On Decidability of LTL+Past Model Checking for Process Rewrite Systems

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1 Electronic Notes in Theoreticl Computer Science 239 (2009) On Decidbility of LTL+Pst Model Checking for Process Rewrite Systems Mojmír Křetínský 1,4 Vojtěch Řehák2,5 Jn Strejček 3,6 Fculty of Informtics, Msryk University Botnická 68, Brno, Czech Republic Abstrct The pper [4] shows tht the model checking problem for (wekly extended) Process Rewrite Systems nd properties given by LTL formule with temporl opertors strict eventully nd strict lwys is decidble. The sme pper contins n open question whether the problem remins decidble even if we extend the set of properties by llowing lso pst counterprts of the mentioned opertors. The current pper gives positive nswer to this question. Keywords: rewrite systems, infinite-stte systems, model checking, decidbility, liner temporl logic 1 Introduction To specify (the clsses of) infinite-stte systems we employ term rewrite systems clled Process Rewrite Systems (PRS) [16]. PRS subsume vriety of the formlisms studied in the context of forml verifiction, e.g. Petri nets (PN), pushdown processes (PDA), nd process lgebrs like PA. Moreover, they re suitble to model current softwre systems with restricted forms of dynmic cretion nd synchroniztion of concurrent processes or recursive procedures or both. The relevnce of PRS (nd their subclsses) for modelling nd nlysing progrms is shown, for exmple, in [7]; for utomtic verifiction we refer to surveys [5,19]. 1 Emil: kretinsky@fi.muni.cz 2 Emil: rehk@fi.muni.cz 3 Emil: strejcek@fi.muni.cz 4 Supported by Ministry of Eduction of the Czech Republic, project No. MSM , nd by the Czech Science Foundtion, grnt No. 201/09/ Supported by the reserch centre Institute for Theoreticl Computer Science (ITI), project No. 1M0545, nd by the Czech Science Foundtion, grnt No. 201/08/P Supported by the Acdemy of Sciences of the Czech Republic, grnt No. 1ET , nd by the Czech Science Foundtion, grnt No. 201/08/P / 2009 Elsevier B.V. Open ccess under CC BY-NC-ND license. doi: /j.entcs

2 106 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) Another merit of PRS is tht the rechbility problem is decidble for PRS [16]. In [13], we hve presented wekly extended PRS (wprs), where finite-stte control unit with self-loops s the only loops is dded to the stndrd PRS formlism (ddition of generl finite-stte control unit mkes PRS lnguge equivlent to Turing mchines). This wek control unit enriches PRS by bilities to model bounded number of rbitrry communiction events nd globl vribles whose vlues re chnged only bounded number of times during ny computtion. We hve shown tht the rechbility problem remins decidble for wprs [12]. One of the minstrems in n utomtic verifiction of progrms is model checking. Here we focus on Liner Temporl Logic (LTL). Recll tht LTL model checking is decidble for both PDA (EXPTIME-complete [1]) nd PN (t lest s hrd s the rechbility problem for PN [6]). Conversely, LTL model checking is undecidble for ll the clsses subsuming PA [2,15]. So fr, there re few positive results for these clsses. Model checking of infinite runs is decidble for the PA clss nd the frgment simple PLTL,see[2], nd lso for the PRS clss nd frgment of LTL expressing exctly firness properties [3]. Recently, the model checking problem hs been shown decidble for (w)prs nd properties given by n LTL frgment LTL(F s, G s ), i.e. tht with opertors strict eventully nd strict lwys only, see [4]. Our contribution: As min result we extend proof technique used in [4] with pst modlities nd show tht the model checking problem stys decidble even for wprs nd LTL(F s, P s ), i.e. n LTL frgment with modlities strict eventully nd eventully in the strict pst (nd where strict lwys nd lwys in the strict pst cn be used s derived modlities). We note tht role of pst opertors in progrm verifiction is dvocted e.g. in [14,9]. Let us mention tht the expressive power of the frgment LTL(F s, P s ) semnticlly coincides with formule of First- Order Mondic Logic of Order contining t most 2 vribles nd no successor predicte (FO 2 [<]), see [8] for effective trnsltions. Thus we lso positively solve the model checking problem for the wprs clss nd FO 2 [<]. 2 Preliminries 2.1 Wekly Extended PRS (wprs) Let Const = {X,...} be set of process constnts. A set T of process terms t is defined by the bstrct syntx t ::= ε X t.t t t, whereε is the empty term, X Const, nd. nd mensequentil nd prllel compositions, respectively. We lwys work with equivlence clsses of terms modulo commuttivity nd ssocitivity of, ssocitivity of., nd neutrlity of ε, i.e. ε.t = t.ε = t ε = t. Let M = {o,p,q,...} be set of control sttes, be prtil ordering on this set, nd Act = {,b,c,...} be set of ctions. AnwPRS (wekly extended process rewrite system) Δ is tuple (R, p 0,t 0 ), where R is finite set of rewrite rules of the form (p, t 1 ) (q, t 2 ), where t 1,t 2 T, t 1 ε, Act,ndp, q M stisfy p q, the pir (p 0,t 0 ) M T forms the distinguished initil stte.

3 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) By Act(Δ), Const(Δ), nd M(Δ) we denote the respective sets of ctions, process constnts, nd control sttes occurring in the rewrite rules or the initil stte of Δ. AwPRSΔ=(R, p 0,t 0 ) induces lbelled trnsition system, whose sttes re pirs (p, t) such tht p M(Δ) nd t is process term over Const(Δ). The trnsition reltion is the lest reltion stisfying the following inference rules: ((p, t 1 ) (q,t 2 )) R (p, t 1 ) (q,t 2 ) (p, t 1 ) (q, t 2 ) (p, t 1 ) (q, t 2 ) (p, t 1 t 1 ) (q, t 2 t 1 ) (p, t 1.t 1 ) (q, t 2.t 1 ) To shorten our nottion we write pt in lieu of (p, t). A stte pt is clled terminl if there is no stte p t nd no ction such tht pt p t. Here, we lwys consider only such systems where the initil stte is not terminl. A (finite or infinite) sequence ( σ = p 0 t 0 0 p1 t n n+1 ) pn+1 t n+1... is clled run of Δ over the word u = n ( n+1...) if it strts in the initil stte nd, provided it is finite, ends in terminl stte. Further, L(Δ) denotes the set of words u such tht there is run of Δ over u. If M(Δ) is singleton, then wprs Δ is clled process rewrite system (PRS) [16]. PRS, wprs, nd their respective subclsses re discussed in more detil in [18]. 2.2 Liner Temporl Logic (LTL) nd the Studied Problems The syntx of Liner Temporl Logic (LTL) [17] is defined s follows ϕ ::= tt ϕ ϕ ϕ Xϕ ϕ U ϕ Yϕ ϕ S ϕ, where X nd U re future modl opertors next nd until, while Y nd S re their pst counterprts previously nd since, nd rnges over Act. The logic is interpreted over infinite nd nonempty finite pointed words of ctions. Given word u = Act Act ω, u denotes the length of the word (we set u = if u is infinite). A pointed word is pir (u, i) of nonempty word u nd position 0 i< u in this word. The semntics of LTL formule is defined inductively s follows: (u, i) = tt (u, i) = iff u = nd i = (u, i) = ϕ iff (u, i) = ϕ (u, i) = ϕ 1 ϕ 2 iff (u, i) = ϕ 1 nd (u, i) = ϕ 2 (u, i) = Xϕ iff i +1< u nd (u, i +1) = ϕ (u, i) = ϕ 1 U ϕ 2 iff k. ( i k< u (u, k) = ϕ 2 j. (i j<k (u, j) = ϕ 1 ) ) (u, i) = Yϕ iff 0 <i nd (u, i 1) = ϕ (u, i) = ϕ 1 S ϕ 2 iff k. ( 0 k i (u, k) = ϕ 2 j. (k <j i (u, j) = ϕ 1 ) )

4 108 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) We sy tht (u, i) stisfies ϕ whenever (u, i) = ϕ. Further, nonempty word u stisfies ϕ, written u = ϕ, whenever (u, 0) = ϕ. Given set L of words, we write L = ϕ if u = ϕ holds for ll u L. Finlly, we sy tht run σ of wprs Δ over wordu stisfies ϕ, written σ = ϕ, whenever u = ϕ. Formule ϕ, ψ re (initilly) equivlent, written ϕ i ψ,iff,forllwordsu, it holds tht u = ϕ u = ψ. Formuleϕ, ψ re globlly equivlent, written ϕ ψ, iff, for ll pointed words (u, i), it holds tht (u, i) = ϕ (u, i) = ψ. Clerly, if two formule re globlly equivlent then they re lso initilly equivlent. The following tble defines some derived future opertors nd their pst counterprts. future modlity mening pst modlity mening Fϕ eventully tt U ϕ Pϕ eventully in the pst tt S ϕ Gϕ lwys F ϕ Hϕ lwys in the pst P ϕ F s ϕ strict eventully XFϕ P s ϕ eventully in the strict pst YPϕ G s ϕ strict lwys F s ϕ H s ϕ lwys in the strict pst P s ϕ F ϕ infinitely often GFϕ Iϕ initilly HPϕ Given set {O 1,...,O n } of modlities, then LTL(O 1,...,O n ) denotes n LTL frgment contining ll formule with modlities O 1,...,O n only. Such frgment is clled bsic if it contins future opertors only or with ech future opertor it contins its pst counterprt. For exmple, the frgment LTL(F, S) is not bsic. Figure 1 shows n expressiveness hierrchy of ll studied bsic LTL frgments. Indeed, every bsic LTL frgment using stndrd 7 modlities is equivlent to one of the frgments in the hierrchy, where equivlence between frgments mens tht every formul of one frgment cn be effectively trnslted into n initilly equivlent formul of the other frgment nd vice vers. We lso mind the result of [9] stting tht ech LTL formul cn be converted to the one which employs future opertors only, i.e. LTL(U, X) i LTL(U, S, X, Y). However note tht LTL(F s, P s, G s, H s ) LTL(F s, P s ) is strictly more expressive thn LTL(F s, G s )scn be exemplified by formul F s (b H s ) i X( U b). We refer to [20] for greter detil. This pper dels with the following two verifiction problems. Let F be n LTL frgment. The model checking problem for F nd wprs is to decide, for ny given formul ϕ Fnd ny given wprs system Δ, whether L(Δ) = ϕ holds. Further, given ny formul ϕ F, ny wprs system Δ, nd ny nonterminl stte pt of Δ, the pointed model checking problem for F nd wprs is to decide whether L(pt, Δ) = ϕ; herel(pt, Δ) denotes the set of ll pointed words (u, i) such tht Δ hs (finite or infinite) run p 0 t 0 0 p1 t 1 i p i t i i... stisfying u = nd pt = p i t i. 7 By stndrd modlities we men the ones defined here nd lso other commonly used modlities like strict until, relese, wek until, etc. However, it is well possible tht one cn define new modlity such tht there is bsic frgment not equivlent to ny of the frgments in the hierrchy.

5 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) LTL(U, X) i FO 3 LTL(U, F s, S, P s ) LTL(F, X, P, Y) FO 2 LTL( U, F s ) LTL(F, X) LTL(F s, P s ) FO 2 [<] LTL(U) LTL( F, X) LTL(F s, G s ) LTL(F, P) LTL(F, G) LTL(X) LTL( F) LTL() Fig. 1. The hierrchy of bsic LTL frgments with respect to the initil equivlence. The dshed line shows the decidbility boundry of the model checking problem for wprs. 3 Min Result In [4], we hve shown tht the model checking problem is decidble for LTL(F s, G s ). Before we prove tht the problem remins decidble even for more expressive frgment LTL(F s, P s ), we recll the bsic structure of the proof for LTL(F s, G s ). First, the proof shows tht every LTL(F s, G s ) formul cn be effectively trnslted into n equivlent disjunction of so-clled α-formule, which re defined below. Note tht LTL() denotes the frgment of formule without ny modlity, i.e. boolen combintions of ctions. In wht follows, we use ϕ 1 U + ϕ 2 to bbrevite ϕ 1 X(ϕ 1 U ϕ 2 ). Let δ = θ 1 O 1 θ 2 O 2...θ n O n θ n+1,wheren>0, ech θ i LTL(), O n is G s, nd, for ech i<n, O i is either U or U + or X. Further, let B LTL() be finite set. An α-formul is defined s α(δ, B) = ( θ 1 O 1 (θ 2 O 2...(θ n O n θ n+1 )...) ) ψ B G s F s ψ. Hence, word u stisfies α(δ, B) iffu cn be written s conctention v 1.v 2...v n+1 of words, where ech word v i consists only of ctions stisfying θ i nd

6 110 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) v i 0ifi = n +1orO i is U, v i > 0ifO i is U +, v i =1ifO i is X or G s, nd v n+1 stisfies G s F s ψ for every ψ B. Second, decidbility of the model checking problem for LTL(F s, G s ) is then direct consequence of the following theorem. Theorem 3.1 ([4]) The problem whether ny given wprs systems hs run stisfying ny given α-formul is decidble. To prove decidbility for LTL(F s, P s ), we show tht every LTL(F s, P s )formulcn be effectively trnslted into disjunction of Pα-formule. Intuitively, Pα-formul is conjunction of n α-formul nd pst version of the α-formul. A forml definition of Pα-formul mkes use of ϕ 1 S + ϕ 2 to bbrevite ϕ 1 Y(ϕ 1 S ϕ 2 ). Definition 3.2 Let η = ι 1 P 1 ι 2 P 2...ι m P m ι m+1,wherem>0, ech ι j LTL(), nd, for ech j<m, P j is either S or S + or Y. Further, let α(δ, B) ben α-formul. Then Pα-formul is defined s Pα(η, δ, B) = ( ι 1 P 1 (ι 2 P 2...(ι m P m ι m+1 )...) ) α(δ, B). Note tht the definition of Pα-formul does not contin ny pst counterprt of ψ B G s F s ψ s every history is finite the semntics of LTL is given in terms of words with fixed beginning. Therefore, pointed word (u, k) = Pα(η, δ, B) if nd only if (u, k)stisfiesα(δ, B) nd 0... k 1 k cn be written s conctention v m+1.v m...v 2.v 1,whereech word v i consists only of ctions stisfying ι i nd v i 0ifi = m +1orP i is S, v i > 0ifP i is S +, v i =1ifP i is Y or H s. The proof of the following lemm is intuitively cler but it is quite technicl exercise, see [18] for some hints. Lemm 3.3 Let ϕ be Pα-formul nd p LTL(). FormuleXϕ, Yϕ, p U ϕ, p S ϕ, F s ϕ, P s (ϕ), s well s, conjunction of Pα-formule cn be effectively converted into globlly equivlent disjunction of Pα-formule. Theorem 3.4 Every LTL(F s, P s ) formul ϕ cn be trnslted into globlly equivlent disjunction of Pα-formule. Proof. As F s, G s nd P s, H s re dul modlities, we cn ssume tht every LTL(F s, G s, P s, H s ) formul contins negtions only in front of ctions. Given n LTL(F s, G s, P s, H s )formulϕ, we construct finite set A ϕ of α-formule such tht ϕ is equivlent to the disjunction of formule in A ϕ. Although our proof looks like by induction on the structure of ϕ, it is in fct by induction on the length of ϕ. Thus, if ϕ LTL(), then we ssume tht for every LTL(F s, G s, P s, H s )formulϕ shorter

7 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) thn ϕ we cn construct the corresponding set A ϕ. In this proof, p represents formul of LTL(). The structure of ϕ fits into one of the following cses. p Cse p: In this cse, ϕ is equivlent to p G s tt. Hence A ϕ = {Pα(tt H s tt,p G s tt, )}. Cse ϕ 1 ϕ 2 : Due to induction hypothesis, we cn ssume tht we hve sets A ϕ1 nd A ϕ2. Clerly, A ϕ = A ϕ1 A ϕ2. Cse ϕ 1 ϕ 2 : Due to Lemm 3.3, A ϕ cn be constructed from the sets A ϕ1 nd A ϕ2. F s Cse F s ϕ 1 : Due to Lemm 3.3, theseta ϕ cn be constructed from the set A ϕ1. P s Cse P s ϕ 1 : Due to Lemm 3.3, theseta ϕ cn be constructed from the set A ϕ1. G s Cse G s ϕ 1 is divided into the following subcses ccording to the structure of ϕ 1 : p Cse G s p: As G s p is equivlent to tt G s p,weseta ϕ = {Pα(tt H s tt, tt G s p, )}. Cse G s (ϕ 2 ϕ 3 ): As G s (ϕ 2 ϕ 3 ) (G s ϕ 2 ) (G s ϕ 3 ), the set A ϕ cn be constructed from A Gsϕ2 nd A Gsϕ3 using Note tht A Gsϕ2 nd A Gsϕ3 cn be constructed becuse G s ϕ 2 nd G s ϕ 3 re shorter thn G s (ϕ 2 ϕ 3 ). F s Cse G s F s ϕ 2 : This cse is gin divided into the following subcses. p Cse G s F s p: As p LTL(), we directly set A ϕ = {Pα(tt H s tt, tt G s tt, {p})}. Cse G s F s (ϕ 3 ϕ 4 ): As G s F s (ϕ 3 ϕ 4 ) (G s F s ϕ 3 ) (G s F s ϕ 4 ), we set A ϕ = A GsFsϕ 3 A GsFsϕ 4. Cse G s F s (ϕ 3 ϕ 4 ): This cse is lso divided into subcses depending on the formule ϕ 3 nd ϕ 4. p Cse G s F s (p 3 p 4 ): As p 3 p 4 LTL(), this subcse hs lredy been covered by Cse G s F s p. Cse G s F s (ϕ 3 (ϕ 5 ϕ 6 )): As G s F s (ϕ 3 (ϕ 5 ϕ 6 )) G s F s (ϕ 3 ϕ 5 ) G s F s (ϕ 3 ϕ 6 ), we set A ϕ = A GsFs(ϕ 3 ϕ 5 ) A GsFs(ϕ 3 ϕ 6 ). F s Cse G s F s (ϕ 3 F s ϕ 5 ): As G s F s (ϕ 3 F s ϕ 5 ) (G s F s ϕ 3 ) (G s F s ϕ 5 ), the set A ϕ cn be constructed from A GsFsϕ 3 nd A GsFsϕ 5 using P s Cse G s F s (ϕ 3 P s ϕ 5 ): As G s F s (ϕ 3 P s ϕ 5 ) (G s F s ϕ 3 ) (G s F s P s ϕ 5 ), the set A ϕ cn be constructed from A GsFsϕ 3 nd A GsFsP sϕ 5 using G s Cse G s F s (ϕ 3 G s ϕ 5 ): As G s F s (ϕ 3 G s ϕ 5 ) (G s F s ϕ 3 ) (G s F s G s ϕ 5 ), the set A ϕ cn be constructed from A GsFsϕ 3 nd A GsFsG sϕ 5 using H s Cse G s F s (ϕ 3 H s ϕ 5 ): As G s F s (ϕ 3 H s ϕ 5 ) (G s F s ϕ 3 ) (G s F s H s ϕ 5 ), the set A ϕ cn be constructed from A GsFsϕ 3 nd A GsFsH sϕ 5 using F s Cse G s F s F s ϕ 3 : As G s F s F s ϕ 3 G s F s ϕ 3,wesetA ϕ = A GsFsϕ 3. P s Cse G s F s P s ϕ 3 : Apointedword(u, i) stisfiesg s F s P s ϕ 3 iff i = u 1oru is n infinite word stisfying Fϕ 3.NotethtG s tt is stisfied only by finite words t their lst position. Further, word u stisfies (F s tt) (G s F s tt) iffu is infinite. Thus, G s F s P s ϕ 3 (G s tt) ϕ where ϕ =(F s tt) (G s F s tt) (ϕ 3 P s ϕ 3 F s ϕ 3 ). Hence, A ϕ = A Gs tt A ϕ where A ϕ is constructed from A Fstt,

8 112 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) A GsFstt, nda ϕ3 A Psϕ 3 A Fsϕ 3 using G s Cse G s F s G s ϕ 3 : Apointedword(u, i) stisfiesg s F s G s ϕ 3 iff i = u 1 or u is n infinite word stisfying F s G s ϕ 3. Thus, G s F s G s ϕ 3 (G s tt) ϕ where ϕ =(F s tt) (G s F s tt) (F s G s ϕ 3 ). Hence, A ϕ = A Gs tt A ϕ where A ϕ is constructed from A Fstt, A GsFstt, nda FsG sϕ 3 using H s Cse G s F s H s ϕ 3 : Apointedword(u, i) stisfiesg s F s H s ϕ 3 iff i = u 1or u is n infinite word stisfying Gϕ 3. Thus, G s F s H s ϕ 3 (G s tt) ϕ where ϕ =(F s tt) (G s F s tt) (ϕ 3 H s ϕ 3 G s ϕ 3 ). Hence, A ϕ = A Gs tt A ϕ where A ϕ is constructed from A Fstt, A GsFstt, A ϕ3, A Hsϕ3,ndA Gsϕ3 using P s Cse G s P s ϕ 2 : Apointedword(u, i) stisfiesg s P s ϕ 2 iff i = u 1or(u, i) stisfies Pϕ 2. Hence, A ϕ = A Gs tt A ϕ2 A Psϕ 2. Cse G s (ϕ 2 ϕ 3 ): According to the structure of ϕ 2 nd ϕ 3, there re the following subcses. p Cse G s (p 2 p 3 ): As p 2 p 3 LTL(), this subcse hs lredy been covered by Cse G s p. Cse G s (ϕ 2 (ϕ 4 ϕ 5 )): As G s (ϕ 2 (ϕ 4 ϕ 5 )) G s (ϕ 2 ϕ 4 ) G s (ϕ 2 ϕ 5 ), the set A ϕ cn be constructed from A Gs(ϕ2 ϕ 4 ) nd A Gs(ϕ2 ϕ 5 ) using F s Cse G s (ϕ 2 F s ϕ 4 ): It holds tht G s (ϕ 2 F s ϕ 4 ) (G s ϕ 2 ) F s (F s ϕ 4 G s ϕ 2 ) G s F s ϕ 4. Therefore, the set A ϕ cn be constructed s A Gsϕ2 A Fs(F sϕ 4 G sϕ 2 ) A GsFsϕ 4,whereA Fs(F sϕ 4 G sϕ 2 ) is obtined from A Fsϕ 4 nd A Gsϕ2 using H s Cse G s (ϕ 2 H s ϕ 4 ): As G s (ϕ 2 H s ϕ 4 ) (G s ϕ 2 )F s (H s ϕ 4 G s ϕ 2 )G s H s ϕ 4. Hence, A ϕ = A Gsϕ2 A Fs(H sϕ 4 G sϕ 2 ) A (GsHsϕ 4 ) where A Fs(H sϕ 4 G sϕ 2 ) cn be obtined from A Hsϕ4 nd A Gsϕ2 using G s, P s There re only the following six subcses (the others fit to some of the previous cses). (i) Cse G s ( ϕ G G sϕ ): It holds tht G s ( ϕ G G sϕ ) (G s tt) ϕ G (XG sϕ ). Therefore, the set A ϕ cn be constructed s A Gs tt ϕ G A XG sϕ where ech A XG sϕ is obtined from A G sϕ using (ii) Cse G s (p 2 ϕ G G sϕ ): As G s (p 2 ϕ G G sϕ ) (G s p 2 ) ϕ G (X(p 2 U (G s ϕ ))), the set A ϕ cn be constructed s A Gsp2 ϕ G A X(p 2 U (G sϕ )) where ech A X(p2 U (G sϕ )) is obtined from A Gsϕ using (iii) Cse G s ( ϕ P P sϕ ): It holds tht G s ( ϕ P P sϕ ) (G s tt) ϕ P (XP sϕ ). Therefore, the set A ϕ cn be constructed s A Gs tt ϕ P A XP sϕ where ech A XP sϕ is obtined from A P sϕ using (iv) Cse G s (p 2 ϕ P P sϕ ): As G s (p 2 ϕ P P sϕ ) (G s p 2 ) ϕ P (X(p 2 U (P s ϕ ))), the set A ϕ cn be constructed s A Gsp2 ϕ P A X(p 2 U (P sϕ )) where ech A X(p2 U (P sϕ )) is obtined from A Psϕ using (v) Cse G s ( ϕ G G sϕ ϕ P P sϕ ): As G s ( ϕ G G sϕ ϕ P P sϕ ) (G s tt) ϕ G (XG sϕ ) ϕ P (XP sϕ ), the set A ϕ cn be constructed s A Gs tt ϕ G A XG sϕ ϕ P A XP sϕ where ech A XG sϕ is obtined from A Gsϕ nd ech A XP sϕ is obtined from A P sϕ using

9 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) (vi) Cse G s (p 2 ϕ G G sϕ ϕ P P sϕ ): As G s (p 2 ϕ G G sϕ ϕ P P sϕ ) (G s p 2 ) ϕ G (X(p 2 U (G s ϕ ))) ϕ P (X(p 2 U (P s ϕ ))), the set A ϕ cn be constructed s A Gsp2 ϕ G A X(p 2 U (G sϕ )) ϕ P A X(p 2 U (P sϕ )) where ech A X(p2 U (G sϕ )) is obtined from A Gsϕ nd ech A X(p2 U (P sϕ )) is obtined from A Psϕ using G s Cse G s G s ϕ 2 : As G s (G s ϕ 2 ) (G s tt) (XG s ϕ 2 ), the set A ϕ cn be constructed s A Gs tt A XGsϕ2 where A XGsϕ2 is obtined from A Gsϕ2 using H s Cse G s H s ϕ 2 : Apointedword(u, i) stisfiesg s (H s ϕ 2 )iffi = u 1or (u, u 1) stisfies H s ϕ 2 or u is infinite nd ll its positions stisfy ϕ 2. Hence, A ϕ = A Gs tt A Fs((Gs tt) (H sϕ 2 )) A (Hsϕ2 ) ϕ 2 (G sϕ 2 ) where A Fs((Gs tt) (H sϕ 2 )) nd A (Hsϕ2 ) ϕ 2 (G sϕ 2 ) re obtined from A Gs tt, A Hsϕ2, A ϕ2,nda Gsϕ2 using H s Cse H s ϕ 1 : This cse is divided into the following subcses ccording to the structure of ϕ 1. p Cse H s p: As H s p is globlly equivlent to tt H s p,weseta ϕ = {Pα(tt H s p, tt G s tt, )}. Cse H s (ϕ 2 ϕ 3 ): As H s (ϕ 2 ϕ 3 ) (H s ϕ 2 ) (H s ϕ 3 ), the set A ϕ cn be constructed from A Hsϕ2 nd A Hsϕ3 using F s Cse H s F s ϕ 2 : Apointedword(u, i) stisfiesh s F s ϕ 2 iff i =0or(u, i) stisfies Fϕ 2. Note tht H s tt is stisfied by (u, i) onlyifi = 0. Therefore, A ϕ = A Hs tt A ϕ2 A Fsϕ 2. P s Cse H s P s ϕ 2 : Apointedword(u, i) stisfiesh s P s ϕ 2 iff i = 0. Therefore, A ϕ = A Hs tt. Cse H s (ϕ 2 ϕ 3 ): According to the structure of ϕ 2 nd ϕ 3, there re the following subcses. p Cse H s (p 2 p 3 ): As p 2 p 3 LTL(), this subcse hs lredy been covered by Cse H s p. Cse H s (ϕ 2 (ϕ 4 ϕ 5 )): As H s (ϕ 2 (ϕ 4 ϕ 5 )) H s (ϕ 2 ϕ 4 ) H s (ϕ 2 ϕ 5 ), the set A ϕ cn be constructed from A Hs(ϕ2 ϕ 4 ) nd A Hs(ϕ2 ϕ 5 ) using P s Cse H s (ϕ 2 P s ϕ 4 ): It holds tht H s (ϕ 2 P s ϕ 4 ) (H s ϕ 2 )P s (P s ϕ 4 H s ϕ 2 ). Therefore, the set A ϕ cn be constructed s A Hsϕ2 A Ps(P sϕ 4 H sϕ 2 ),where A Ps(P sϕ 4 H sϕ 2 ) is obtined from A Psϕ 4 nd A Hsϕ2 using G s Cse H s (ϕ 2 G s ϕ 4 ): As H s (ϕ 2 G s ϕ 4 ) (H s ϕ 2 ) P s (G s ϕ 4 H s ϕ 2 ), A ϕ is constructed s A Hsϕ2 A Ps(G sϕ 4 H sϕ 2 ) where A Ps(G sϕ 4 H sϕ 2 ) is obtined from A Gsϕ4 nd A Hsϕ2 ) using F s, H s There re only the following six subcses (the others fit to some of the previous cses). (i) Cse H s ( ϕ F F sϕ ): It holds tht H s ( ϕ F F sϕ ) (H s tt) ϕ F (YF sϕ ). Therefore, the set A ϕ cn be constructed s A Hs tt ϕ F A YF sϕ where ech A YF sϕ is obtined from A F sϕ using (ii) Cse H s (p 2 ϕ F F sϕ ): As H s (p 2 ϕ F F sϕ ) (H s p 2 ) ϕ F (Y(p 2 S (F s ϕ ))), the set A ϕ cn be constructed s A Hsp2

10 114 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) ϕ F A Y(p 2 S (F sϕ )) where ech A Y(p2 S (F sϕ )) is obtined from A Fsϕ using (iii) Cse H s ( ϕ H H sϕ ): It holds tht H s ( ϕ H H sϕ ) (H s tt) ϕ H (YH sϕ ). Therefore, the set A ϕ cn be constructed s A Hs tt ϕ H A YH sϕ where ech A YH sϕ is obtined from A H sϕ using (iv) Cse H s (p 2 ϕ H H sϕ ): As H s (p 2 ϕ H H sϕ ) (H s p 2 ) ϕ H (Y(p 2 S (H s ϕ ))), the set A ϕ cn be constructed s A Hsp2 ϕ H A Y(p 2 S (H sϕ )) where ech A Y(p2 S (H sϕ )) is obtined from A Hsϕ using (v) Cse H s ( ϕ F F sϕ ϕ H H sϕ ): As H s ( ϕ F F sϕ ϕ H H sϕ ) (H s tt) ϕ F (YF sϕ ) ϕ H (YH sϕ ), the set A ϕ cn be constructed s A Hs tt ϕ F A YF sϕ ϕ H A YH sϕ where ech A YF sϕ is obtined from A Fsϕ nd ech A YH sϕ is obtined from A H sϕ using (vi) Cse H s (p 2 ϕ F F sϕ ϕ H H sϕ ): As H s (p 2 ϕ F F sϕ ϕ H H sϕ ) (H s p 2 ) ϕ F (Y(p 2 S (F s ϕ ))) ϕ H (Y(p 2 S (H s ϕ ))), the set A ϕ cn be constructed s A Hsp2 ϕ F A Y(p 2 S (F sϕ )) ϕ H A Y(p 2 S (H sϕ )) where ech A Y(p2 S (F sϕ )) is obtined from A Fsϕ nd ech A Y(p2 S (H sϕ )) is obtined from A Hsϕ using G s Cse H s G s ϕ 2 : Apointedword(u, i) stisfiesh s (G s ϕ 2 )iffi =0or(u, 0) stisfies G s ϕ 2. Hence, A ϕ = A Hs tt A Ps((Hs tt) (G sϕ 2 )) where A Ps((Hs tt) (G sϕ 2 )) is obtined from A Hs tt nd A Gsϕ2 using H s Cse H s H s ϕ 2 : As H s (H s ϕ 2 ) (H s tt) (YH s ϕ 2 ), the set A ϕ cn be constructed s A Hs tt A YHsϕ2 where A YHsϕ2 is obtined from A Hsϕ2 using Remrk 3.5 In other words, we hve just shown tht LTL(F s, P s )issemntic subset (with respect to globl equivlence) of every formlism tht is (i) ble to express p, G s p, H s p,ndg s F s p,wherep LTL(); nd (ii) is closed under disjunction, conjunction, nd pplictions of X, Y, p U,ndp S,wherep LTL(). Now, using Theorem 3.1, we cn esily solve the problem dul to the model checking problem, i.e. given ny wprs system nd ny Pα-formul, to decide whether the system hs run stisfying the formul. Theorem 3.6 The problem whether ny given wprs system hs run stisfying given Pα-formul is decidble. Proof. A run over nonempty (finite or infinite) word u = stisfies formulϕ iff (u, 0) = ϕ. Moreover, (u, 0) = Pα(η, δ, B) iff ( 0, 0) = η nd (u, 0) = α(δ, B). Let η = ι 1 P 1 ι 2 P 2...ι m P m ι m+1. It follows from the semntics of LTL tht ( 0, 0) = η if nd only if ( 0, 0) = ι m nd P i = S for ll i<m. Therefore, the problem is to check whether P i = S for ll i<mnd whether the given wprs system hs run stisfying ι m α(δ, B). As ι m α(δ, B) cn be esily trnslted into disjunction of α-formule, Theorem 3.1 finishes the proof.

11 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) As LTL(F s, P s ) is closed under negtion, Theorem 3.4 nd Theorem 3.6 give us the following. Corollry 3.7 The model checking problem for wprs nd LTL(F s, P s ) is decidble. Moreover, we cn show tht the pointed model checking problem is decidble for wprs nd LTL(F s, P s ) s well. Agin, we solve the dul problem. Theorem 3.8 Let Δ be wprs nd pt be rechble nonterminl stte of Δ. The problem whether L(pt, Δ) contins pointed word (u, i) stisfying ny given Pα-formul is decidble. Proof. Let Δ = (M,,R,p 0,t 0 )bewprsndpt be rechble nonterminl stte of Δ. We construct wprs Δ =(M,,R,p 0,t 0.X) wherex Const(Δ) is fresh process constnt, f Act(Δ) is fresh ction, R = R {(p(t.x) px ), (px f py ), (py p t ) pt p t }, nd X,Y Const(Δ) re fresh process constnts for ech Act(Δ). It is esy to see tht (u, i) isinl(pt, Δ) iff u = i 1 i.f. i. i+1... is in L(Δ ). Hence, for ny given Pα-formul ϕ =Pα(η, δ, B) weconstructpα-formul ϕ =Pα(η, tt Xf Xδ, B). We get tht L(pt, Δ) = Pα(η, δ, B) L(Δ ) = F(Pα(η, tt Xf Xδ, B)) nd due to Lemm 3.3 nd Theorem 3.6 the proof is done. As LTL(F s, P s ) is closed under negtion nd Theorem 3.4 works with globl equivlence, Theorem 3.8 give us the following. Corollry 3.9 The pointed model checking problem is decidble for wprs nd LTL(F s, P s ). 4 Conclusion We hve exmined the model checking problem for bsic LTL frgments with both future nd pst modlities nd the PRS clss, i.e. the clss of infinite stte system generted by Process Rewrite Systems (PRS), possibly enriched with wek finite control unit (wekly extended PRS wprs). We hve proved tht the problem is decidble for wprs nd LTL(F s, P s ), i.e. the frgment with modlities strict eventully, eventully in the strict pst, nd derived modlities strict lwys nd lwys in the strict pst. 8 However, both these problems re t lest s hrd s the rechbility problem for PN [6] (EXPSPACE-hrd without ny elementry upper bound known). Note tht the expressive power of the frgment LTL(F s, P s ) semnticlly coincides with formule of First-Order Mondic Logic of Order contining t most 2 vribles 8 In fct, we hve shown tht the problem is decidble even for more expressive frgment contining negtions of disjunctions of so-clled Pα-formule (see Definition 3.2).

12 116 M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) nd no successor predicte (FO 2 [<]), nd tht First-Order Mondic Logic of Order contining t most 2 vribles (FO 2 ) coincides with n LTL(F, X, P, Y) frgment[8]. Further, let us recll our undecidbility results for model checking of PA systems ( subclss of PRS) nd frgments LTL( F, X)ndLTL(U ), respectively (the former with modlities infinitely often nd next only, the ltter with until s the only modlity), see [4]. Thus, we hve locted the borderline between decidbility nd undecidbility of the problem for wprs nd the LTL frgments, s well s for wprs nd First-Order Mondic Logic of Order: it is decidble for FO 2 [<] nd undecidble for FO 2. For the ske of completeness, we note tht the First-Order Mondic Logic of Order contining t most 3 vribles (FO 3 ) coincides with the set of ll LTL formule s well s with the full First-Order Mondic Logic of Order [11,10]. Finlly, we note tht the decidbility results re new for the PRS clss too nd they re illustrted by the decidbility border in Figure 1. References [1] Boujjni, A., J. Esprz nd O. Mler, Rechbility Anlysis of Pushdown Automt: Appliction to Model-Checking, in: Proc.ofCONCUR 97, LNCS1243, 1997, pp [2] Boujjni, A. nd P. Hbermehl, Constrined Properties, Semiliner Systems, nd Petri Nets, in: Proc. of CONCUR 96, LNCS1119 (1996), pp [3] Bozzelli, L., Model checking for process rewrite systems nd clss of ction-bsed regulr properties, in: Proc. of VMCAI 05, LNCS3385 (2005), pp [4] Bozzelli, L., M. Křetínský, V. Řehák nd J. Strejček, On decidbility of LTL model checking for process rewrite systems, in: FSTTCS 2006, LNCS4337 (2006), pp [5] Burkrt, O., D. Cucl, F. Moller nd B. Steffen, Verifiction on infinite structures, in: Hndbook of Process Algebr (2001), pp [6]Esprz,J.,On the Decidbility of Model Checking for Severl mu-clculi nd Petri Nets, in:caap, LNCS 787 (1994), pp [7] Esprz, J., Grmmrs s Processes, in: Forml nd Nturl Computing, LNCS 2300 (2002), pp [8] Etessmi, K., M. Y. Vrdi nd T. Wilke, First-order logic with two vribles nd unry temporl logic, Informtion nd Computtion 179 (2002), pp [9] Gbby, D., The Declrtive Pst nd Impertive Future: Executble Temporl Logic for Interctive Systems, in: Temporl Logic in Specifiction, LNCS398, 1987, pp [10] Gbby, D., A. Pnueli, S. Shelh nd J. Stvi, On the Temporl Anlysis of Firness, in:conference Record of the 7th Annul ACM Symposium on Principles of Progrmming Lnguges (POPL 80) (1980), pp [11] Kmp, J. A. W., Tense Logic nd the Theory of Liner Order, Ph.D. thesis, University of Cliforni, Los Angeles (1968). [12] Křetínský, M., V. Řehák nd J. Strejček, Extended Process Rewrite Systems: Expressiveness nd Rechbility, in: Proceedings of CONCUR 04, LNCS3170 (2004), pp [13] Křetínský, M., V. Řehák nd J. Strejček, On Extensions of Process Rewrite Systems: Rewrite Systems with Wek Finite-Stte Unit, in: Proceedings of INFINITY 03, Electr. Notes Theor. Comput. Sci. 98 (2004), pp [14] Lichtenstein, O., A. Pnueli nd L. D. Zuck, The Glory of The Pst, in:logic of Progrms, LNCS193, 1985, pp

13 [15] Myr, R., Decidbility nd Complexity of Model Checking Problems for Infinite-Stte Systems, Ph.D. thesis, Technische Universität München (1998). [16] Myr, R., Process rewrite systems, Informtion nd Computtion 156 (2000), pp [17] Pnueli, A., The Temporl Logic of Progrms, in:proc. 18th IEEE Symposium on the Foundtions of Computer Science, 1977, pp [18] M. Křetínský et l. / Electronic Notes in Theoreticl Computer Science 239 (2009) Řehák, V., On Extensions of Process Rewrite Systems, Ph.D. thesis, Fculty of Informtics, Msryk University, Brno (2007). [19] Srb, J., Rodmp of Infinite Results, Current Trends In Theoreticl Computer Science, The Chllenge of the New Century Vol2:FormlModelsndSemntics, World Scientific Publishing Co., 2004, pp. [20] Strejček, J., Liner Temporl Logic: Expressiveness nd Model Checking, Ph.D. thesis, Fculty of Informtics, Msryk University, Brno (2004).