utomt. One-counter utomt re pushdown utomt over singleton stck lphbet. This model hs been extensively studied in the verifiction community; we cn nme, e.g., [2, 5, 6, 3, 26] s recent works. Wek bisimilrity for one-counter processes is shown to be undecidble in [15], vi reduction from the emptiness problem of Minsky mchines. For strong bisimilrity the third uthor estblished decidbility in [9], however without providing ny precise complexity bounds. In n unpublished rticle [29] Yen nlyses the pproch of [9], deriving triply exponentil spce upper bound. A PSPACE lower bound for bisimilrity is proven by Srb [24]. This lower bound lredy holds over one-counter utomt tht cnnot test for zero nd whose ctions cn moreover be restricted to be visible (so clled visibly one-counter nets), i.e. tht the lbel of the ction determines if the counter is incremented, decremented, or not modified respectively. For visibly one-counter utomt it is proven in [24] tht strong bisimilrity is in PSPACE vi reduction to the model checking problem of the modl µ-clculus over one-counter processes [20]. For bisimilrity on generl one-counter processes, in prticulr when dropping the visibility restriction, the sitution is surely more involved. Our min result closes the complexity gp for bisimilrity of one-counter processes from bove, thus estblishing PSPACE-completeness. In nutshell, we provide nondeterministic lgorithm implementble in polynomil spce which genertes bisimultion reltion on-the-fly. The lgorithm uses polynomil-time procedure which, given pir p(m), q(n) of processes, either gives definite nswer surely bisimilr or surely non-bisimilr, or declres the pir s cndidte. For ech fixed m there re (only) polynomilly mny cndidtes (p(m), q(n)), nd the lgorithm processes ech m = 0, 1, 2,... in turn, guessing the bisimilrity sttus of ll respective cndidtes nd verifying the (locl) consistency of the guesses. A crucil point is tht it is sufficient to stop the processing fter exponentilly mny steps, since then certin periodicity is gurnteed, which would enble to successfully continue forever. We lso consider the problem of deciding regulrity (finiteness w.r.t. bisimilrity) which sks if, for given one-counter process, there is bisimilr stte in some finite system. Decidbility of this problem ws proven in [9] nd ccording to [24] it follows

3 from [1] nd [21] tht the problem is lso hrd for P. We give simpler P-hrdness proof, but we lso show tht the regulrity problem is in P, thus estblishing its P- completeness. It is pproprite to dd tht Kučer [12] showed polynomil lgorithm deciding bisimilrity between one-counter process nd (given) finite system stte. The pper is orgnized s follows. Section 2 contins the bsic notions, definitions, nd reclls some uxiliry results. Section 3 reclls nd enhnces some useful notions which were used in [9] nd elsewhere. Section 4 contins the crucil technicl results, which hve enbled to replce the decision lgorithm from [9] with polynomil spce lgorithm. The lgorithm is elborted in Section 5 nd its correctness is shown in Section 6. Section 7 then shows PTIME-completeness of -regulrity. 2 Preliminries N denotes the set {0, 1, 2,...}. For set X, by X we denote its crdinlity. Trnsition systems. A (lbelled) trnsition system is structure T = (S, A, { A}), where S is set of sttes, A set of ctions, nd S S is set of -lbeled trnsitions, for ech ction A. We define = A, nd prefer to use the infix nottion s 1 s 2 (resp. s 1 s 2 ) insted of (s 1, s 2 ) (resp. (s 1, s 2 ) ). T is finite trnsition system if S nd A re finite; we then define the size of T s T = S + A + A. Bisimultion equivlence. Let T = (S, A, { A}) be trnsition system. A binry reltion R S S is bisimultion if for ech (s 1, s 2 ) R the following bisimultion condition holds: for ech s 1 S, A, where s 1 s 1, there is some s 2 S such tht s 2 nd (s 1, s 2 ) R, nd for ech s 2 S, A, where s 2 s 2, there is some s 1 S such tht s 1 nd (s 1, s 2) R. s 2 s 1 We sy tht sttes s 1 nd s 2 re bisimilr, bbrevited by s 1 s 2, if there is bisimultion R contining (s 1, s 2 ). Bisimilrity is obviously n equivlence. We lso note tht the union of bisimultions is bisimultion, nd tht is the mximl bisimultion on S. Bisimilrity is nturlly defined lso between sttes of different trnsition systems (by considering their disjoint union). One-counter utomt. A one-counter utomton is tuple M = (Q, A, δ =0, δ >0 ), where Q is finite nonempty set of control sttes, A is finite set of ctions, δ =0 Q {0, 1} A Q is finite set of zero trnsitions, nd δ >0 Q { 1, 0, 1} A Q is finite set of positive trnsitions. (There re no ε-steps in M.) The size of M is defined s M = Q + A + δ =0 + δ >0. Ech one-counter utomton M = (Q, A, δ =0, δ >0 ) defines the trnsition system T M = (Q N, A, { A}),

4 where (q, n) (q, n + i) iff either n = 0 nd (q, i,, q ) δ =0, or n > 0 nd (q, i,, q ) δ >0. A one-counter net is one-counter utomton, where δ =0 δ >0. A stte (q, m) of T M is lso clled configurtion of M, or one-counter process; we usully write it s q(m). Elements of δ =0 δ >0 re clled trnsitions. The notion of pth p(m) q(n), where is sequence of trnsitions, is defined in the nturl wy. A trnsition sequence β in (δ >0 ) + is clled n elementry cycle if it induces n elementry cycle in the control stte set Q, i.e., if β = (q 1, i 1, 1, q 2 ), (q 2, i 2, 2, q 3 ),... (q m, i m, m, q m+1 ) where q i q j for 1 i < j m nd q m+1 = q 1. We note tht such cycle hs length t most Q, nd its effect (i.e., the cused chnge) on the counter vlue is in the set { Q, Q + 1,..., Q }. Decision problems. We re interested in the following two decision problems. BISIMILARITY ON OCA INPUT: A one-counter utomton M with two sttes p 0 (m 0 ) nd q 0 (n 0 ) of T M, where both m 0 nd n 0 re given in binry. QUESTION: Is p 0 (m 0 ) q 0 (n 0 )? We sy tht one-counter process q(n), i.e. configurtion q(n) of one-counter utomton M, is -regulr (or finite up to bisimilrity) if there is finite trnsition system with some stte s such tht q(n) s. REGULARITY ON OCA INPUT: A one-counter utomton M nd stte q(n) of T M (n given in binry). QUESTION: Is q(n) -regulr? Strtified bisimilrity. Given trnsition system T = (S, A, { A}), on S we define the fmily of i-equivlences, i N, s follows. We put 0 = S S, nd we hve s 1 i+1 s 2 if the following two conditions hold: for ech s 1 S, A, where s 1 nd s 1 i s 2 ; for ech s 2 S, A, where s 2 nd s 1 i s 2. s 1, there is some s 2 S such tht s 2 s 2, there is some s 1 S such tht s 1 The following proposition is n instnce of the result for imge finite systems [16]. Proposition 1. On sttes of T M we hve = i 0 i. s 2 s Some useful observtions The next proposition cptures the loclity of the bisimultion condition for one-counter utomt, implied by the fct tht the counter vlue cn chnge by t most 1 in move.

5 Proposition 2. Given one-counter utomton M = (Q, A, δ =0, δ >0 ) nd reltion R (Q N) (Q N), for checking if pir (p(m), q(n)) R stisfies the bisimultion condition it suffices to know the restriction of R to the set NEIGHBOURS(m, n) = { (p (m ), q (n )) m m 1, n n 1 }. Stndrd prtition rguments [11, 18] imply the following proposition for finite systems. Proposition 3. Given finite trnsition system F = (Q, A, { A}), where k = Q, we hve k 1 = k = on Q. Moreover, (the prtition of Q corresponding to) cn be computed in polynomil time. 3 Underlying finite utomton nd the set INC Most of the notions, clims nd ides in this section ppered in [9] nd elsewhere; nevertheless, we present (nd extend) them in concise self-contined wy. If the context does not indicte otherwise, in wht follows we (often implicitly) ssume fixed one-counter utomton M = (Q, A, δ =0, δ >0 ), using k for Q. We strt by observing tht if the counter vlue is lrge, then M behves, for long time, like (nondeterministic) finite utomton. By F M we denote the finite trnsition system underlying M; we put F M = (Q, A, { A}), where = {(q 1, q 2 ) Q Q i : (q, i,, q ) δ >0 }. (F M thus behves s if the counter is positive, ignoring the counter chnges.) In wht follows, p, q, r Q re viewed s control sttes of M or s sttes of F M, depending on context. Our observtion is formlized by the next proposition (which is obvious, by induction on m). Proposition 4. If m m then p(m ) m p. (Here p(m ) is stte of T M, wheres p is stte of F M.) This implies, e.g., tht if p q (i.e., p k q by Proposition 3) nd m, n k, then p(m) k q(n) (nd thus p(m) q(n)), since p(m) k p, q(n) k q nd k is n equivlence. If p q then we cn hve p(m) q(n), due to the possibility of reching zero. For mking this more precise, we define the following set INC = { r(l) q Q : r(l) k q }. The configurtions in INC re incomptible with F M in the sense tht they re not bisimilr up to k moves with ny stte of F M. Proposition 5. If r(l) INC then l < k. Moreover, INC cn be constructed in polynomil time. Proof. If l k then r(l) k r, nd thus r(l) INC. To construct INC, we cn strt with the set contining ll k 3 pirs (r(l), q), where l < k ; ll such pirs belong to 0. We then delete the pirs not belonging to 1, then those not belonging to 2, etc., until k. The configurtions r(l) for which no pir (r(l), q) survived, re in INC. (This process cn be done simultneously for the pirs (p, q) in F M ; we lso use the fct r(k) k r.)

6 The rguments of the previous proof lso induce the following useful proposition. Proposition 6. The question if p(m) k q(n) cn be decided in polynomil time. We note tht if p(m) INC nd q(n) INC then p(m) q(n) (in fct, p(m) k q(n)). More generlly, if two one-counter processes re bisimilr then they must gree on the distnce to INC; this is formlized by the next lemm. We define dist(p(m)) s the length of the shortest trnsition sequence such tht p(m) INC (i.e., p(m) r(l) for some r(l) INC); we put dist(p(m)) = ω if there is no such sequence, i.e., when INC is unrechble, denoted p(m) INC. Lemm 7. If p(m) q(n) then dist(p(m)) = dist(q(n)). Proof. For the ske of contrdiction, suppose tht p(m) q(n) nd d = dist(p(m)) < dist(q(n)), for the lest d ; necessrily d > 0, since we cnnot hve p(m) INC, q(n) INC. Thus there is move p(m) p (m ) with dist(p (m )) = d 1, which must be mtched by some q(n) q (n ) where p (m ) q (n ). Necessrily d 1 = dist(p (m )) < dist(q (n )), which contrdicts the minimlity of d. The next lemm clrifies the opposite direction in the cse of infinite distnces. Lemm 8. If dist(p(m)) = ω then p(m) r for some r Q. Thus if dist(p(m)) = dist(q(n)) = ω then p(m) q(n) iff there is some r Q such tht p(m) k r k q(n). Proof. If dist(p(m)) = ω, i.e. p(m) INC, then in prticulr p(m) INC, nd there is thus r Q such tht p(m) k r. We cn esily check tht R = { (p(m), r) p(m) k r, p(m) INC } is bisimultion: if p(m) p (m ) nd r r where p (m ) k 1 r, then p (m ) INC nd the fct p (m ) INC implies tht r k p (m ) k 1 r for some r Q; hence r k 1 r nd thus r k r (by Proposition 3), which mens p (m ) k r. In the next section we look in more detil t the function dist(p(m)), which provides useful constrint on bisimilr pirs. But before tht, we prtition the set (Q N) (Q N) into three ctegories. We sy tht pir (p(m), q(n)) is surely-positive if dist(p(m)) = dist(q(n)) = ω nd p(m) k q(n) (nd thus surely p(m) q(n), by Lemm 8), surely-negtive if p(m) k q(n) or dist(p(m)) dist(q(n)) (nd thus surely p(m) q(n)), cndidte otherwise, i.e., if p(m) k q(n) nd dist(p(m)) = dist(q(n)) < ω. By SUREPOS we denote the set of ll surely-positive pirs, nd we note the following obvious proposition.

7 Proposition 9. SUREPOS is bisimultion. It will be lso useful to view the set CAND of ll cndidte pirs s the union CAND = CAND 0 CAND 1 CAND 2 where CAND i contins the cndidte pirs t level i, i.e. the pirs (p(m), q(n)) CAND with m = i. 4 Distnce to INC In this section we look t the distnce function dist(p(m)) in more detil (Lemm 10) nd derive some consequences (Lemm 11) which will be useful for the design nd nlysis of our lter lgorithm. We strt with sketching some intuition which is then formlized in Lemm 10. To rech INC from p(m) most quickly, for lrge m, one uses suitble prefix rriving β t most effective elementry cycle q( ) q( ) (which decreses the counter by k = Q t most), let us cll it d-cycle, then repets the d-cycle sufficiently mny times, nd finishes with suffix rriving t INC. It is not difficult to nticipte tht one cn bound the length (nd thus lso the counter chnge) of the prefix nd the suffix by (smll degree) polynomil pol(k). We now stte the lemm. For technicl resons, we do not require explicitly tht the d-cycle is elementry; it is sufficient to bound its length by k. Lemm 10. There is polynomil pol : N N (independent of M) such tht for ny p(m) with dist(p(m)) < ω there is shortest pth p(m) INC with the trnsition sequence of the form = αβ i γ where length(αγ) pol(k) nd β is decresing cycle of length k. Proof. To give complete forml proof requires some technicl work. Since the essence of the clim is not originl nd similr technicl results pper in the previous works on one-counter utomt, we do not provide self-contined proof, but we use Lemm 2 from n older pper [27]; in our nottion, this lemm is formulted s follows: Clim. If there is positive pth (using positive trnsitions) from p(m) to q(n) nd m n k 2 nd n k 2 then there is shortest pth p(m) q(n) such tht = αβ i γ where length(αγ) < k 2 nd length(β) k. (Although [27] studies deterministic one-counter utomt, the lemm obviously pplies to our nondeterministic cse s well, since we cn view the trnsitions themselves s the ctions.) We note tht if m n k 2 + k then β is necessrily decresing cycle (i 2 in this cse). It is lso cler tht the (shortest) pth p(m) q(n) in the Clim does not visit ny q (n ) with n m + k 2 + k; we sy tht the pth moves in the < (m+k 2 +k) re (note tht the prefix α moves in the < (m+k 2 ) re nd the suffix γ moves in the < (n+k 2 ) re ). Reclling tht l < k for ech r(l) INC, we note tht ny shortest pth p(m) INC either moves in the < k 2 -re, in which cse its length is bounded

8 by k 3 (since no configurtion is visited twice), or cn be presented in the form p(m) 1 q1 (k 2 ) 2 q2 (k 2 ) 3 m qm (k 2 ) m+1 INC where 1 m k nd q 1 (k 2 ), q 2 (k 2 ),..., q m (k 2 ) re ll configurtions on the pth which hve the counter vlue k 2. By the bove considertions, the segment q i (k 2 ) i+1 moves in the < (3k 2 + k) re, nd its length is thus bounded by k (3k 2 +k). The segment p(m) 1 q 1 (k 2 ) either moves in the < 2k 2 re, in which cse its length is bounded by 2k 3, or it cn be written p(m) 1 p (m ) 2 q 1 (k 2 ) where m 2k 2 nd 1 (which might be empty) is bounded by 2k 3. The sttement of our Lemm thus follows from the bove Clim pplied to the segment p (m ) 2 q 1 (k 2 ). The next lemm lists some importnt consequences. A min point is to clrify the distribution of the set CAND. Informlly speking, the cndidte pirs re contined inside polynomilly mny liner belts, ech belt hving rtionl slope, being frction of polynomilly bounded integers, s well s polynomilly bounded (verticl) thickness. Remrk. It is helpful to think in geometricl notions. Every reltion R (Q N) (Q N) cn be viewed s coloring χ R : Q Q N N {, }; for ech p, q Q it prescribes blck-white coloring of the plne (grid) N N. This ws more formlized in [9]; here we just informlly use Figure 1. Lemm There is polynomil-time lgorithm computing dist(p(m)) for ny p, m; here the size of the input is M + log m (m is written in binry). 2. If dist(p(m)) < ω then dist(p(m)) = c 1 c 2 (m + d 1 ) + d 2 = c 1 c 2 m + ψ for some integers 0 c 1 k, 1 c 2 k, d 1 pol 1 (k), 0 d 2 pol 1 (k) where pol 1 is polynomil (independent of M); the vlues c 1, c 2, d 1, d 2 generlly depend on p, m. Moreover, for the rtionl number ψ = c1 c 2 d 1 + d 2 we hve ψ (k+1) pol 1 (k). 3. If dist(p(m)) = dist(q(n)) < ω then n = ρ m + ξ where (the slope) ρ is either 0 or of the form c1c 2 c 2c, for c 1, c 2, c 1, c 2 {1, 2,..., k}, 1 nd ξ is bounded by polynomil pol 2 (k). (This formlizes the bove nnounced polynomilly mny belts, with the verticl thickness pol 2 (k).) 4. There is polynomil pol 4 such tht for ech m pol 4 (k) we hve ρ 1 m + pol 2 (k) + 1 < ρ 2 (m 1) pol 2 (k), where ρ 1 < ρ 2 re (different) slopes from Point 3, pol 2 lso being tken from there. (I.e., for levels m pol 4 (k) the belts re seprted, in the sense tht no two pirs from different belts re neighbours.)

9 5. There is polynomil-time lgorithm which, given i (in binry), computes the set CAND i of ll cndidte pirs t level i (ll pirs (p(i), q(n)) such tht p(i) k q(n) nd dist(p(i)) = dist(q(n)) < ω). We hve CAND i pol 3 (k) for polynomil pol If is multiple of the effects of ll decresing cycles of length k (the bsolute vlues of the effects re in the set {1, 2,..., k}) then for ech m k + pol(k), where pol is tken from Lemm 10, we hve: 7. If m, n k + pol(k) then p(m) INC iff p(m + ) INC. (p(m), q(n)) SUREPOS i, j N : (p(m + i ), q(n + j )) SUREPOS (where pol nd re s in Point 6). Proof. Point 1. By Lemm 10 we know tht shortest pth p(m) INC (if there is ny) is of the form α β β β p(m) q(m+e 1 ) q(m+e 1 c 2 ) q(m+e 1 2c 2 ) β q(l e 2 +c 2 ) β q(l e 2 ) γ r(l) INC where e 1 is the effect (the counter chnge) of the prefix α, c 2 is the bsolute vlue of the effect of the d-cycle β, nd e 2 is the effect of the suffix γ ; we put c 1 = length(β), c 3 = length(α), c 4 = length(γ). Let us recll tht 0 c 2 c 1 k nd tht the bsolute vlues of other integers re bounded by pol(k) from Lemm 10. (Independently of p, m,) we thus hve polynomilly mny possibilities (in k) for the tuple q, e 1, c 1, c 2, c 3, c 4, e 2, r, l; these possible tuples cn be processed in turn. For ech tuple we cn check if (m+e 1 ) (l e 2 ) is divisible by c 2 nd then verify if the tuple is relizble by some pproprite α, β, γ; this verifiction is done by using strightforwrd grph rechbility lgorithms. (Regrding the d-cycle, it is sufficient to verify the relizbility of the first segment q(m+e 1 ) q(m+e 1 c 2 ) nd of the finl segment q(l e 2 +c 2 ) q(l e 2 ).) With ( ech relizble tuple we ssocite the vlue c 3 + c 4 when c 2 = 0 nd c 3 + c 4 + c1 c (m+e1 2 ) (l e 2 ) ) when c 2 > 0. We ssocite ω with ech non-relizble tuple. The vlue dist(p(m)) is obviously the miniml vlue ssocited with the bove tuples. Point 2. This follows immeditely from the nlysis in the proof of Point 1. (Since d 2 = c 3 + c 4, d 1 = e 1 l+e 2, it suffices to tke pol 1 (k) = k + pol(k), for pol from Lemm 10. The consequence for ψ is obvious.) Point 3. From dist(p(m)) = c1 c 2 m + ψ = c 1 c + ψ 2n = dist(q(n)), we derive n = c1/c2 c 1 /c 2 m + ψ ψ c. If c 1 = 0 or c 1 /c 1 2 = 0 then dist(p(m)) = dist(q(n)) (k+1) pol 1 (k), nd thus n < k + (k+1) pol 1 (k) (nd we cn put ρ = 0). We cn thus tke pol 2 (k) = 2 (k+1) pol 1 (k) k. Point 4. Reclling the slopes from Point 3, we note tht ρ 1 <ρ 2 implies ρ 2 ρ Since ρ 1 k 2, it is sufficient to hve pol 2 (k) + 1 < 1 k m k k pol 4 2 (k). Point 5. Given i, for ech p Q in turn we compute z = dist(p(i)) nd ll polynomilly mny n such tht c1 c 2 (n+d 1 )+d 2 = z, where c 1, c 2, d 1, d 2 stisfy the constrints k 4.

10 from Point 2. For ech such n nd ech q Q we check if (p(i), q(n)) CAND i, i.e., if dist(q(n)) = z nd p(i) k q(n); Point 1 nd Proposition 6 show tht this cn be done in polynomil time. Point 6. Since m k + pol(k), the length of (ech) such tht p(m) INC is greter thn pol(k). Incresing or decresing the number of repeting the d-cycle does the job. Point 7. From Point 6 we know tht for m k+pol(k) we hve dist(p(m)) = ω iff dist(p(m+i )) = ω for ll i N. Thus for m, n k+pol(k) we hve p(m+i ) k q(n + j ) nd dist(p(m + i )) = dist(q(n + j )) = ω if nd only if p k q nd dist(p(m)) = dist(q(n)) = ω. 5 A polynomil spce lgorithm The next lemm follows from Lemm 11, Point 1, nd Proposition 6. Lemm 12. There is polynomil-time lgorithm which, given (M nd) pir (p(m), q(n)), decides if the pir is in SUREPOS, or in CAND, or is surely-negtive. We might be tempted to try to resolve the question of bisimilrity of the cndidte pirs by looking for dditionl polynomilly checkble conditions. But the PSPACE-hrdness result for (visibly) one-counter processes [24] discourges us from doing so; we should be stisfied with solving our problem in polynomil spce. Thus nondeterministic lgorithm working in polynomil spce is sufficient (since PSPACE=NPSPACE by Svitch s Theorem). We strt with noting the following two obvious propositions; this will give rise to min lgorithmic ide. Proposition 13. For cndidte pir (p 0 (m 0 ), q 0 (n 0 )) CAND we hve: p 0 (m 0 ) q 0 (n 0 ) iff there is subset B CAND such tht (p 0 (m 0 ), q 0 (n 0 )) B nd B SUREPOS is bisimultion. The following (infinite) lgorithm builds certin B CAND s the union of (nondeterministiclly chosen) sets B 0 CAND 0, B 1 CAND 1, B 2 CAND 2,..., while checking the bisimultion condition for their elements on the fly (recll the loclity cptured by Proposition 2). If its computtion does not fil, then it is infinite nd the respective set B CAND (which would result s the limit) stisfies tht B SUREPOS is bisimultion. We strt with putting m = 0, compute the set CAND 0 nd (nondeterministiclly) choose set B 0 CAND 0. Then we successively process m = 0, 1, 2..., where processing m mens the following: Compute CAND m+1 (recll Point 5 of Lemm 11) nd (nondeterministiclly) choose B m+1 CAND m+1. Verify tht (ech pir in) B m is (loclly) correct, using B m 1 (when m > 0) nd B m+1, nd the polynomil procedure deciding membership in SUREPOS (cf. Lemm 12).

11 (If B m is not correct, the computtion fils.) If we force the lgorithm to include the input pir (p 0 (m 0 ), q 0 (n 0 )) into B m0 then n infinite run is possible if nd only if p 0 (m 0 ) q 0 (n). We lso note tht it is sufficient for the lgorithm to keep only the current number m, nd the sets B m 1 (if m > 0), B m, B m+1 in memory. (By Point 5 of Lemm 11 this consists of t most 3 pol 3 (k) pirs, while the bit-size of the numbers is polynomil in k nd in the bit-size of m, i.e. in log m.) A finl crucil point is tht the lgorithm, getting p 0 (m 0 ), q 0 (n 0 ) in the input, will hlt (nswering p 0 (m 0 ) q 0 (n 0 )) fter it hs successfully processed the following levels. m = 0, 1, 2,..., z where z = m 0 + pol 4 (k) + 2 pol 5 (k) 2 3k log k (1) Here pol 4 is from Point 4 of Lemm 11, nd we put pol 5 (k) = 2 k 2 (1 + 2 pol 2 (k)), where pol 2 is from Point 3 of Lemm 11. With this hlting condition, the lgorithm obviously runs in polynomil spce (when given M nd pir (p 0 (m 0 ), q 0 (n 0 ))). Wht remins to show is the correctness of the hlting condition. 6 Correctness of (the hlting condition of) the lgorithm Recll Points 3 nd 4 of Lemm 11; the cndidte pirs re contined inside polynomilly mny liner belts with verticl thickness (1 + 2 pol 2 (k)), which re seprted for m pol 4 (k). Informlly speking, if the lgorithm (successfully) processes sufficiently mny (exponentilly mny) numbers m fter processing m 0, then the pigeonhole principle gurntees tht certin pumpble segment ppers inside ech belt (this is visulized in Figure 1). At tht time we re gurnteed tht the reltion R = {(p(m), q(n)) B m SUREPOS m m 0 } cn be extended with certin pirs (p (m ), q (n )), with m > m 0, so tht the resulting reltion is bisimultion. (These pirs (p (m ), q (n )), m > m 0, my differ from those which were ctully included in B m by the lgorithm.) We now mke this informl rgument more precise. Suppose tht our lgorithm successfully hlts for the input pir (p 0 (m 0 ), q 0 (n 0 )), nd consider the following subsequence of the sequence (1). m 0, m 0 + 3, m , m ,..., m 0 + 2pol 5 (k) 3 (2) where m 0 = mx{m 0, pol 4 (k)} nd = k! ; hence k k, nd so 3 2 3k log k. Remrk. We hve chosen so tht Points 6 nd 7 of Lemm 11 cn be pplied. The chosen period 3 hs the following useful property. We re gurnteed tht ρ 3 is multiple of for ech slope ρ = c1c 2 c 2c (c 1, c 2, c 1, c 2 {1, 2,..., k}) from Point 3 of 1 Lemm 11; by Point 7 of Lemm 11 we thus lso get for ech m pol 4 (k): (p(m), q(n)) SUREPOS i N : (p(m+i 3 ), q(n+iρ 3 )) SUREPOS. (3)

12 n m 1 m 2 (q k, q k )... (q 1, q 2) (q 1, q 1) Fig. 1. Two isomorphic belt cuts in coloring m (In the proof of Lemm 11, we hve ctully derived pol 4 stisfying pol 4 (k) k+pol(k). But ny polynomil pol 4 stisfying Point 4 could be replced with bigger one to stisfy lso pol 4 (k) k+pol(k) nywy.) For reltion R (Q N) (Q N) nd belt, identified with its slope ρ from Point 3 of Lemm 11, we define the R-cut of the belt ρ t level m s CUT ρ m(r) = { (p(m), q(n)) R ρm pol 2 (k) n ρm + pol 2 (k) }. Figure 1 illustrtes two cuts CUT ρ m 1 (R), CUT ρ m 2 (R) (the blck points representing elements of R, the white points being non-elements); the depicted cuts re the sme in the sense tht one rises by shifting the other. Our choice of the subsequence (2) gurntees repet of 2-thick cut : Proposition 14. For every R nd belt ρ there re m 1, m 2 in (2), where m 1 < m 2, m 2 = m 1 + c 3, such tht (p(m 1 ), q(n)) CUT ρ m 1 (R) (p(m 2 ), q(n + ρc 3 )) CUT ρ m 2 (R), (p(m 1 +1), q(n)) CUT ρ m 1+1 (R) (p(m 2 +1), q(n+ρc 3 )) CUT ρ m 2+1 (R). Proof. We first note tht our choice of lso gurntees tht ρc 3 is integer. Describing CUT ρ m (R) nd CUTρ m+1 (R) (for ny m) obviously mounts to determine

13 (blck-point) subset of set with (t most) 2 k 2 (1+2 pol 2 (k)) elements; this is how we defined pol 5 (k) in the hlting condition of our lgorithm (cf. (1)). There re 2 pol 5 (k) such subsets; thus the clim follows by the pigeonhole principle. Our im is to define some reltion R so tht R SUREPOS is bisimultion nd it coincides with B SUREPOS for the pirs (p(m), q(n)) with m m 0 ; the set B consists of the cndidte pirs included by (the successfully hlting computtion of) our lgorithm into B m, for m = 0, 1, 2,..., z s in (1). Let us now consider prticulr belt ρ. Let m 1, m 2, where m 1 < m 2 = m 1 +c 3, be the levels gurnteed by Proposition 14 for the reltion R = B SUREPOS. Inside the belt ρ, the suggested R will coincide with R for ll levels m m For ll levels m = m 2 +2, m 2 +3, m 2 +4,..., we define R inside the belt ρ by the following inductive definition: for ech m, n, where m > m 2 +1 nd n ρm pol 2 (k): (p(m), q(n)) R iff (p(m c 3 ), q(n ρc 3 )) R. We note tht this condition is, in fct, stisfied lso for m {m 2, m 2 +1}, due to our choice of m 1, m 2. We get the whole R fter hving defined it inside ll belts. Proposition 15. R SUREPOS is bisimultion. Proof. Suppose there is pir (p(m), q(n)) R SUREPOS which does not stisfy the bisimultion condition (which is determined by the restriction to NEIGHBOURS(m, n); recll Proposition 2). We tke such pir with the lest m. It is cler tht (p(m), q(n)) SUREPOS (recll Proposition 9); moreover, the restriction of R SUREPOS to NEIGHBOURS(m, n) cnnot be the sme s for B SUREPOS (where the lgorithm verified the bisimultion condition). Hence (m, n) lies in belt ρ, nd m m 2 +1 for the respective m 2 = m 1 +c 3. Then the pir (p(m c 3 ), q(n ρc 3 )) belongs to R nd stisfies the bisimultion condition; moreover, this pir enbles the sme trnsitions s the pir (p(m), q(n)). So there must be some (p (m ), q (n )) NEIGHBOURS(m, n) such tht (p (m ), q (n )) R SUREPOS nd (p (m c 3 ), q (n ρc 3 )) R SUREPOS. But this contrdicts the definition of R or the equivlence (3). Our hlting condition is thus correct, nd we hve proved: Theorem 16. There is polynomil spce lgorithm which, given one-counter utomton M nd pir p 0 (m 0 ), q 0 (n 0 ), decides if p 0 (m 0 ) q 0 (n 0 ). Remrk. As in [9], we could derive tht the bisimilrity (i.e., the mximl bisimultion) is belt-regulr. Our results here show tht nturl (finite) description of this (semiliner) reltion cn be written in exponentil spce. 7 -Regulrity We cn esily derive the next lemm, which tells us tht p(m) is not -regulr iff it llows to rech sttes with rbitrrily lrge finite distnces to INC. Lemm 17. Given p(m) for one counter utomton M, p(m) is not -regulr iff for ny d N there is q(n) such tht p(m) q(n) nd d dist(q(n)) < ω.

14 The next proposition gives more convenient chrcteriztion. Proposition 18. p(m) is not -regulr iff p(m) q(m+2k) INC for some q Q. (Recll tht k = Q for the set Q of control sttes of M.) Proof. Only if is obvious. On ny pth p(m) 1 q(m + 2k) 2 INC we hve to cross the level (m + k) when going up s well s when going down to INC (recll tht l < k for ny r(l) INC). The elementry cycles, which must necessrily pper when going up nd down, cn be suitbly pumped to show the condition in Lemm 17. Lemm 19. Deciding -regulrity of one-counter processes is in PTIME. Proof. We check the condition from Proposition 18. Given p(m), we cn compute ll q(m+ 2k) which hve finite distnces to INC by polynomil lgorithm (recll Point 1 of Lemm 11). When m = 0, the rechbility of suitble q(2k) (q(2k) INC) cn be checked strightforwrdly. So we cn compute ll p such tht p (0) is not -regulr. Thus p(m) is not -regulr iff it cn rech one of the computed q(m+2k) nd p (0) by positive trnsitions. The polynomility follows by the ides similr to those discussed in the proof of Lemm 10. Lemm 20. Deciding -regulrity (even) of one-counter nets is PTIME-hrd. Proof. We use logspce reduction from bisimilrity on finite trnsition systems which is PTIME-complete [1]. Given finite trnsition system (Q, A, { } A ) nd f, g Q, we construct one counter net which hs the following behviour: in s 0 (m), m > 0, b it hs trnsitions s 0 (m) s 0 (m + 1), s 0 (m) s 0 (m 1), s 0 (m) f(m), b b s 0 (m) g(m). In s 0 (0) we only hve s 0 (0) s 0 (1) nd s 0 (0) f(0). Any stte f(n) just mimicks f (not chnging the counter); similrly g(n) mimicks g. It is esy to verify tht s 0 (n) is regulr iff f g. Theorem 21. Deciding -regulrity of one-counter processes is PTIME-complete. Acknowledgements. We thnk the nonymous reviewers for useful comments nd suggestions. References 1. J. L. Blcázr, J. Gbrró, nd M. Snth. Deciding Bisimilrity is P-Complete. Forml Asp. Comput., 4(6A): , T. Brázdil, V. Brozek, K. Etessmi, A. Kučer, nd D. Wojtczk. One-Counter Mrkov Decision Processes. In Proc. of SODA, pges IEEE, S. Demri nd A. Sngnier. When Model-Checking Freeze LTL over Counter Mchines Becomes Decidble. In Proc. of FOSSACS, volume 6014 of LNCS, pges Springer, R. v. Glbbeek. The Liner Time Brnching Time Spectrum I; The Semntics of Concrete, Sequentil Processes. In J. Bergstr, A. Ponse, nd S. Smolk, editors, Hndbook of Process Algebr, chpter 1, pges Elsevier, 2001.

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