On Decidbility of LTL Model Checking for Process Rewrite Systems Lur Bozzelli 1, Mojmír Křetínský 2, Vojtěch Řehák 2, nd Jn Strejček 2 1 Diprtimento di Mtemtic e Aplliczioni, Università degli Studi di Npoli Federico II, Vi Cinti, 80126 Npoli, Itly, lur.bozzelli@dm.unin.it 2 Fculty of Informtics, Msryk University, Botnická 68, 60200 Brno, Czech Republic, kretinsky,rehk,strecek@fi.muni.cz Abstrct. We estblish decidbility boundry of the model checking problem for infinite-stte systems defined by Process Rewrite Systems (PRS) or wekly extended Process Rewrite Systems (wprs), nd properties described by bsic frgments of ction-bsed Liner Temporl Logic (LTL) with both future nd pst opertors. It is known tht the problem for generl LTL properties is decidble for Petri nets nd for pushdown processes, while it is undecidble for PA processes. We show tht the problem is decidble for wprs if we consider properties defined by LTL formule with only modlities strict eventully, strict lwys, nd their pst counterprts. Moreover, we show tht the problem remins undecidble for PA processes even with respect to the LTL frgment with the only modlity until or the frgment with modlities next nd infinitely often. 1 Introduction Automtic verifiction of current softwre systems often needs to model them s infinite-stte systems. One of the most powerful formlisms for finite description of infinite-stte systems (except formlisms which re lnguge equivlent to Turing mchines) is clled Process Rewrite Systems (PRS) [My00]. The PRS frmework, bsed on term rewriting, subsumes mny formlisms studied in the context of forml verifiction, e.g. Petri nets (PN), pushdown processes (PDA), nd process lgebrs like BPA, BPP, or PA. PRS cn be dopted s forml model for progrms with recursive procedures nd restricted forms of dynmic cretion nd synchroniztion of concurrent processes. A substntil merit of PRS is tht some importnt verifiction problems re decidble for the whole PRS clss. In prticulr, Myr [My00] proved tht the following problems re decidble for PRS: the rechbility problem - whether given stte is rechble, the rechble property problem - whether there is rechble stte where some given ctions re enbled nd some given ctions re disbled. In [KŘS04b], 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 Some of the results presented in this pper hve been lredy published in [BKŘS06] nd [KŘS07].
(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 proved tht the rechbility problem remins decidble for wprs [KŘS04] nd tht the problem clled rechbility Hennessy Milner property (whether there is rechble stte stisfying given Hennessy Milner formul) is decidble for wprs s well [KŘS05]. Note tht the ltter problem is strictly more generl thn the rechble property problem. The hierrchy of ll PRS nd wprs clsses is depicted in Figure 1. Concerning the model checking problem, brod overview of (un)decidbility results for subclsses of PRS nd vrious temporl logics cn be found in [My98]. Here we focus exclusively on Liner Temporl Logic (LTL). It is known tht LTL model checking of PDA is EXPTIME-complete [BEM97]. LTL model checking of PN is lso decidble, but t lest s hrd s the rechbility problem for PN [Esp94] (the rechbility problem is EXPSPACE-hrd [My84,Lip76] nd no primitive recursive upper bound is known). If we consider only infinite runs, then the problem for PN is EXPSPACEcomplete [Hb97,My98]. Conversely, LTL model checking is undecidble for ll the clsses subsuming PA [BH96,My98]. So fr, there re only two positive results for these clsses. Boujjni nd Hbermehl [BH96] hve identified frgment clled simple PLTL¾ for which model checking of infinite runs is decidble for PA (strictly speking, simple PLTL¾ is not frgment of LTL s it cn express lso some non-regulr properties, while LTL cnnot). Only recently, Bozzelli [Boz05] hs demonstrted tht model checking of infinite runs is decidble for PRS nd the frgment of LTL cpturing exctly firness properties. Our contribution: This pper contins severl results on decidbility of LTL model checking. In prticulr, we completely locte the decidbility boundry of the model checking problem for ll subclsses of PRS (nd wprs) nd ll bsic LTL frgments, where bsic LTL frgment is set of ll LTL formule contining only given subset of stndrd temporl modlities nd closed under boolen connectives. The boundry is depicted in Figure 2. To locte the boundry, we demonstrte the following results. 1. We introduce new LTL frgment A. Then we prove tht the problem whether given wprs hs (finite or infinite) run stisfying given formul of A is decidble. The proof employs our results presented in [Boz05,KŘS04,KŘS05] to reduce the problem to LTL model checking for PDA nd PN. This result directly implies decidbility of the model checking problem for wprs nd negted formule ofa. 2. We show tht every formul of the bsic frgment LTL µ (i.e. the frgment with modlities strict eventully nd strict lwys only) cn be effectively trnslted intoa. As LTL µ is closed under negtion, we cn lso trnslte LTL µ formule into negtions of A formule. This trnsltion yields decidbility of the model checking problem for wprs nd LTL µ. Note tht LTL µ is strictly more expressive thn the Lmport logic (i.e. the bsic frgment with modlities eventully nd lwys), which is gin strictly more expressive thn the
mentioned frgment of firness properties nd lso thn the regulr prt of simple PLTL¾. 3. We define pst extension PA of the frgment A. Using the result for A, we show tht the model checking problem for wprs nd negted formule of PA remins decidble. Further, we prove tht every formul of the bsic frgment LTL È À µ (LTL µ extended with the pst counterprts of nd ) cn be effectively trnslted into PA. Hence, we get decidbility of the model checking problem for wprs nd LTL È À µ. We note tht LTL È À µ is strictly more expressive thn LTL µ (for exmple, the formul b À µ is not equivlent to ny LTL µ formul) nd semnticlly equivlent to First-Order Mondic Logic of Order restricted to 2 vribles nd without successor predicte (Ç 2, see [EVW02] for effective trnsltions). Thus we lso positively solve the model checking problem for wprs nd Ç 2. 4. We demonstrte tht the model checking problem remins undecidble for PA even if we consider the bsic frgment with modlity until or the bsic frgment with modlities next nd infinitely often (which is strictly less expressive thn the one with next nd eventully). The pper lso presents two results tht re not connected to the decidbility boundry. 5. We introduce more generl pointed model checking problem (whether ll runs of given wprs system going through given stte stisfy given formul in the given stte). We show tht this problem is decidble for wprs nd LTL È À µ. 6. Finlly, we show tht negted formule of LTL det (the frgment known s the common frgment of CTL nd LTL [Mi00]) cn be effectively trnslted intoa. As consequence we get tht the model checking problem is decidble for wprs nd LTL det. Structure of the pper: The following section reclls bsic definitions. Sections 3, 4, 5, nd 6 correspond, respectively, to the first four items listed bove. Section 5 lso covers the results on the pointed model checking problem. Section 7 dels with the model checking problem for LTL det. The lst section summrizes our results nd tries to give n intuitive explntion of the found decidbility border loction. 2 Preliminries 2.1 PRS nd Wekly Extended PRS Let Const X be set of process constnts. The set of process terms t is defined by the bstrct syntx t :: ε X tt tt, where ε is the empty term, X ¾ Const, nd nd men sequentil 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. εt tε tε t. We distinguish four clsses of process terms s: 1 terms consisting of single process constnt, in prticulr, ε ¾ 1, S sequentil terms - terms without prllel composition, e.g. XYZ,
P prllel terms - terms without sequentil composition, e.g. XYZ, G generl terms - terms without ny restrictions, e.g. X YZµµW. Let M o pq be set of control sttes, be prtil ordering on this set, nd Act bc be set of ctions. Let αβ ¾ 1SPG be clsses of process terms such tht α β. An αβµ-wprs (wekly extended process rewrite system) is triple R p 0 t 0 µ, where R is finite set of rewrite rules of the form pt 1 µ qt 2 µ, where t 1 ¾ α, t 1 ε, t 2 ¾ β, ¾ Act, nd pq ¾ M stisfy p q, the pir p 0 t 0 µ ¾ M β forms the distinguished initil stte. By Act µ, Const µ, nd M µ we denote the respective sets of ctions, process constnts, nd control sttes occurring in the rewrite rules or the initil stte of. A wprs R p 0 t 0 µ induces lbelled trnsition system, whose sttes re pirs ptµ such tht p ¾ M µ nd t is process term over Const µ. The trnsition reltion is the lest reltion stisfying the following inference rules: pt 1 µ qt 2 µµ ¾ R pt 1 µ qt 2 µ pt 1 µ qt 2 µ pt 1 µ qt 2 µ pt 1 t1 ¼ µ qt 2 t1 ¼ µ pt 1 t1 ¼ µ qt 2 t1 ¼ µ Sometimes we write insted of if is cler from the context. The trnsition reltion cn be extended to finite words over Act in stndrd wy. To shorten our nottion we write pt in lieu of ptµ. A stte pt is rechble from stte p ¼ t ¼ if there exists word u such tht p ¼ t ¼ u pt. We sy tht stte is rechble if it is rechble from the initil stte p 0 t 0. Further, stte pt is clled terminl if there is no stte p ¼ t ¼ nd no ction such tht pt p ¼ t ¼. In this pper we lwys consider only systems where the initil stte is not terminl. A (finite or infinite) sequence σ p 1 t 1 1 p 2 t 2 2 n p n 1 t n 1 n 1 is clled derivtion over the word u 1 2 n n 1 µ in. Finite derivtions re lso denoted s p 1 t 1 u p n 1 t n 1, infinite ones s p 1 t 1 u. A derivtion in is clled run of if it strts in the initil stte p 0 t 0 nd it is either infinite, or its lst stte is terminl. Further, L µ denotes the set of words u such tht there is run of over u. An (αβ)-wprs where M µ is singleton is clled αβµ-prs (process rewrite system) [My00]. In such systems we omit the single control stte from rules nd sttes. Some clsses of (αβ)-prs correspond to widely known models, nmely finite-stte systems (FS), bsic process lgebrs (BPA), bsic prllel processes (BPP), process lgebrs (PA), pushdown processes (PDA), nd Petri nets (PN). The other clsses hve been nmed s PAD, PAN, nd PRS [My00]. The reltions between (αβ)-prs nd the mentioned formlisms nd nmes re indicted in Figure 1. Insted of (αβ)-wprs we juxtpose the prefix w- with the cronym corresponding to the (αβ)-prs clss. For exmple, we use wbpa rther thn (1S)-wPRS. Figure 1 shows the expressiveness hierrchy of ll the clsses mentioned bove, where expressive power of clss is mesured by the set of trnsition systems tht re definble (up to the strong bisimultion
wprs Õ Õ ÕÕÕÕ PRS Õ ÕÕÕÕ (GG)-PRS Õ ÕÕÕÕ Õ ÕÕ Õ ÕÕÕÕ wpad wpan Õ ÕÕÕÕ Ö ÖÖÖÖ PAD PAN (SG)-PRS Ö ÖÖÖÖ (PG)-PRS Ö ÖÖÖÖ wpa Õ Õ ÕÕÕÕ wpda=pda PA Õ ÕÕÕÕ wpn=pn (SS)-PRS (1G)-PRS Õ ÕÕÕÕ Õ ÕÕ (PP)-PRS Õ ÕÕÕÕ wbpa wbpp Õ ÕÕÕÕ BPA BPP (1S)-PRSÇ ÇÇÇÇÇ (1P)-PRS Ç Ô ÔÔÔÔ ÇÇÇÇÇÇ Ô ÔÔÔÔÔÔ wfs=fs (11)-PRS Fig. 1. The hierrchy of PRS nd wprs subclsses. equivlence [Mil89]) by the clss. This hierrchy is strict, with possible exception concerning the clsses wprs nd PRS, where the strictness is just our conjecture. For detils see [KŘS04b]. For technicl resons, we define norml form of wprs systems. A rewrite rule is prllel or sequentil if it hs one of the following forms: Prllel rules: px 1 X 2 X n qy 1 Y 2 Y m Sequentil rules: px qyz pxy qz px qy px qε where XYX i Y j Z ¾ Const, pq ¾ M, n 0, m 0, nd ¾ Act. A rule is clled trivil if it is both prllel nd sequentil, i.e. it hs the form px qy or px qε. A wprs R p 0 t 0 µ is in norml form if t 0 is process constnt nd R contins only prllel nd sequentil rewrite rules. PRS, wprs, other extensions of PRS, nd their respective subclsses re discussed in more detil in [Řeh07]. 2.2 Liner Temporl Logic The syntx of Liner Temporl Logic (LTL) [Pnu77] is defined s follows ϕ :: tt ϕ ϕ ϕ ϕ ϕíϕ ϕ ϕëϕ, where nd Í re the future modl opertors next nd until, while nd Ë 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 0 1 2 ¾ Act Act ω, u denotes the length of the word (we set u if u is infinite). A pointed word is pir uiµ of nonempty word u nd position 0 i u in this word. The semntics of LTL formule is defined inductively s follows: uiµ tt uiµ iff u 0 1 2 nd i uiµ ϕ iff uiµ ϕ uiµ ϕ 1 ϕ 2 iff uiµ ϕ 1 nd uiµ ϕ 2 uiµ ϕ iff i 1 u nd ui 1µ ϕ uiµ ϕ 1 Íϕ 2 iff k i k u ukµ ϕ 2 j i j k µ u jµ ϕ 1 µ uiµ ϕ iff 0 i nd ui 1µ ϕ uiµ ϕ 1 Ëϕ 2 iff k 0 k i ukµ ϕ 2 j k j i µ u jµ ϕ 1 µ We sy tht uiµ stisfies ϕ whenever uiµ ϕ. Further, nonempty word u stisfies ϕ, written u ϕ, whenever u0µ ϕ. Given set L of words, we write L ϕ if u ϕ holds for ll u ¾ L. Finlly, we sy tht run σ of wprs over word u stisfies ϕ, written σ ϕ, whenever u ϕ. Formule ϕψ re (initilly) equivlent, written ϕ i ψ, iff, for ll words u, it holds tht u ϕ µ u ψ. Formule ϕψ re globlly equivlent, written ϕ ψ, iff, for ll pointed words uiµ, it holds tht uiµ ϕ µ uiµ ψ. Clerly, if two formule re globlly equivlent then they re lso initilly equivlent. Moreover, two formule without pst modlities re globlly equivlent if nd only if they re initilly equivlent. Therefore we do not distinguish between initil nd globl equivlence when we tlk bout formule without pst. The following tble defines some derived future opertors nd their pst counterprts. future modlity mening pst modlity mening ϕ eventully ttíϕ Èϕ eventully in the pst tt Ëϕ ϕ lwys ϕ Àϕ lwys in the pst Èϕ ϕ strict eventully ϕ È ϕ eventully in the strict pst Èϕ ϕ strict lwys ϕ À ϕ lwys in the strict pst È ϕ ϕ infinitely often ϕ ϕ initilly ÀÈϕ Given set O 1 O n of modlities, LTL O 1 O n µ denotes the LTL frgment (closed under boolen connectives) contining ll formule with modlities O 1 O n only. Such frgment is clled bsic if either it contins future opertors only, or for ech included future opertor, it contins its pst counterprt nd vice vers. For exmple, the frgment LTL ˵ is not bsic.
ß ßßß ß ßßß ß ßßß LTL Í µ i Ç 3 ß ßßß ß ßß Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ LTL Í Ë È µ ß ßßß Þ ÞÞÞ LTL È µ Ç 2 Ë Þ ÞÞÞ LTL Í Ë µ Ë Û Þ ËËËËËËËËËËËË ÞÞÞ Ð ß ßßß Þ ÞÞÞ ß LTL ßßß µ LTL È µ Ç 2 Ê ß ßßß LTL ͵ É ß ÊÊÊÊÊÊÊÊÊÊÊÊÊ ßßß ß ÉÉÉÉÉÉÉÉÉÉÉ Ê ßßß LTL LTL µ µ LTL ȵ ÊÞ É Ð ÉÉÉÉÉÉÉÉÉÉÉÉ Ö LTL µ ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ LTL µ LTL µ Û ÛÛÛ Û ÛÛÛ Û ÛÛÛ Û ÛÛÛ Þ ÞÞ Û LTL µ Fig. 2. 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: the problem is decidble for ll the frgments below the line, while it is undecidble for ll the frgments bove the line (even if we consider PA systems only). Figure 2 shows n expressiveness hierrchy of ll studied bsic LTL frgments. Indeed, every bsic LTL frgment using stndrd 3 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. In prticulr, LTL È À µ is equiv- 3 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.
lent to LTL È µ. 4 We lso mind the result of [Gb87] stting tht ech LTL formul cn be converted into one which employs future opertors only, i.e. LTL ͵ i LTL Í˵. The hierrchy is lso strict: solid line between two frgments indictes tht every formul of the lower frgment is initilly equivlent to some formul of the upper frgment, but the opposite reltion does not hold. We refer to [Str04] for detils bout the expressiveness of LTL frgments. 2.3 Studied Problems Let F be n LTL frgment nd C be clss of wprs systems. This pper dels with the following three verifiction problems. 1. The model checking problem forf ndc is to decide, for ny given formul ϕ ¾F nd ny given system ¾C, whether L µ ϕ holds. 2. We lso consider the problem clled model checking of infinite runs, where L µ Act ω ϕ is exmined. 3. The pointed model checking problem for F nd wprs is to decide whether given formul ϕ ¾ F, given wprs system, nd given nonterminl stte pt of stisfy L pt µ ϕ, where L pt µ is the set of ll pointed words uiµ such tht hs run p 0 t 0 0 p 1 t 1 1 i 1 p i t i i stisfying u 0 1 2 nd pt p i t i. 3 Model Checking for NegtedA This section strts with the definition of the LTL frgmenta. The rest of the section is devoted to decidbility of the model checking problem for wprs nd negted formule of this frgment. Recll tht LTL µ denotes the frgment of formule without ny modlity, i.e. boolen combintions of ctions. In the following we use ϕ 1 Í ϕ 2 to bbrevite ϕ 1 ϕ 1 Íϕ 2 µ. Definition 1. Let δ θ 1 O 1 θ 2 O 2 θ n O n θ n 1, where n 0, ech θ i ¾ LTL µ, O n is, nd, for ech i n, O i is either Í or Í or. Further, letb LTL µ be finite set. An α-formul is defined s α δbµ θ1 O 1 θ 2 O 2 θ n O n θ n 1 µµ ψ The frgmenta consists of ll finite disjunctions of α-formule. Hence, word u stisfies α δbµ iff u cn be written s conctention u 1 u 2 u n 1, where ech word u i consists only of ctions stisfying θ i nd u i 0 if i n 1 or O i is Í, 4 As nd È À re pirs of dul opertors, the frgments LTL È À µ nd LTL È µ re in fct equivlent even with respect to the globl equivlence. ψ¾b
u i 0 if O i is Í, u i 1 if O i is or, u n 1 stisfies ψ for every ψ ¾B. In the following we use the fct tht finite disjunctions of α-formule re closed under conjunction. Lemm 2. A conjunction of α-formule cn be effectively converted into n equivlent disjunction of α-formule. The proof is strightforwrd but quite technicl exercise, see [Řeh07] for some hints. To support n intuition, we provide n exmple of conjunction of two simple α- formule nd n equivlent disjunction. Exmple 3. A conjunction α θ 1 Íθ 2 θ 3 Bµα θ ¼ 1 Íθ¼ 2 θ ¼ 3B ¼ µ is equivlent to the following disjunction. α θ 1 θ ¼ 1 µí θ 2 θ ¼ 2 µ θ 3 θ ¼ 3 µb B ¼ µ α θ 1 θ ¼ 1 µí θ 2 θ ¼ 1 µ θ 3 θ ¼ 1 µí θ 3 θ ¼ 2 µ θ 3 θ ¼ 3 µb B ¼ µ α θ 1 θ ¼ 1 µí θ 1 θ ¼ 2 µ θ 1 θ ¼ 3 µí θ 2 θ ¼ 3 µ θ 3 θ ¼ 3 µb B ¼ µ In order to show tht the model checking problem for wprs nd negted formule ofa is decidble, we prove decidbility of the dul problem, i.e. whether given wprs system hs run stisfying given formul of A. Finite nd infinite runs re treted seprtely. Theorem 4. The problem whether given wprs system hs finite run stisfying given α-formul is decidble. Proof. Let be wprs system nd α δbµ be n α-formul. Note tht formul ψ is stisfied by finite nonempty word if nd only if the length of the word is 1. Therefore, ifb /0 then it is esy to check whether there is finite run of stisfying α δbµ. In wht follows we ssumeb /0. Let δ θ 1 O 1 θ 2 O 2 θ n O n θ n 1. We construct wprs system ¼ with control sttes M µ 12n 1 nd the following four types of trnsition rules. 1. For ny 1 i n nd every rule pt 1 qt 2 of such tht n ction stisfies θ i, we dd the rule piµt 1 qi 1µt 2 to ¼. Moreover, if O i is Í or Í then we lso dd the rule piµt 1 qiµt 2. 2. Let e be fresh ction. For every p ¾ M µ, X ¾ Const µ, nd for ll i, 1 i n, such tht O i Í, we dd the rule piµx e pi 1µX to ¼. 3. For every rule pt 1 qt 2 of such tht stisfies θ n 1, we dd the rule pn 1µt 1 qn 1µt 2 to ¼. 4. For every rule pt 1 qt 2 of we dd the rule pn 1µt 1 pn 1µt 1 to ¼. Loosely speking, the rules of type 1 3 llow ¼ to simulte ll the runs of which stisfy α δ /0µ. The rules of type 4 ssure tht stte pn 1µt of ¼ is terminl if nd only if the stte pt of is terminl.
Let p 0 t 0 be the initil stte of. There is finite run p 0 t 0 u qt stisfying α δ /0µ v if nd only if there is finite run p 0 1µt 0 ¼ qn 1µt. Hence, we need to decide whether there exists stte of the form qn 1µt tht is terminl nd rechble from p 0 1µt 0. To tht end, for every p ¾ M µ we dd to ¼ the rule pn 1µZ end pn 1µε, where end ¾ Act µ is fresh ction nd Z ¾ Const µ is fresh process constnt. Now, it holds tht hs finite run stisfying α δ /0µ if nd only if there exists stte of ¼, which is rechble from p 0 1µ t 0 Zµ nd the only enbled ction in this stte is end. This lst condition on the stte cn be expressed by formul ϕ endtt Î ¾Act µ tt of the Hennessy Milner logic. As rechbility of stte stisfying given Hennessy Milner formul is decidble for wprs (see [KŘS05] for detils), we re done. ÙØ The problem for infinite runs is more complicted. In order to solve it, we introduce more terminology nd nottion. At first we define β-formule nd regulr lnguges clled γ-lnguges. Let w 1 O 1 2 O 2 n O n, where n 0, 1 n ¾ Act re pirwise distinct ctions nd ech O i is either Í or. Further, let B ActÖ 1 n be nonempty finite set of ctions nd C B. A β-formul β wbcµ nd γ-lnguge γ wcµ re defined s β wbcµ 1 O 1 2 O 2 n O n bµµ b b bµ b¾b b¾c b¾böc where o i if O i Í γ wcµ o 1 1 o 2 2 o n n L, nd L 1 if O i Ì b¾cc bc ε if C /0 otherwise. Roughly speking, β-formul is more restrictive version of n α-formul nd in the context of β-formule we consider infinite words only. Contrry to δ of n α-formul, w of β-formul employs ctions rther thn LTL µ formule. While til of n infinite word stisfying n α-formul is specified by θ n 1, in the definition of β-formule we use set B contining exctly ll the ctions of the til nd its subset C of exctly ll those ctions occurring infinitely mny times in the til. Remrk 5. Note tht n infinite word stisfies formul β wbcµ if nd only if it cn be divided into prefix u ¾ γ wbµ nd suffix v ¾ C ω such tht v contins infinitely mny occurrences of every c ¾ C. Let B, C, nd w 1 O 1 2 O 2 n O n be defined s bove. We sy tht finite derivtion σ over word u stisfies γ wcµ if nd only if u ¾ γ wcµ. We write w ¼ B ¼ µ Ú wbµ whenever B ¼ B nd w ¼ i1 O i1 i2 O i2 ik O ik for some 1 i 1 i 2 i k n. Moreover, we write w ¼ B ¼ C ¼ µ Ú wbcµ whenever w ¼ B ¼ µ Ú wbµ, B ¼ is nonempty, nd C ¼ C B ¼. Remrk 6. If u is n infinite word stisfying β wbcµ nd v is n infinite subword of u (i.e. it rises from u by omitting some letters), then there is exctly one triple w ¼ B ¼ C ¼ µ Ú wbcµ such tht v β w ¼ B ¼ C ¼ µ. Further, for ech finite subword v of u, there is exctly one pir w ¼ B ¼ µ such tht w ¼ B ¼ µ Ú wbµ nd v ¾ γ w ¼ B ¼ µ.
Given PRS in norml form, by tri µ, pr µ, nd seq µ we denote the system restricted to trivil, prllel, nd sequentil rules, respectively. A derivtion in tri µ is clled trivil derivtion in. In wht follows we write simply triprseq s is lwys clerly determined by the context. Definition 7. Let be PRS in norml form nd β wbcµ be β-formul. The PRS is in flt wbcµ-form if nd only if, for ech XY ¾ Const µ, ech w ¼ B ¼ C ¼ µ Ú wbcµ, nd ech B ¼¼ B, the following conditions hold: 1. If there is finite derivtion X u Y stisfying γ w ¼ B ¼¼ µ, then there is lso finite derivtion X v tri Y stisfying γ w ¼ B ¼¼ µ. 2. If there is term t nd finite derivtion X u t stisfying γ w ¼ B ¼¼ µ, then there is lso constnt Z nd finite derivtion X v tri Z stisfying γ w ¼ B ¼¼ µ. 3. If w ¼ ε nd there is n infinite derivtion X u stisfying β w ¼ B ¼ C ¼ µ, then there is lso n infinite derivtion X v tri stisfying β w ¼ B ¼ C ¼ µ. 4. If there is n infinite derivtion X u pr stisfying β w ¼ B ¼ C ¼ µ, then there is lso n infinite derivtion X v tri stisfying β w ¼ B ¼ C ¼ µ; 5. If there is n infinite derivtion X u seq stisfying β w ¼ B ¼ C ¼ µ, then there is lso n infinite derivtion X v tri stisfying β w ¼ B ¼ C ¼ µ. Intuitively, the system is in flt wbcµ-form if, for every derivtion of one of the listed types there is n equivlent trivil derivtion. All conditions of the definition cn be checked due to the following lemm, results of [Boz05], nd decidbility of LTL model checking for PDA nd PN. Lemm 9 sys tht every PRS in norml form cn be trnsformed into n equivlent flt system. Finlly, Lemm 12 sys tht if PRS system in flt wbcµ-form hs n infinite derivtion stisfying β wbcµ, then it hs lso trivil infinite derivtion stisfying β wbcµ. Note tht it is esy to check whether such trivil derivtion exists. Lemm 8. Given γ-lnguge γ wcµ, PRS system, nd constnts XY, the following problems re decidble: (i) Is there ny derivtion X u Y stisfying γ wcµ? (ii) Is there ny derivtion X u t such tht t is term nd u ¾ γ wcµ? Proof. The two problems cn be reduced to the rechbility problem for wprs (i.e. to decide whether given sttes p 1 t 1 p 2 t 2 of given wprs system ¼ stisfy p 1 t 1 v ¼ p 2 t 2 for some v), which is known to be decidble [KŘS04]. with the set of control sttes (i) Let w 1 O 1 n O n. We construct wprs ¼ 12n 2 C. Intuitively, control sttes 12n re used to check tht the ctions 1 2 n pper in the right order nd quntity due to w, while the other ctions re not llowed. After tht, the control sttes in 2 C re used to check tht every ction in C ppers t lest once. The set of rewrite rules is defined s follows. For the ske of compctness, we use n 1µ s nother nme for the control stte /0. For every 1 i n nd every rule t 1 i t 2 of, we dd to ¼ the rule it 1 i 1µt 2 nd if O i Í then lso the rule it 1 i it 2. i
b For every b ¾ C, every D C, nd every rule t 1 t 2 of, we dd to ¼ the rule Dt b 1 D bµt 2. Obviously, word u ¾ Act stisfies 1X u ¼ CY if nd only if it stisfies both X u Y nd u ¾ γ wcµ. As we cn decide whether 1X u ¼ CY holds for some u, we cn decide Problem (i). (ii) We construct wprs ¼ s in the previous cse. Moreover, for every Z ¾ Const µ, we dd to ¼ the rule CZ e Cε. It is esy to see tht if word u ¾ γ wcµ stisfies X u t for some t, then 1X uem ¼ Cε holds for some m 0. Conversely, if 1X v ¼ Cε holds for some v, then some prefix u of v stisfies both u ¾ γ wcµ nd X u t for some t. As we cn decide whether, for some v, 1X v ¼ Cε holds, we cn decide Problem (ii). ÙØ The proof of the following lemm contins the lgorithmic core of this section. Lemm 9. Let be PRS in norml form nd β wbcµ be β-formul. One cn construct PRS ¼ in flt wbcµ-form such tht, for ech w ¼ B ¼ C ¼ µ Ú wbcµ nd ech X ¾ Const µ, ¼ hs n infinite derivtion strting from X nd stisfying β w ¼ B ¼ C ¼ µ if nd only if hs n infinite derivtion strting from X nd stisfying β w ¼ B ¼ C ¼ µ. Proof. In order to obtin ¼, we describe n lgorithm extending with trivil rewrite rules in ccordnce with Conditions 1 5 of Definition 7. All the conditions of Definition 7 cn be checked for ech XY ¾ Const µ, ech w ¼ B ¼ C ¼ µ Ú wbcµ, nd ech B ¼¼ B. For Conditions 1 nd 2, this follows from Lemm 8. The problem whether there is n infinite derivtion X u stisfying β εb ¼ C ¼ µ is specil cse of the firness problem, which is decidble due to [Boz05]. Finlly, Conditions 4 nd 5 cn be checked due to decidbility of LTL model checking for PDA [BEM97] nd PN [Esp94]. If there is non-stisfied condition, we dd some trivil rules forming the missing derivtion. Let us ssume tht Condition 3 (or 4 or 5, respectively) is not stisfied, i.e. there exists n infinite derivtion X u (or X u pr or X u seq, respectively) stisfying β w ¼ B ¼ C ¼ µ for some w ¼ B ¼ C ¼ µ Ú wbcµ nd violting the condition. Remrk 5 implies tht C ¼ is nonempty nd there is finite derivtion X v t stisfying γ w ¼ B ¼ µ. Hence, there exists n ordering of B ¼ b 1 b 2 b m such tht (*) for ech 1 j m, there is finite derivtion in strting from X nd stisfying γ w ¼ b 1 b j µ. We cn effectively select such n ordering out of ll orderings of B ¼ using Lemm 8. Further, let w ¼ 1 O 1 2 O 2 n O n nd let C ¼ c 1 c 2 c k. Then, we dd the trivil rule Z i 1 i Z i for ech 1 i n, the trivil rule Z n j 1 b j Zn j for ech 1 j m, nd the trivil rule Z n m j 1 c j Zn m j for ech 1 j k, where Z 0 X, Z 1 Z n m k 1 re fresh process constnts, nd Z n m k Z n m. These dded rules form n infinite derivtion using only trivil rules, strting from X, nd stisfying β w ¼ B ¼ C ¼ µ. Similrly, if there re X, Y, nd γ w ¼ B ¼¼ µ with w ¼ 1 O 1 2 O 2 n O n such tht Condition 1 or 2 of Definition 7 is violted, then we first compute n ordering
b 1 b m of B ¼¼ stisfying (*), nd then we dd the trivil rule Z i 1 i Z i for ech 1 i n, nd the trivil rule Z n j 1 b j Zn j for ech 1 j m, where Z 0 X nd Z 1 Z n m re fresh process constnts (with exception of Z n m which is Y in the cse of Condition 1). The dded trivil rules generte derivtion X 1 n b 1 b m Z n m stisfying γ w ¼ B ¼¼ µ. Let ¼¼ be the PRS extended with the new rules. The condition (*) ensures tht, for ech X ¾ Const µ nd ech w ¼ B ¼ C ¼ µ Ú wbcµ, the system ¼¼ is equivlent to with respect to the existence of n infinite derivtion strting from X nd stisfying β w ¼ B ¼ C ¼ µ. If ¼¼ is not in flt wbcµ-form, then the lgorithm repets the procedure described bove on the system ¼¼ with the difference tht X nd Y rnge over the constnts of the originl system. The lgorithm eventully termintes s the number of itertions is bounded by the number of pirs of process constnts XY of, times the number of triples w ¼ B ¼ C ¼ µ stisfying w ¼ B ¼ C ¼ µ Ú wbcµ, nd times the number of subsets B ¼¼ B. Let ¼ be the resulting PRS. We clim tht ¼ is in flt wbcµform. For the process constnts of the originl system, by construction ¼ stisfies ll conditions of Definition 7. For the dded constnts, it is sufficient to observe tht ny derivtion in ¼ strting from such constnt either is trivil or hs trivil prefix leding to constnt of. Hence, ¼ is the desired PRS system. ÙØ Definition 10 (Subderivtion). Let be PRS in norml form nd σ 1 be (finite or infinite) derivtion s 1 1 s 2 2, where s 1 1 s 2 hs the form X 1 YZ nd, for ech i 2, if s i is not the lst stte of the derivtion, then it hs the form s i t i Z with t i ε. Then σ 1 is clled subderivtion of derivtion σ if σ hs suffix σ ¼ stisfying the following: 1. every trnsition step in σ ¼ is of the form s i t ¼ i si 1 t ¼ or s i t ¼ b si t ¼¼, where t ¼ b t ¼¼, 2. in σ ¼, if we replce every step of the form s i t ¼ i si 1 t ¼ by the step s i i si 1 nd we skip every step of the form s i t ¼ b si t ¼¼, we get precisely σ 1. Further, if σ 1 nd σ re finite, the lst term of σ 1 is process constnt, nd σ is prefix of derivtion σ ¼, then σ 1 is lso subderivtion of σ ¼. Remrk 11. Let be PRS in norml form nd σ be derivtion of hving suffix σ ¼ of the form σ ¼ u Xt YZµt. Then, there is subderivtion of σ whose first trnsition step X YZ corresponds to the first trnsition step of σ ¼. Intuitively, the subderivtion cptures the behviour of the subterm YZ since its emergence until it is possibly reduced to term without ny sequentil composition. Due to the norml form of, the subterm YZ behves independently on the rest of the term (s long s it contins sequentil composition). Lemm 12. Let be PRS in flt wbcµ-form. Then, for ech X ¾ Const µ nd ech w ¼ B ¼ C ¼ µ Ú wbcµ, the following condition holds: If there is n infinite derivtion X u stisfying β w ¼ B ¼ C ¼ µ, then there is lso n infinite derivtion X v tri stisfying β w ¼ B ¼ C ¼ µ.
A sketch of the proof. Given n infinite derivtion σ stisfying formul β σµ β w ¼ B ¼ C ¼ µ where w ¼ B ¼ C ¼ µ Ú wbcµ, by trivil equivlent of σ we men n infinite trivil derivtion strting with the sme term s σ nd stisfying β σµ. Similrly, given finite derivtion σ stisfying some γ σµ γ w ¼ B ¼ µ where w ¼ B ¼ µ Ú wbµ, by trivil equivlent of σ we men finite trivil derivtion σ ¼ such tht σ ¼ strts with the sme term s σ, it stisfies γ σµ, nd if the lst term of σ is process constnt, then the lst term of σ ¼ is the sme process constnt. The lemm is proven by contrdiction. We ssume tht there exist some infinite derivtions violting the condition of the lemm. Let σ be one of these derivtions such tht the number of trnsition steps of σ generted by sequentil non-trivil rules with ctions ¾ B is miniml (note tht this number is lwys finite s we consider derivtions stisfying β w ¼ B ¼ C ¼ µ for some w ¼ B ¼ C ¼ µ Ú wbcµ). First, we prove tht every subderivtion of σ hs trivil equivlent. Then we replce ll subderivtions of σ by the corresponding trivil equivlents. This step is techniclly nontrivil becuse σ my hve infinitely mny subderivtions. By the replcement we obtin n infinite derivtion σ ¼ stisfying β σµ nd strting with the sme process constnt s σ. Moreover, σ ¼ hs no subderivtions nd hence it does not contin ny sequentil opertor. Flt wbcµform of (Condition 4) implies tht σ ¼ hs trivil equivlent. This is lso trivil equivlent of σ which mens tht σ does not violte the condition of our lemm. Proof. In this proof, by β-formul we lwys men formul of the form β w ¼ B ¼ C ¼ µ where w ¼ B ¼ C ¼ µ Ú wbcµ. We lso consider only infinite derivtions stisfying some of these β-formule. Remrk 6 implies tht such n infinite derivtion σ stisfies exctly one β-formul. We denote this β-formul by β σµ. Further, by SEQ σµ we denote the number of trnsition steps t i ti 1 of σ generted by sequentil non-trivil rule nd such tht ¾ B. Note tht SEQ σµ is lwys finite due to the restrictions on considered infinite derivtions. Given n infinite derivtion σ, by its trivil equivlent we men n infinite trivil derivtion strting with the sme term s σ nd stisfying β σµ. Similrly, we consider only finite derivtions stisfying some γ w ¼ B ¼ µ where w ¼ B ¼ µ Ú wbµ. Remrk 6 implies tht such finite derivtion σ stisfies exctly one γ-lnguge, which is denoted by γ σµ. Given finite derivtion σ, by its trivil equivlent we men finite trivil derivtion σ ¼ such tht σ ¼ strts with the sme term s σ, it stisfies γ σµ, nd if the lst term of σ is process constnt, then the lst term of σ ¼ is the sme process constnt. Using the introduced terminology, the lemm sys tht every infinite derivtion strting with process constnt hs trivil equivlent. For the ske of contrdiction, we ssume tht the lemm does not hold. Let Σ be the nonempty set of infinite derivtions violting the lemm nd let k minseq σµ σ ¾ Σ. First of ll, we prove two clims. Clim 1 Let σ be n infinite derivtion stisfying SEQ σµ k. Then every subderivtion of σ hs trivil equivlent. Proof of the clim For finite subderivtions, the existence of trivil equivlents follows directly from the flt wbcµ-form of (Conditions 1 nd 2). Let σ 1 be n infinite subderivtion of σ. It hs the form σ 1 X seq YZ b1 t 1 Z b2 t 2 Z b3 where t 1 t 2 re nonempty terms. There re two cses:
If ¾ B, then β σ 1 µ hs the form β εb ¼ C ¼ µ. Hence, σ 1 hs trivil equivlent due to the flt wbcµ-form of (Condition 3). If ¾ B, then the first step X seq YZ of σ 1 is counted in SEQ σ 1 µ nd the corresponding step Xt ¼ seq YZt ¼ of σ is counted in SEQ σµ. Hence, 0 SEQ σµ. Let σ 2 be the derivtion σ 2 Y b1 b t 2 b 1 t 3 2. As SEQ σ 2 µ SEQ σ 1 µ k, the definition of k implies tht σ 2 hs trivil equivlent σ ¼ 2 Y c1 c tri Y 2 1 tri c Y 3 2 tri. Further, s σ ¼ 2 stisfies β σ 2µ, the derivtion σ ¼ 1 X seq YZ c1 tri Y 1 Z c2 tri Y 2 Z c3 tri stisfies β σ 1 µ. Moreover, the flt wbcµ-form of (Condition 5) implies tht σ ¼ 1 hs trivil equivlent. Obviously, it is lso trivil equivlent of σ 1. ÙØ Clim 2 Let σ be n infinite derivtion such tht SEQ σµ k, it strts with prllel term p, nd it stisfies formul β w ¼ B ¼ C ¼ µ. Then there is n infinite derivtion p u pr p ¼ v such tht p¼ is prllel term, u ¾ γ w ¼ B ¼ µ, nd v stisfies β εc ¼ C ¼ µ. Proof of the clim Remrk 5 implies tht σ cn be written s p u1 u t 2 where p u1 t is the miniml prefix of σ stisfying γ w ¼ B ¼ u µ nd such tht t 2 stisfies β εc ¼ C ¼ µ. Let SEQ σµ denote the number of trnsition steps in the prefix p u 1 t generted by sequentil non-trivil rules (note tht SEQ σµ SEQ σµ s in SEQ σµ we do not count trnsition steps lbelled with ctions of B). We prove the clim by induction on SEQ σµ. The bse cse SEQ σµ 0 is obvious. Now, ssume tht SEQ σµ 0. Since p is prllel term nd is in norml form, the first trnsition step of p u1 t counted in SEQ σµ hs the form Yp ¼ WZµp¼ nd it corresponds to the first trnsition step Y WZ of subderivtion σ 1. In σ, we replce the subderivtion σ 1 with its trivil equivlent (whose existence is gurnteed by Clim 1) nd we obtin new derivtion σ ¼¼ strting with p, stisfying β σµ nd such tht SEQ σ ¼¼ µ SEQ σµ. Hence, the second clim directly follows from the induction hypothesis. In the following, we describe the replcement of such subderivtion. u Let σ 1 Y nd σ ¼ 1 Y v tri be its trivil equivlent. Let β σ 1 µ β c 1 O 1 c 2 O 2 c n O n B ¼¼ C ¼¼ µ. Then uv ¾ c 1 c c 2 n B ω. Recll tht c 1 c 2 c n re pirwise distinct nd B ActÖc 1 c n. Intuitively, for every 1 i n, we replce the first trnsition step of σ 1 lbelled with c i by the sequence of trnsition steps of σ ¼ 1 lbelled with c i, nd then we cncel the other trnsition steps of σ 1 lbelled with c i. 5 Further, the first trnsition step of σ 1 lbelled with n ction of B is replced with the miniml prefix of the remining prt of σ ¼ 1 stisfying γ εb¼¼ µ. Finlly, the remining 5 By replcement of trnsition step s 1 s2 of σ 1 by sequence Y 1 v ¼ tri Y 2 of trnsition steps of σ ¼ 1 we men tht the corresponding trnsition step s 1t ¼ s2 t ¼ of σ is replced by Y 1 t ¼ v ¼ tri Y 2 t ¼, nd ll immeditely succeeding steps s 2 t ¼¼ b s2 t ¼¼¼ of σ re replced by Y 2 t ¼¼ b Y2 t ¼¼¼. Further, by cncelltion of trnsition step s 1 c i s2 of σ 1 we men tht the corresponding trnsition step s 1 t ¼ c i s2 t ¼ of σ is replced by Y 2 t ¼, where Y 2 is the lst process constnt of σ ¼ 1 such tht trnsition under c i leds to Y 2, nd ll immeditely succeeding steps s 2 t ¼¼ b s2 t ¼¼¼ of σ re replced by Y 2 t ¼¼ b Y2 t ¼¼¼.
trnsition steps of σ 1 re orderly replced with the remining trnsition steps of σ ¼ 1. The cse when σ 1 nd its trivil equivlent σ ¼ 1 re finite is similr. It is esy to see tht the described replcement opertion preserves the fulfilment of β σµ nd the obtined derivtion σ ¼¼ stisfies SEQ σ ¼¼ µ SEQ σµ. ÙØ With this clim, we cn esily derive contrdiction. Let σ X u be n infinite derivtion such tht SEQ σµ k nd it hs no trivil equivlent. Further, let β σµ w ¼ B ¼ C ¼ µ. Note tht C ¼ is nonempty. Clim 2 sys tht there is derivtion X u1 pr v p 1 1 where p 1 is prllel term, u 1 ¾ γ w ¼ B ¼ µ, nd v 1 stisfies β εc ¼ C ¼ µ. Applying v this clim on the suffix p 1 u 1, we get derivtion p 2 v 1 pr p 2 2 where p 2 is prllel term, u 2 ¾ γ εc ¼ µ, nd v 2 stisfies β εc ¼ C ¼ µ. Iterting this rgument, we get u i 1 sequence p i pr p i 1 µ i¾æ of derivtions stisfying γ εc ¼ µ. These derivtions re nonempty s C ¼ is nonempty. Let us consider the derivtion σ ¼ X u1 pr p 1 u 2 pr p 2 u 3 pr p 3 u 4 pr Flt wbcµ-form of (Condition 4) implies tht σ ¼ hs trivil equivlent. However, this is lso trivil equivlent of σ s both σσ ¼ strt with X nd σ ¼ stisfies β σµ. This is contrdiction. ÙØ Theorem 13. The problem whether given PRS in norml form hs n infinite run stisfying given formul β wbcµ is decidble. Proof. Due to Lemmt 9 nd 12, the problem cn be reduced to the problem whether there is n infinite derivtion X v tri stisfying β wbcµ. This problem corresponds to LTL model checking of finite-stte systems, which is decidble. ÙØ The following three theorems show tht Theorem 13 holds even for wprs nd α- formule. Theorem 14. The problem whether given PRS in norml form hs n infinite run stisfying given α-formul is decidble. Proof. Let be PRS in norml form nd α θ 1 O 1 θ n O n ξbµ be n α-formul. For b i every θ i nd every rule t 1 t 2 such tht b stisfies θ i, we dd rule t 1 t 2, where i is fresh ction corresponding to θ i. Similrly, for every ψ ¾B ξ nd every rule b ψ t 2 such tht b stisfies ψ ξ, we dd rule t 1 t2, where ψ is fresh ction. t 1 Let ¼ be the resulting PRS system. Note tht ¼ is lso in norml form. Obviously, hs n infinite run stisfying the originl α-formul if nd only if ¼ hs n infinite run stisfying α 1 O 1 n O n ξ Ï b¾c bµcµ, where C ψ ψ ¾B. It is n esy exercise to show tht this new α-formul cn be effectively trnsformed into disjunction of β-formule which is equivlent with respect to infinite words. Hence, the problem is decidble due to Theorem 13. ÙØ Theorem 15. The problem whether given PRS hs n infinite run stisfying given α-formul is decidble.
Proof. Let be PRS, α δbµ be n α-formul, nd e ¾ Act µ be fresh ction. First of ll, we describe our modifiction of the stndrd lgorithm [My00] tht trnsforms into PRS in norml form. Let t 0 be the initil stte of. If t 0 is not process constnt, we replce it by fresh process constnt X 0 nd we dd rewrite rule X 0 t for ech ction nd ech term t such tht t 0 t. Note tht the number of dded rules is lwys finite. If is still not in norml form, then there exists rule r which is neither prllel nor sequentil; r hs one of the following forms: 1. r t t 1 t 2 (resp., r t 1 t 2 t) where t or t 1 or t 2 is not prllel term. Let e Z 1 Z 2 Z ¾ Const µ be fresh process constnts. We replce r with the rules t Z, Z e e e e Z 1 Z 2, Z 1 t 1, nd Z 2 t 2 (resp., t 1 Z 1, t 2 Z 2, Z 1 Z 2 Z, nd Z e t). 2. r t t 1 t 2 t 3 µ (resp., r t 1 t 2 t 3 µ t). Let Z ¾ Const µ be fresh process constnt. We modify in two steps. First, we replce t 2 t 3 by Z in left-hnd nd right-hnd sides of ll rules of. Then, we dd the rules Z e e t 2 t 3 nd t 2 t 3 Z. 3. r t 1 t 2 X (resp., r t 2 X t 1 ) where t 1 or t 2 is not process constnt. Let Z 1 Z 2 ¾ Const µ be fresh process constnts. We replce r with the rules t 1 e e Z 2 X, nd Z 2 t 2 (resp., t 2 Z 2, Z 2 X e Z 1, nd Z 1 t 1 ). Z 1 e Z 1, After finite number of pplictions of this procedure (with the sme ction e), we obtin PRS ¼ in norml form. We define formul α δ ¼ B ¼ µ, where B ¼ B Ï ¾Act µ nd δ ¼ rises from δ θ 1 O 1 θ n O n ξ by the following substitution for every i, 1 i n. If O i is Í, then replce the pir θ i Í by the pir e θ i µí. If O i is Í, then replce the pir θ i Í by the sequence e θ i µíθ i Í. If O i is, then replce the pir θ i by the sequence eíθ i. θ n O n θ n is replced by the sequence eíθ n. ξ is replced by ξ eµ. Let us note tht the construction ofb ¼ ensures tht ny word with suffix e ω does not stisfy α δ ¼ B ¼ µ. Observe tht u ¼ α δ ¼ B ¼ µ if nd only if u α δbµ, where u is obtined from u ¼ by eliminting ll occurrences of ction e. Clerly, hs n infinite run stisfying α δbµ if nd only if ¼ hs n infinite run stisfying α δ ¼ B ¼ µ. As ¼ is in norml form, we cn now pply Theorem 14. ÙØ Theorem 16. The problem whether given wprs system hs n infinite run stisfying given α-formul is decidble. Proof. Let be wprs with the initil stte p 0 t 0 nd α δbµ be n α-formul. We construct PRS ¼ with the initil stte t 0 which cn simulte. We lso define set of formule recognizing correct simultions. The system ¼ is very similr to. We chnge only ctions of rules to hold informtion bout control sttes in the rules nd then we remove ll control sttes. To be more p precise, for every rule of the form pt 1 pt 2 of, we dd the rule t 1 t 2 to ¼, nd for pq every rule of the form pt 1 qt 2 of, we dd the rule t 1 t 2 to ¼.
Further, we modify the formul α δbµ in such wy tht every occurrence of ech ction is replced by Ï q¾m µ q Ï pq pq µ. Let α δ ¼ B ¼ µ be the resulting formul. Moreover, for every nonempty subset p 1 p 2 p k M µ of control sttes stisfying p 1 p 2 p k nd p 1 p 0, we define n α-formul ϕ p1 p k α θ p 1 Íθ p1 p 2 θ p2 Íθ p2 p 3 θ pk 1 p k θ pk /0µ where θ pi Ï ¾Act µ pi nd θ pi p j Ï ¾Act µ pi p j. It is esy to see tht there is n infinite run of stisfying α δbµ if nd only if there is n infinite run of ¼ stisfying α δ ¼ B ¼ µ nd ϕ p1 p 2 p k for some control sttes p 1 p 2 p k such tht p 1 p 2 p k nd p 1 p 0. As the number of such sequences is finite nd ech ϕ p1 p 2 p k is n α-formul, Theorem 15 nd Lemm 2 imply tht the considered problem is decidble. ÙØ Theorems 4 nd 16 imply the following corollry. Corollry 17. The model checking problem for wprs nd negted formule of A is decidble. 4 Model Checking for LTL µ This section focuses on the frgment LTL µ: we show tht formule of this frgment cn be trnslted intoa nd thus the model checking problem for LTL µ nd wprs is decidble. Theorem 18. Every LTL µ formul cn be trnslted into n equivlent disjunction of α-formule. Proof. As nd re dul modlities, we cn ssume tht every LTL µ formul contins negtions only in front of ctions. Given n LTL µ formul ϕ, 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 ¾ LTL µ, then we ssume tht for every LTL µ formul ϕ ¼ shorter thn ϕ we cn construct the corresponding set A ϕ ¼. In this proof, p represents formul of LTL µ. The structure of ϕ fits into one of the following cses. p Cse p: In this cse, ϕ is equivlent to p tt. Hence A ϕ α p tt /0µ. 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 2, the set A ϕ cn be constructed from the sets A ϕ1 nd A ϕ2. Cse ϕ 1 : As α 1 α 2 µ α 1 µ α 2 µ nd α φµ αµ φµ, we set A ϕ α tt Í δbµ α δbµ ¾ A ϕ1. Cse ϕ 1 : This cse is divided into the following subcses ccording to the structure of ϕ 1.
Æp Cse p: As p is equivlent to tt p, we set A ϕ α tt p /0µ. Æ Cse ϕ 2 ϕ 3 µ: As ϕ 2 ϕ 3 µ ϕ 2 µ ϕ 3 µ, the set A ϕ cn be constructed from A ϕ 2 nd A ϕ 3 using Lemm 2. Note tht A ϕ 2 nd A ϕ 3 cn be constructed becuse ϕ 2 nd ϕ 3 re shorter thn ϕ 2 ϕ 3 µ. Æ Cse ϕ 2 : This cse is gin divided into the following subcses. p Cse p: As p ¾ LTL µ, we directly set A ϕ α tt ttpµ. Cse ϕ 3 ϕ 4 µ: As ϕ 3 ϕ 4 µ ϕ 3 µ ϕ 4 µ, we set A ϕ A ϕ 3 A ϕ 4. Cse ϕ 3 ϕ 4 µ: This cse is lso divided into subcses depending on the formule ϕ 3 nd ϕ 4. p Cse p 3 p 4 µ: As p 3 p 4 ¾ LTL µ, this subcse hs lredy been covered by Cse p. Cse ϕ 3 ϕ 5 ϕ 6 µµ: As ϕ 3 ϕ 5 ϕ 6 µµ ϕ 3 ϕ 5 µ ϕ 3 ϕ 6 µ, we set A ϕ A ϕ 3 ϕ 5 µ A ϕ 3 ϕ 6 µ. Cse ϕ 3 ϕ 5 µ: As ϕ 3 ϕ 5 µ ϕ 3 µ ϕ 5 µ, the set A ϕ cn be constructed from A ϕ 3 nd A ϕ 5 using Lemm 2. Cse ϕ 3 ϕ 5 µ: As ϕ 3 ϕ 5 µ ϕ 3 µ ϕ 5 µ, the set A ϕ cn be constructed from A ϕ 3 nd A ϕ 5 using Lemm 2. Cse ϕ 3 : As ϕ 3 ϕ 3, we set A ϕ A ϕ 1. Cse ϕ 3 : A word u stisfies ϕ 3 iff u 1 or u is n infinite word stisfying ϕ 3. Note tht tt is stisfied only by finite words of length one. Further, word u stisfies ttµ ttµ iff u is infinite. Thus, ϕ 3 ttµ ϕ ¼ where ϕ ¼ ttµ ttµ ϕ 3 µ. Hence, A ϕ A tt A ϕ ¼ where A ϕ ¼ is constructed from A tt, A tt, nd A ϕ 3 using Lemm 2. Æ Cse ϕ 2 ϕ 3 µ: According to the structure of ϕ 2 nd ϕ 3, there re the following subcses. p Cse p 2 p 3 µ: As p 2 p 3 ¾ LTL µ, this subcse hs lredy been covered by Cse p. Cse ϕ 2 ϕ 4 ϕ 5 µµ: As ϕ 2 ϕ 4 ϕ 5 µµ ϕ 2 ϕ 4 µ ϕ 2 ϕ 5 µ, the set A ϕ cn be constructed from A ϕ 2 ϕ 4 µ nd A ϕ 2 ϕ 5 µ using Lemm 2. Cse ϕ 2 ϕ 4 µ: It holds tht ϕ 2 ϕ 4 µ ϕ 2 µ ϕ 4 ϕ 2 ϕ 2 µ ϕ 4. Therefore, the set A ϕ cn be constructed s A ϕ 2 α tt Í δbµ α δbµ ¾ A ϕ4 ϕ 2 ϕ 2 A ϕ 4, where A ϕ4 ϕ 2 ϕ 2 is constructed from A ϕ4, A ϕ2, nd A ϕ 2 due to Lemm 2. Cse ϕ 2 ϕ 4 µ: There re only the following two subcses (the others fit to some of the previous cses). iµ Cse Ïϕ ¼ ¾G ϕ ¼ µ: It holds tht Ïϕ ¼ ¾G ϕ ¼ µ ttµ Ï Ë ϕ¼ ¾G ϕ ¼ µ. Therefore, the set A ϕ cn be constructed s A tt ϕ ¼ ¾Gα tt δbµ α δbµ ¾ A ϕ ¼. iiµ Cse p 2 Ï ϕ 1 ¾G ϕ 1 µ: As p 2 Ï ϕ ¼ ¾G ϕ ¼ µ p 2 µ Ï Ë ϕ¼ ¾G p 2 Í ϕ ¼ µµ, the set A ϕ cn be constructed s A p 2 ϕ ¼ ¾Gα tt p 2 ÍδBµ α δbµ ¾ A ϕ ¼.
Æ Cse ϕ 2 µ: As ϕ 2 µ ttµ ϕ 2 µ, the set A ϕ cn be constructed s A tt α tt δbµ α δbµ ¾ A ϕ 2. ÙØ As LTL µ is closed under negtion, Theorem 18 nd Corollry 17 give us the following. Corollry 19. The model checking problem for wprs nd LTL µ is decidble. This problem is EXPSPACE-hrd due to EXPSPACE-hrdness of the model checking problem for LTL µ nd PN [Hb97]. Our decidbility proof does not provide ny primitive recursive upper bound s it employs rechbility for PN (for exmple, it is used in decision procedure for rechbility for wprs [KŘS04]), for which no primitive recursive upper bound is known. 5 Model Checking for LTL È À µ This section extends the results of the previous two sections to hndle pst modlities eventully in the strict pst nd lwys in the strict pst s well. We strt with pst extension of α-formule clled Pα-formule. Intuitively, Pαformul is conjunction of n α-formul nd pst version of the α-formul. A forml definition of Pα-formul mkes use of ϕ 1 Ë ϕ 2 to bbrevite ϕ 1 ϕ 1 Ëϕ 2 µ. Definition 20. Let η ι 1 P 1 ι 2 P 2 ι m P m ι m 1, where m 0, ech ι j ¾ LTL µ, P m is À, nd, for ech j m, P j is either Ë or Ë or. Further, let α δbµ be n α-formul. Then Pα-formul is defined s Pα ηδbµ ι1 P 1 ι 2 P 2 ι m P m ι m 1 µµ α δbµ The frgment PA consists of ll finite disjunctions of Pα-formule. Note tht the definition of Pα-formul does not contin ny pst counterprt of ψ¾b ψ s every history is finite. Therefore, pointed word ukµ, where u 0 1 2, stisfies Pα ηδbµ if nd only if 0 1 k cn be written s conctention v m 1 v m v 2 v 1, where ech word v i consists only of ctions stisfying ι i nd v i 0 if i m 1 or P i is Ë, v i 0 if P i is Ë, v i 1 if P i is or À. The following lemm sys tht the frgment PA is semnticlly closed under conjunction nd ppliction of some temporl opertors. As in the cse of Lemm 2, the proof is intuitively cler but some prts re quite technicl. We refer to [Řeh07] for some hints. Lemm 21. Let ϕ be Pα-formul nd p ¾ LTL µ. Formule ϕ, ϕ, píϕ, pëϕ, ϕ, È ϕ, nd lso ny conjunction of Pα-formule cn be effectively converted into globlly equivlent disjunction of Pα-formule.
The next step is to show tht we cn decide whether given wprs system hs run stisfying given Pα-formul. The proof utilizes Corollry 17. Theorem 22. The problem whether given wprs system hs run stisfying given Pα-formul is decidble. Proof. A run over nonempty (finite or infinite) word u 0 1 2 stisfies formul ϕ iff u0µ ϕ. Moreover, u0µ Pα ηδbµ iff 0 0µ η nd u0µ α δbµ. 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 0µ η if nd only if 0 0µ ι m nd P i Ë for ll i m. Therefore, the problem is to check whether P i Ë for ll i m nd whether the given wprs system hs run stisfying ι m α δbµ. As ι m α δbµ cn be esily trnslted into disjunction of α-formule, Corollry 17 finishes the proof. ÙØ It remins to show tht every LTL È À µ formul cn be trnslted into PA formul. The proof uses the sme pproch s the one of Theorem 18: it proceeds by thorough nlysis of the structure of trnslted formul. The full proof is in Appendix A. Theorem 23. Every LTL È À µ formul ϕ cn be trnslted into globlly equivlent disjunction of Pα-formule. As LTL È À µ is closed under negtion, Theorems 23 nd 22 give us the following. Corollry 24. The model checking problem for wprs nd LTL È À µ is decidble. Moreover, we cn show tht the pointed model checking problem is decidble for wprs nd LTL È À µ s well. Agin, we solve the dul problem for Pαformule. Theorem 25. Let be wprs nd pt be rechble nonterminl stte of. The problem whether L pt µ contins pointed word uiµ stisfying given Pα-formul is decidble. Proof. Let R p 0 t 0 µ be wprs nd pt be rechble nonterminl stte of. We construct wprs ¼ R ¼ p 0 t 0 X µ where X ¾ Const µ is fresh process constnt, R ¼ R p tx µ px µ px f py µ py p ¼ t ¼ µ pt p ¼ t ¼, f ¾ Act µ is fresh ction, nd X Y ¾ Const µ re fresh process constnts for ech ¾ Act µ. Let u 0 1 2 be word. It is esy to see tht uiµ is in L pt µ iff 0 1 i 1 i f i i 1 is in L ¼ µ. Hence, for ny given Pα-formul ϕ Pα ηδbµ we construct Pα-formul ϕ ¼ Pα ηtt f δbµ. We get tht L pt µ Pα ηδbµ µ L ¼ µ Pα ηtt f δbµµ nd due to Lemm 21 nd Theorem 22 the proof is done. ÙØ