Contents. Bibliography 25

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1 Contents 1 Bisimultion nd Logic pge Introduction Modl logic nd bisimilrity Bisimultion invrince Modl mu-clculus Mondic second-order logic nd bisimultion invrince Bibliogrphy 25 1

2 1 Bisimultion nd Logic 1.1 Introduction Bisimultion is rich concept which ppers in vrious res of theoreticl computer science s this book testifies. Besides its origin by Prk [P81] s smll refinement of the behviourl equivlence originlly defined by Hennessy nd Milner between bsic concurrent processes [HM80, HM85], it ws independently, nd erlier, defined nd developed in the context of the model theory of modl logic (under the nmes of p-reltions nd zigzg reltions) by Vn Benthem [vb84] to give n exct ccount of which subfmily of first-order logic is definble in modl logic. Interestingly, to mke their definition of process equivlence more pltble, Hennessy nd Milner introduced modl logic to chrcterize it. For more detils of the history of bisimultion see Chpter [DAVIDE:HIST]. A lbelled trnsition system (LTS) is triple (Pr, Act, ), see Chpter [DA- VIDE:INTRO], where Pr is non-empty set of sttes or processes, Act is set of lbels nd (Pr Act Pr) is the trnsition reltion. As usul, we write P Q when (P,, Q). A trnsition P Q indictes tht P cn perform ction nd become Q. In logicl presenttions, there is often extr structure in trnsition system, lbelling of sttes with tomic propositions (or colours): let Prop be set of propositions with elements p, q. Formlly, this extr component is vlution, function V : Prop (Pr) tht mps ech p Prop to set V (p) Pr (those sttes coloured p). An LTS with vlution is often clled Kripke model 1. We recll the importnt definition of bisimultion nd bisimilrity, see Chpter [DAVIDE:INTRO] 2. 1 Trditionlly, Kripke model hs unlbelled trnsitions of the form P Q representing tht stte Q is ccessible to P. 2 This is reference to Volume 1. 2

3 1.1 Introduction 3 b P 3 Q 2 b Q 3 P 1 P 2 Q 1 c P 2 Q 4 c Q 5 R 1 c S 1 c c b R 2 b R 3 S 2 S 3 c c c S 4 S 5 b Fig Exmples of bisimilr nd non-bisimilr processes Definition A binry reltion R on sttes of n LTS is bisimultion if whenever P 1 R P 2 nd A, (1) for ll P 1 with P 1 P 1, there is P 2 such tht P 2 P 2 nd P 1 R P 2 ; (2) for ll P 2 with P 2 P 2, there is P 1 such tht P 1 P 1 nd P 1 R P 2. P 1 is bisimilr to P 2, P 1 P 2, if there is bisimultion R with P 1 R P 2. In the cse of n enriched LTS with vlution V there is n extr cluse in the definition of bisimultion tht it preserves colours. (0) for ll p Prop, P 1 V (p) iff P 2 V (p). Definition ssumes tht bisimultion reltion is between sttes of single LTS. Occsionlly, we lso llow bisimultions between sttes of different LTSs ( minor relxtion becuse the disjoint union of two LTSs is n LTS). Exmple In Figure 1.1, R 1 S 1 becuse the following reltion R is bisimultion {(R 1, S 1 ), (R 2, S 2 ), (R 2, S 4 ), (R 3, S 3 ), (R 3, S 5 )}. For instnce, tke the pir (R 3, S 3 ) R; we need to show it obeys the hereditry conditions of c c c Definition R 3 R 1 nd R 3 R 2 ; however, S 3 S 1 nd (R 1, S 1 ) R; c lso, S 3 S 4 nd (R 2, S 4 ) R. If this trnsition system were enriched with V (p) = {R 2, S 4 } then R 1 nd S 1 would no longer be bisimilr. Furthermore,

4 4 1 Bisimultion nd Logic P 1 Q 1 becuse P 2 cn engge in both b nd c trnsitions wheres Q 2 nd Q 4 cnnot. In the reminder of this chpter, we shll describe key reltionships between logics nd bisimultion. In Section 1.2, we exmine Henessy-Milner s modl chrcteriztion of bisimilrity. In Section 1.3 we prove vn Benthem s expressiveness result tht modl logic corresponds to the frgment of first-order logic tht is bisimultion invrint. These results re then extended in Sections 1.4 nd 1.5 to modl mu-clculus, tht is, modl logic with fixed-points, nd to the bisimultion invrint frgment of mondic second-order logic. 1.2 Modl logic nd bisimilrity Let M be the following modl logic where rnges over Act. φ ::= tt φ φ 1 φ 2 φ A formul is either the true formul tt, the negtion of formul, φ, disjunction of two formuls, φ 1 φ 2, or modl formul, φ, dimond φ. M is often clled Hennessy-Milner logic s it ws introduced by Hennessy nd Milner to clrify process equivlence [HM80, HM85]. Unlike stndrd presenttion of modl logic t tht time, such s [Ch80], it is multi-modl, involving fmilies of modl opertors, one for ech element of Act, nd it voids tomic propositions. The inductive stipultion below defines when stte P Pr of LTS L hs modl property φ, written P = L φ; however we drop the index L. P = tt P = φ iff P = φ P = φ 1 φ 2 iff P = φ 1 or P = φ 2 P = φ iff P = φ for some P with P P The criticl cluse here is the interprettion of s fter some -trnsition ; for instnce, Q 1 = b tt, where Q 1 is in Figure 1.1, becuse Q 1 Q 2 nd Q 2 = b tt. In the context of full propositionl modl logic over n enriched LTS with vlution V one dds propositions p Prop, with semntic cluse P = p iff P V (p). Other connectives re introduced s follows: flse, ff = tt, conjunction, φ 1 φ 2 = ( φ 1 φ 2 ), impliction, φ 1 φ 2 = φ 1 φ 2 nd the dul modl opertor box, []φ = φ. Derived semntic cluses for these defined connectives re s follows.

5 1.2 Modl logic nd bisimilrity 5 P = ff P = φ 1 φ 2 iff P = φ 1 nd P = φ 2 P = φ 1 φ 2 iff if P = φ 1 then P = φ 2 P = []φ iff P = φ for every P with P P So, [] mens fter every -trnsition ; for exmple P 1 = [] b tt wheres Q 1 = [] b tt, where these re in Figure 1.1, becuse Q 1 Q 4 nd Q 4 = b tt. Exercise Show the following using the inductive definition of the stisfction reltion = where the processes re depicted in Figure 1.1. (1) S 2 = []( b tt c tt) (2) S 1 = []( b tt c tt) (3) S 2 = [b][c]( tt c tt) (4) S 1 = [b][c]( tt c tt) A nturl notion of equivlence between sttes of n LTS is induced by the modl logic (with or without tomic propositions). Definition P nd P hve the sme modl properties, written P M P, if {φ M P = φ} = {φ M P = φ}. Bisimilr sttes hve the sme modl properties. Theorem If P P then P M P. Proof By structurl induction on φ M we show for ny P, P if P P then P = φ iff P = φ. The bse cse is when φ is tt which is cler (s is the cse p Prop when considering n enriched LTS). For the inductive step, there re three cses when φ = φ 1, φ = φ 1 φ 2 nd φ = φ 1, ssuming the property holds for φ 1 nd for φ 2. We just consider the lst of these three nd leve the other two s n exercise for the reder. Assume P = φ 1. So, P P 1 nd P 1 = φ 1 for some P 1. However, P P nd so P P 1 for some P 1 such tht P 1 P 1. By the induction hypothesis, if Q Q then Q = φ 1 iff Q = φ 1. Therefore, P 1 = φ 1 becuse P 1 = φ 1 nd so P = φ 1, s required. A symmetric rgument pplies if P = φ. The converse is true in the circumstnce tht the LTS is imge-finite: tht is, when the set {P P P } is finite for ech P Pr nd Act, see Chpter [DAVIDE:INTRO]. Theorem If the LTS is imge-finite nd P M P then P P.

6 6 1 Bisimultion nd Logic Proof By showing tht the binry reltion M is bisimultion. Assume P M P. If the LTS is enriched then, clerly, P = p iff P = p for ny p Prop. Assume P P 1. We need to show tht P P i such tht P 1 M P i. Since P = tt lso P = tt nd, so, the set {P i P P i } is non-empty. As the LTS is imge-finite, this set is finite, sy {P 1,..., P n}. If P 1 M P i for ech i : 1 i n then there re formuls φ 1,..., φ n of M where P 1 = φ i nd P i = φ i nd so P 1 = φ nd P i = φ for ech i when φ = φ 1... φ n. But this contrdicts P M P s P = []φ nd P = []φ. So, for some P i, 1 i n, P 1 M P i. The proof for the cse P P 1 is symmetric. Theorems nd together re known s the Hennessy-Milner Theorem, the modl chrcteriztion of bisimilrity. Modl formuls cn, therefore, be witnesses for inequivlent (imge-finite) processes; n exmple is tht [b]ff distinguishes Q 1 nd P 1 of Figure 1.1. Exercise Sets of formuls of M cn be strtified ccording to their modl depth. The modl depth of φ M, md(φ), is defined inductively: md(tt) = 0; md( φ) = md(φ); md(φ 1 φ 2 ) = mx{md(φ 1 ), md(φ 2 )}; md( φ) = md(φ)+1. Let n M men hving the sme modl properties with modl depth t most n nd recll the strtified bisimilr reltions n defined in Chpter [DAVIDE:INTRO]. Wht Hennessy nd Milner showed is P n P iff P n M P. (1) Prove by induction on n, P n P iff P n M P. (2) Therefore, show tht the restriction to imge-finite LTSs in Theorem is essentil. (3) Assume n LTS where Act is finite nd which need not be imge-finite. Show tht for ech P Pr nd for ech n 0, there is formul φ of modl depth n such tht P = φ iff P n P. (Hint: if Act is finite then for ech n 0 there re only finitely mny inequivlent formuls of model depth n.) Exercise Let M be modl logic M with rbitrry countble disjunction (nd, therefore, conjunction becuse of negtion). If Φ is countble set of formuls then Φ is formul whose semntics is: P = Φ iff P = φ for some φ Φ. Prove tht if Pr is countble set then P Q iff P M Q. Next, we identify when process hs the Hennessy-Milner property [BRV01]. Definition P Pr hs the Hennessy-Milner property iff if P M then P P. P

7 1.2 Modl logic nd bisimilrity 7 P 1 P 2 P 3 b P 4 Fig The trnsition grph for Exmple Exercise Pr is modlly sturted if for ech Act, P Pr nd Φ M if for ech finite set Φ Φ there is Q {Q P Q} nd Q = φ for ll φ Φ then there is Q {Q P Q} such tht Q = φ for ll φ Φ. Show tht if Pr is modlly sturted then ech P Pr hs the Hennessy-Milner property. (See, for instnce, [BRV01] for the notion of modl sturtion nd how to build LTSs with this feture using ultrfilter extensions.) A formul φ is chrcteristic for process P (with respect to bisimilrity) provided tht P = φ nd if P = φ then P P. An LTS is cyclic if its trnsition grph does not contin cycles; tht is, if P Pr nd P P nd P 1 2 P1... n Pn for n 0 then P P n. Proposition Assume n cyclic LTS (Pr, Act, ) where Pr, Act nd Prop re finite. If P Pr then there is formul φ M tht is chrcteristic for P. Proof Assume n cyclic LTS with finite sets Pr, Act nd Prop. For ech P Pr we define propositionl formul PROP(P ) nd for ech Act modl formul MOD(, P ). Then FORM(P ) = PROP(P ) {MOD(, P ) Act} is the chrcteristic formul for P. PROP(P ) = {p Prop P = p} { p Prop P = p} MOD(, P ) = { FORM(P ) P P } [] {FORM(P ) P P } where s usul = tt nd = ff. We need to show tht PROP(P ) is indeed well-defined nd modl formul; nd tht it is chrcteristic for P. The first depends on the fct tht the LTS is cyclic nd tht the sets Pr, Act nd Prop re finite; why? The proof tht FORM(P ) is chrcteristic for P is lso left s n exercise for the reder.

8 8 1 Bisimultion nd Logic Exmple The LTS in Figure 1.2 is cyclic with Pr = {P 1,..., P 4 }, Act = {, b} nd Prop =. Now, FORM(P 2 ) = FORM(P 4 ) = []ff [b]ff FORM(P 3 ) = FORM(P 4 ) []FORM(P 4 ) [b[ff FORM(P 1 ) = FORM(P 2 ) FORM(P 3 ) [](FORM(P 2 ) FORM(P 3 )) [b]ff Here, we construct the formuls strting from the nodes P 2 nd P 4 tht hve no outgoing trnsitions; then we construct the formul for P 3 ; nd then finlly for P 1. Exercise Give n exmple of finite-stte P such tht no formul of M is chrcteristic for P. Exercise Recll tht trce equivlence equtes two sttes P nd Q if they cn perform the sme finite sequences of trnsitions, see Chpter [DA- VIDE:INTRO]. (1) Show tht Proposition lso holds for trce equivlence. Tht is, ssume n cyclic LTS where Pr nd Act re finite nd Prop is empty. Prove tht if P Pr then there is formul φ M tht is chrcteristic for P with respect to trce equivlence. (2) Construct the chrcteristic formul for P 1 nd Q 1 of Figure 1.1. (3) OTHER EQUIVALENCES, PREORDERS? 1.3 Bisimultion invrince An lterntive semntics of modl logic emphsizes properties. Reltive to LTS nd vlution V, let φ = {P P = φ}: we cn think of φ s the property expressed by φ on the LTS. In the cse of the LTS in Figure 1.2, tt b tt = {P 1, P 3 }. Exercise Define φ on LTS directly by induction on φ (without ppeling to the stisfction reltion =). Another wy of understnding Theorem is tht properties of sttes of n LTS expressed by modl formuls re bisimultion invrint: if P φ nd P P then P φ. There re mny kinds of properties tht re not bisimultion invrint. Exmples include counting of successor trnsitions, hs 3 -trnsitions, or invoctions of finiteness such s is finite-stte or behviourl cyclicity, hs sequence of trnsitions tht is eventully cyclic :

9 P 1 P Bisimultion invrince 9 Q 1 Q 2 Q 3 Q 4 P 3 Q 1 Q 3 P 4 Fig More trnsition grphs ech of these properties distinguishes P 1 nd Q 1 in Figure 1.3 even though P 1 Q 1. The definition of invrince is neither restricted to mondic properties nor to prticulr logic within which properties of LTSs re expressed. Definition Assume Pr n is (Pr... Pr) n-times, n 1. (1) An nry property, n 1, of LTS is set Γ Pr n. (2) Property Γ Pr n is bisimultion invrint if whenever (P 1,..., P n ) Γ nd P i P i for ech i : 1 i n, then lso (P 1,..., P n) Γ. Exercise (1) Prove tht the property {(P, Q) P, Q re trce equivlent} is bisimultion invrint. More generlly, show tht if is behviourl equivlence between processes such tht P Q implies P Q, then is bisimultion invrint. (2) A property Γ Pr n is sfe for bisimultion if whenever (P 1,..., P n ) Γ nd P 1 P 1 then (P 1,..., P n) Γ for some P 2,..., P n ( notion due to vn Benthem [vb98]). Show tht the generl trnsition reltions, w w Act *, Chpter [DAVIDE:INTRO] re sfe for bisimultion. (3) Show tht if Γ is bisimultion invrint then it is sfe for bisimultion. Another logic within which to express properties of LTS is first-order logic, FOL. It hs countble fmily of vribles Vr typiclly represented s x, y, z nd binry reltion E for ech Act (nd mondic predicte p for ech p Prop when the LTS is enriched). Formuls of FOL hve the following form. φ ::= p(x) xe y x = y φ φ 1 φ 2 x. φ To interpret formuls with free vribles we need vlution σ : Vr Pr tht ssocites stte with ech vrible. Also, we use stndrd updting

10 10 1 Bisimultion nd Logic nottion: σ{p 1 /x 1,..., P n /x n } is the vlution tht is the sme s σ except tht its vlue for x i is P i, 1 i n (where ech x i is distinct). We inductively define when FOL formul φ is true on n LTS L with respect to vlution σ s σ = L φ, where gin we drop the index L. σ = p(x) iff σ(x) V (p) σ = xe y iff σ(x) σ(y) σ = x = y iff σ(x) = σ(y) σ = φ iff σ = φ σ = φ 1 φ 2 iff σ = φ 1 or σ = φ 2 σ = x. φ iff σ{p/x} = φ for some P Pr The universl quntifier, the dul of x, is introduced s x. φ = φ. Its derived semntic cluse is: σ = x. φ iff σ{p/x} = φ for ll P Pr. Exmple Assume σ(x 1 ) = P 1 nd σ(x 2 ) = Q 1 of Figure 1.3. Then the following pir hold. (1) σ = x. y. z. (x 1 E x x 1 E y x 1 E z x y x z y z) (2) σ = y. z. (x 2 E y ye z z x 2 ) There is recognized trnsltion of modl formuls into first-order formuls, for instnce, see [BRV01]. Definition The FOL trnsltion of modl formul φ reltive to vrible x is T x (φ) which is defined inductively. T x (p) = p(x) T x (tt) = x = x T x ( φ) = T x (φ) T x (φ 1 φ 2 ) = T x (φ 1 ) T x (φ 2 ) T x ( φ) = y. xe y T y (φ) Exercise (1) For ech of the following formuls φ, present its FOL trnsltion T x (φ). () [] b tt (b) p [] p (c) []([]p p) []p (2) FOL 2 is first-order logic when Vr is restricted to two vribles {x, y} which cn be reused in formuls. Show tht modl formuls cn be trnslted into FOL 2.

11 1.3 Bisimultion invrince 11 The trnsltion of modl formuls into FOL, Definition 1.3.5, is clerly correct s it imittes the semntics. Proposition P = φ iff σ{p/x} = T x (φ) Proof By structurl induction on φ M. For the bse cses, first P = p iff P V (p) iff σ{p/x} = p(x) iff σ{p/x} = T x (p). Similrly, for the other bse cse, P = tt iff σ{p/x} = x = x iff σ{p/x} = T x (tt). For the inductive step we only exmine the interesting cse when φ = φ 1. P = φ iff P = φ 1 for some P where P P iff σ{p /z} = T z (φ 1 ) for some P where P P, by the induction hypothesis, iff σ{p/x} = z. xe z T z (φ 1 ) iff σ{p/x} = T x (φ). A FOL formul with free vribles is bisimultion invrint if the property it expresses is bisimultion invrint. Definition Formul φ FOL whose free vribles belong to {x 1,..., x n } is bisimultion invrint if {(P 1,..., P n ) σ{p 1 /x 1,..., P n /x n } = φ} is bisimultion invrint. Corollry Any first-order formul T x (φ) is bisimultion invrint. Not ll first-order formuls re bisimultion invrint. The two formuls in Exmple refexmp3 re cses; the first sys tht x 1 hs t lest three different -trnsitions. Vn Benthem introduced the notion of bisimultion (s p- reltion nd zig-zg reltion) to identify which formuls φ(x) FOL with one free vrible re equivlent to modl formuls [vb96]. Definition A FOL formul φ(x) is equivlent to modl φ M provided tht for ny LTS nd for ny stte P, σ{p/x} = φ iff P = φ. Vn Benthem proved Proposition , FOL formul φ(x) is equivlent to modl formul iff it is bisimultion invrint. The proof utilises some model theory. Some nottion first: if Φ is set of first-order formuls then Φ = ψ provided tht for ny LTS nd vlution σ, if for ll φ Φ, σ = φ then σ = ψ. The compctness theorem for first-order logic sttes tht if Φ = ψ then there is finite set Φ Φ such tht Φ = ψ. Next we stte further property of first-order logic tht will lso be used. Fct If Φ is set of first-order formuls ll of whose free vribles belong to {x 1,..., x n } nd σ{p 1 /x 1,..., P n /x n } = φ for ll φ Φ, then there is LTS nd processes P 1,..., P n Pr such tht σ{p 1 /x 1,..., P n/x n } = φ for ll φ Φ nd ech P i hs the Hennessy-Milner property (Definition 1.2.7).

12 12 1 Bisimultion nd Logic Proposition A FOL formul φ(x) is equivlent to modl formul iff φ(x) is bisimultion invrint. Proof If φ(x) is equivlent to modl formul φ then {P σ{p/x} = φ} = φ which is bisimultion invrint. For the converse property, ssume tht φ(x) is bisimultion invrint. Consider the following fmily Φ = {T x (ψ) ψ M nd {φ(x)} = T x (ψ)}. We prove Φ = φ(x) nd, therefore, by the compctness theorem, φ(x) is equivlent to modl formul ψ such tht T x (ψ ) Φ. Assume σ{p/x} = ψ for ll ψ Φ. We need to show tht σ{p/x} = φ. We choose P with the Hennessy-Milner property by Fct Let Ψ = {T x (ψ) P = ψ}. First, Φ Ψ. Next we show tht Ψ {φ} is stisfible. For suppose otherwise, Ψ = φ nd so by the compctness theorem there is finite subset Ψ = {T x (ψ 1 ),..., T x (ψ k )} Ψ such tht Ψ = φ. But then φ = T x (ψ ) where ψ is the modl formul ψ 1... ψ k nd so T x (ψ ) Φ which contrdicts tht Φ Ψ. Therefore, for some Q, σ{q/x} = ψ for ll ψ Ψ nd σ{q/x} = φ. However, Q P nd becuse φ is bisimultion invrint, σ{p/x} = φ s required. Exercise Prove tht FOL formul φ(x 1,..., x n ) is bisimultion invrint iff it is equivlent to boolen combintion of formuls of the following form T x1 (ψ 11 ),..., T x1 (ψ 1k1 ),..., T xn (ψ n1 ),..., T xn (ψ nkn ) for some k 1,..., k n 0. An lterntive proof of Proposition ppels to tree (or forest) models. A LTS is forest if it is cyclic nd the trget of ech trnsition is unique; if P Q nd R b Q then P = R nd = b. The trnsition grph tht is rooted t Q 1 in Figure 1.3 is tree ( forest with single tree). Given LTS there is wy of unfolding P Pr nd ll its rechble processes into tree rooted t P which is clled unrvelling. Definition Assume LTS L = (Pr, Act, ) with P 0 Pr. The k- unrvelling of P 0, for k 0, is the following LTS, L k = (Pr k, Act, k ) where (1) Pr k = {P 0 1 k 1 P 1... n k n P n n 0, 0 k i k, P 1 0 P1... Pn }; (2) if P P nd P is the finl stte in π Pr k then π k πk P for ech 0 k k; (3) if V is the vlution for L then V k is the vlution for L k where V k (p) = {π Pr k P is finl in π nd P V (p)}. The ω-unrvelling of P 0, the LTS L ω, permits ll indices k 0: so, Pr ω includes ll sequences P 0 1 k 1 P 1... n k n P n such tht P 0 1 P1... n Pn nd ech k i 0. n

13 1.4 Modl mu-clculus 13 π 1 π 3 π 1 π 2 π 3 π 4 π 1 π 3 Fig Unrvelled LTS Exmple The 0-unrvelling of P 1 of Figure 1.3 is presented in Figure 1.4 where π 1 = P 1, π 2(i+1) = π 2i+1 0P 1, π 2i+1 = π 2i 0P 1 π 2i+1 = π 2i+10P 3 nd π 2i+1 = π 2i+10P 4. The reder is invited to describe the 2-unrvelleing nd the ω-unrvelling of P 1. Proposition For ny LTS nd k : 0 k ω, if P Pr nd π Pr k nd the finl stte in π is P, then P π. Proof Clerly, the binry reltion R Pr Pr k contining ll pirs (P, π) when the finl stte of π is P is bisimultion becuse first, P V (p) iff π V k (p) nd second, P P iff π k πk P for k k. Exercise Let R 1 nd S 1 be the processes depicted in Figure 1.1. (1) Define the 0-unrvellings of R 1 nd S 1. (2) Define the ω-unrvelling of R 1 nd S 1 nd show tht they re isomorphic. (3) Assume L is LTS contining P nd Q nd P Q. Show tht the ω-unrvellings of P nd Q re isomorphic. (4) Reprove Proposition using ω-unrvelled LTSs. 1.4 Modl mu-clculus Modl logic M of Section 1.2 is not very expressive. For instnce, temporl properties of sttes of LTS, such s liveness, this desirble property will eventully hold, nd sfety, this defective property never holds, re not expressible in M. (Prove this; hint, use Exercise ) Such properties hve been found to be very useful when nlysing the behviour of concurrent systems. (Reference to Chpter [DAVIDE:INTRO]?) Modl mu-clculus, µm, modl logic with fixpoints, introduced by Kozen [Ko83], hs the required extr expressive power.

14 14 1 Bisimultion nd Logic The setting for µm is the complete lttice generted by the powerset construction (Pr) where the ordering is, join is union nd meet is intersection, is the bottom element nd Pr is the top element, see Chpter [DAVIDE:INTRO]. Exercise Consider LTS nd recll the definitions of monotone nd continuous function f on the powerset (Pr): f is monotone provided tht if S S then f(s) f(s ); f is continuous just in cse if S 1,..., S n,... is n incresing sequence of subsets of Pr, (tht is, if i j then S i S j Pr), then f( i S i) = i f(s i); see Chpter [DAVIDE:INTRO]. (1) Define the semntic functions nd [] on (Pr) such tht for ny φ M, φ = φ nd [] φ = []φ. (2) Show tht these functions nd [] re monotone. (3) Prove tht is continuous iff the LTS is imge-finite with respect to the lbel ; tht is, if for ech P Pr, the set {P P P } is finite. The new constructs of µm over nd bove those of M re φ ::= X... µx. φ where X rnges over fmily of propositionl vribles. The semntics for formul φ of µm is the set φ V Pr where V is vlution tht not only mps elements of Prop but lso propositionl vribles to (Pr). As usul we employ updting nottion: φ V {S/X} uses vlution V like V except tht V (X) = S. Exercise Assume tht φ is formul of M when extended with propositionl vribles. Prove tht if ll free occurrences of X in φ re within the scope of n even number of negtions nd V is vlution then the function f : (Pr) (Pr) such tht f(s) = φ V {S/X} is monotone. Therefore, show the following (see Chpter [DAVIDE:INTRO]) (1) the lest fixed point lfp(f) exists nd is the intersection of ll pre-fixed points, {S f(s) S}; (2) the gretest fixed point gfp(f) exists nd is the union of post-fixed points, {S S f(s)}. In the cse of µx. φ there is, therefore, the restriction tht ll free occurrences of X in φ re within the scope of n even number of negtions (to gurntee monotonicity). This formul expresses the lest fixed point lfp of the semntic function induced by φ. Its dul, νx. φ, expresses the gretest fixed point gfp nd is derived construct in µm: νx. φ = µx φ{ X/X}. Here re the semntics for µm formuls.

15 1.4 Modl mu-clculus 15 p V = V (p) Z V = V (Z) φ V = Pr φ V φ 1 φ 2 V = φ 1 V φ 2 V φ V = { P Pr for some Q. P Q nd Q φ V } µz. φ V = {S Pr φ V {S/Z} S} Exercise Extend the first prt of Exercise by proving tht if ll free occurrences of X in φ µm re within the scope of n even number of negtions nd V is vlution then the function f on (Pr) such tht f(s) = φ V {S/X} for S Pr is monotone. Derived semntic cluses for other connectives re below. φ 1 φ 2 V = φ 1 V φ 2 V []φ V = {P Pr for ll Q. if P Q then Q φ V } νz. φ V = {S Pr S φ V {S/Z} } P stisfies the µm formul φ reltive to vlution V, P = V φ, iff P φ V ; s usul we omit V wherever possible. The stndrd theory of fixpoints tells us, see Chpter [DAVIDE:INTRO], tht if f is monotone function on lttice, we cn construct lfp(f) by pplying f repetedly on the lest element of the lttice to form n incresing chin, whose limit is the lest fixed point. Similrly, gfp(f) is constructed by pplying f repetedly on the lrgest element to form decresing chin, whose limit is the gretest fixed point. The stges of these itertions µ α X. φ nd ν α X. φ cn be be defined s M formuls, see Exercise 1.2.6, inductively s follows. µ 0 X. φ = ff ν 0 X. φ = tt µ β+1 X. φ = φ{µ β X. φ/x} ν β+1 X. φ = φ{ν β X. φ/x} µ λ X. φ = β<λ µβ X. φ ν λ X. φ = β<λ νβ X. φ So for miniml fixpoint formul µx. φ, if P stisfies the fixpoint, it stisfies some iterte, sy the β + 1 th so tht P = µ β+1 X. φ. Now if we unfold this formul once, we get P = φ{µ β X. φ/x}. Therefore, the fct tht P stisfies the fixpoint depends, vi φ, on the fct tht other sttes in Pr stisfy the fixpoint t smller itertes thn P does. So if one follows chin of dependencies, the chin

16 16 1 Bisimultion nd Logic termintes. Therefore, µ mens finite looping, which, with little refinement, is sufficient to understnd the logic µm. On the other hnd, for mximl fixpoint νx. φ, there is no such decresing chin: P = νx. φ iff P = ν β X. φ for every iterte β iff P = φ{ν β X. φ/x} for every iterte β iff P = φ{νx. φ/x}, nd so we my loop for ever. Exmple Assume P 1 is the process in Figure 1.3, which cn repetedly do n trnsition. P 1 fils to hve the property µx. []X (which expresses tht there cnnot be n infinite sequence of trnsitions). Consider its itertes, µ 1 X. []X = []ff, so P 3 nd P 4 hve this property; µ 3 X. []X is [][][]ff nd µ ω X. []X is n 0 []n ff where [] 0 ff = ff nd [] i+1 ff = [][] i ff. Consequently, P 1 = νx. X. Itertes of this formul include ν ω X. X = n 0 n tt where i is i-times. Exercise Wht properties re expressed by the following formuls? (1) µx. p []X (2) µx. q (p X) (3) νx. p []X (4) µx. νy. (p []X) ( p []Y ) Definition of M, hving the sme modl properties, is extended to µm; so, P µm P mens P nd P hve the sme µm properties, s expressed by closed formuls of µm (tht is, formuls without free vribles). Bisimilr sttes hve the sme µm properties. Theorem If P P then P µm P. Proof The proof of this uses Exercise tht M chrcterizes bisimilrity nd the observtion bove tht closed formuls of µm cn be trnslted into M. Theorem If the LTS is imge-finite nd P µm P then P P. Proof Becuse µm contins M this follows directly from Theorem Is imge-finiteness still necessry in Theorem 1.4.7? In Exercise the reltionship between strtified bisimilrity, n, nd formuls of M with modl depth n is explored. It is possible P Q but P n Q for ll n 0 nd so, P M Q. For instnce, let P be i 0 P i nd Q = P + R where P j+1 P j, P 0 hs no trnsitions nd R R. Unlike P, Q hs n infinite sequence of

17 1.4 Modl mu-clculus 17 trnsitions: so, P µm Q (becuse P = µx. []X). So, more sophisticted exmple is needed for the presence of imge-finiteness. Exmple The following exmple is from [BS07]. It uses key property of µm, the finite model property : if P = φ then there is finite LTS nd P within it with P = φ. Let φ 1, φ 2,... be n enumertion of ll closed µm formuls over the finite lbel set {, b} tht re true t some stte of some LTS. Let Pr i, with initil stte P i, be finite LTS such tht P i = φ i, with ll Pr i disjoint. Let Pr 0 be constructed by tking n initil stte P 0 nd mking P 0 P i for ll i > 0. Similrly, let Pr 0 be constructed from initil stte P 0 with trnsitions P 0 P i for ll i > 0 nd P 0 P 0. Clerly, P 0 P 0 becuse in Pr 0 it is possible to defer indefinitely the choice of which Pr i to enter. On the other hnd, suppose tht ψ is closed µm formul, nd w.l.o.g. ssume the topmost opertor is modlity. If the modlity is [b], ψ is true of both P 0 nd P 0 ; if it is b, ψ is flse of both; if ψ is ψ, then ψ is flse t both P 0 nd P 0 iff ψ is unstisfible, nd true t both otherwise; if ψ is []ψ, then ψ is true t both P 0 nd P 0 iff ψ is vlid, nd flse t both otherwise. Consequently, P 0 µm P 0. Definition cn be extended to µm formuls: P Pr hs the extended Hennessy-Milner property provided tht if P µm P then P P. Little is known bout this property except tht, if P hs the Hennessy-Milner property then it lso hs the extended Hennessy-Milner property. Exercise In Exercise modlly sturted LTS ws defined. This notion does not redily extend to µm formuls. A set of µm formuls is unstisfible if there is not LTS nd process P belonging to it such tht P stisfies every formul in the set. Show tht there is n unstisfible set Φ µm such tht every finite subset Φ Φ is stisfible. Show tht this is equivlent to showing tht µm fils the compctness theorem. Another indiction tht µm is more expressive thn M is tht it contins chrcteristic formuls with respect to bisimilrity for finite-stte processes. So, the restriction to cyclic LTSs in Proposition cn be relxed. Proposition Assume (Pr, Act, ) where Pr, Act nd Prop re finite. If P Pr then there is formul φ µm tht is chrcteristic for P. Proof Let (Pr, Act, ) be LTS with finite sets Act, Prop nd Pr. Assume we wnt to define chrcteristic formul for P Pr. Let P 1,..., P n be the distinct elements of Pr with P = P 1 nd let X 1,..., X n be distinct propositionl

18 18 1 Bisimultion nd Logic vribles. We define modl eqution X i = φ i (X 1,..., X n ) for ech i which cptures the behviour of P i. X(i) = PROP(P i ) {MOD (, P ) Act} where PROP(P i ) = {p Prop P = p} { p Prop P = p} MOD (, P i ) = { X j P i P j } [] {X j P i P j } where s usul = tt nd = ff. We now define the chrcteristic formul for P 1 s ψ 1 where ψ n = νx n. φ n (X 1,..., X n ).. ψ j = νx j. φ j (X 1,..., X j, ψ j+1,..., ψ n ).. ψ 1 = νx 1. φ 1 (X 1, ψ 2,..., ψ n ) The proof tht ψ 1 is chrcteristic for P is left s n exercise for the reder. Exmple Let R 1, R 2 nd R 3 be the processes in Figure 1.1 nd ssume Prop =. The modl equtions re s follows. X 1 = φ 1 (X 1, X 2, X 3 ) = ( X 2 X 3 ) [](X 2 X 3 ) [b]ff [c]ff X 2 = φ 2 (X 1, X 2, X 3 ) = []ff b X 3 [b]x 3 [c]ff X 3 = φ 3 (X 1, X 2, X 3 ) = []ff [b]ff c X 1 c X 2 [c](x 1 X 2 ) So, ψ 3 is νx 3. φ 3 (X 1, X 2, X 3 ), nd ψ 2 is νx 2. φ 2 (X 1, X 2, ψ 3 ) nd ψ 1 is the following formul νx 1. ( ψ 2 ψ 3 ) [](ψ 2 ψ 3 ) [b]ff [c]ff The reder cn check tht S 1 = ψ 1 where S 1 is lso in Figure 1.1. Exercise Provide chrcteristic formul for P 1 of Figure 1.3 nd show tht Q 1 in the sme figure stisfies it. The proof of Proposition shows tht chrcteristic formul for finite stte process only uses gretest fixpoints. Furthermore, there is more succinct representtion if simultneous fixpoints re llowed 1. One ppliction of chrcteristic formuls is the reduction of equivlence checking (whether two given processes re equivlent) to model checking (whether given process hs given property). This is especilly useful in the cse when only one of the two 1 Insted of defining ψ i itertively in the proof of Proposition , they re defined t the sme time in vectoril form.

19 1.5 Mondic second-order logic nd bisimultion invrince 19 given processes is finite stte, see [KJ06] for survey of known results which lso covers wek bisimilrity nd preorder checking. A simple corollry of Theorem is tht µm hs the tree model property. If µm formul hs model, it hs model tht is tree. Just 0-unrvel, see Definition , the originl model, thereby preserving bisimultion. This cn be strengthened to the bounded brnching degree tree model property (just cut off ll the brnches tht re not ctully required by some dimond subformul; this leves t most (number of dimond subformuls) brnches t ech node). Clerly we cnnot trnslte µm into FOL becuse of the fixpoints. (See Exercise ) However, it cn be trnslted into mondic second-order logic. 1.5 Mondic second-order logic nd bisimultion invrince MSO, mondic second-order logic of LTSs, extends FOL in Section 1.3 by llowing quntifiction over subsets of Pr. The new constructs over nd bove those of FOL re φ ::= X(x)... X. φ where X rnges over fmily of mondic predicte vribles, nd X. φ quntifies over such predictes. To interpret formuls with free predicte nd individul vribles we extend vlution σ to include mpping from predicte vribles to sets of sttes. We inductively define when MSO formul φ is true on n LTS L with respect to vlution σ s σ = L φ, where gin we drop the index L. The new cluses re s follows. σ = X(x) iff σ(x) σ(x) σ = X. φ iff σ{s/x} = φ for some S Pr The universl mondic quntifier, the dul of X, is X. φ = X φ. derived semntic cluse is: σ = X. φ iff σ{s/x} = φ for ll S Pr. Exmple Given LTS with Act = {} the property tht it is three colourble is expressible in MSO s follows X. Y. Z. x. φ(x, X, Y, Z) y. z. ψ(y, z, X, Y, Z) where φ(x, X, Y, Z) expresses x hs unique colour X, Y or Z (X(x) Y (x) Z(x)) ( X(x) Y (x) Z(x)) ( X(x) Y (x) Z(x)) nd ψ(y, z, X, Y, Z) confirms tht if there is n trnsition from y to z then they re not coloured the sme ye z (X(y) X(z)) (Y (y) Y (z)) (Z(y) Z(z)) Its

20 20 1 Bisimultion nd Logic There is trnsltion of µm formuls into MSO tht extends Definition Definition The MSO trnsltion of µm formuls φ reltive to vrible x is T + x (φ) which is defined inductively. T x + (p) = p(x) T x + (X) = X(x) T x + (tt) = x = x T x + ( φ) = T x + (φ) T x + (φ 1 φ 2 ) = T x + (φ 1 ) T x + (φ 2 ) T x + ( φ) = y. xe y T y + (φ) T x + (µx. φ) = X. ( y. (T y + (φ) X(y))) X(x) The trnsltion of lest fixpoint formul uses quntifiction nd impliction to cpture tht x belongs to every pre-fixed point. Exercise For ech of the following formuls φ, present its MSO trnsltion T + x (φ). (1) µx. p []X (2) µx. q (p X) (3) νx. p []X (4) µx. νy. (p []X) ( p []Y ) The trnsltion of µm formuls into MSO, Definition 1.5.2, is correct. Proposition If for ech vrible Z, V (Z) = σ(z) then P = V φ iff σ{p/x} = T x + (φ). Proof By structurl induction on φ M. The proofs for the modl nd boolen cses follow Proposition There re just the two new cses. P = V X iff P V (X) iff P σ(x) iff σ{p/x}(x) σ{p/x}(x) iff σ{p/x} = T x + (X). P = V µx. φ iff for ll S, if φ V {S/X} S then P S iff for ll S, if y, y = V {S/X} φ implies y S then P S iff for ll S, if y, σ{s/x} = T y + (φ) by the induction hypothesis where σ obeys tht for ll Z, σ(z) = V (Z) iff σ{p/x} = X. ( y. (T y + (φ) X(y))) X(x) iff σ{p/x} = T x + (µx. φ). A corollry of Theorem is tht if φ is closed µm formul then the MSO formul ψ(x) = T + x (φ) with one free vrible is bisimultion invrint. As with FOL there re formuls of MSO which re not bisimultion invrint.

21 1.5 Mondic second-order logic nd bisimultion invrince 21 Therefore, it is nturl to sk the question whether vn Benthem s theorem, Proposition , cn be extended to MSO formuls. The following result ws shown by Jnin nd Wlukiewicz [JW96]. Proposition A MSO formul φ(x) is equivlent to closed µm formul iff φ(x) is bisimultion invrint. However, its proof utilizes utomt (nd gmes) which we shll provide flvour of. The im is now to think of different chrcteriztion of logics on LTSs using utomt or gmes which operte loclly on the LTS (compre Chpter [DA- VIDE:INTRO]). A prticulr logicl formul of MSO or µm cn only mention finitely mny different elements of Prop nd finitely mny different elements of Act; therefore, we ssume now tht these sets re finite in ny given LTS. They will constitute finite lphbets for utomt; let Σ 1 = Act nd Σ 2 = Prop. Let us begin with the notion of n utomton fmilir from introductory computer science courses. Definition An utomton A = (S, Σ, δ, s 0, F ) consists of finite set of sttes S, finite lphbet Σ, trnsition function δ, n initil stte s 0 S nd n cceptnce condition F. Trditionlly, A does not operte on LTSs but on words, recognizing lnguge, subset of Σ *. Assuming A is nondeterministic, its trnsition function δ : S Σ S. Given word w = 1... n Σ *, run of A on w is sequence of sttes s 0... s n tht trverses w, so s i+1 δ(s i, i+1 ) for ech i : 0 i < n. The run is ccepting if the sequence s 0... s n obeys F : clssiclly, F S is the subset of ccepting sttes nd s 0... s n is ccepting if the lst stte s n F. There my be mny different runs of A on w, some ccepting the others rejecting, or no runs t ll. The lnguge recognized by A is the set of words for which there is t lest one ccepting run. Exmple Let A = ({s 0, s 1 }, {}, δ, s 0, {s 0 }) with δ(s 0, ) = {s 1 } nd δ(s 1, ) = {s 0 }. The lnguge ccepted by A is the set { 2n n 0} of even length words. A simple extension is recognition of infinite length words. A run of A on w = 1... i... is n infinite sequence of sttes π = s 0... s i... tht trvels over w, so s i+1 δ(s i, i+1 ), for ll i 0; it is ccepting if it obeys the condition F. Let inf (π) S contin exctly the sttes tht occur infinitely often in π. Clssiclly, F Q nd π is ccepting if inf (π) F which is the Büchi cceptnce condition.

22 22 1 Bisimultion nd Logic Büchi utomt re n lterntive nottion for chrcterizing infinite pths of LTS. There re different choices ccording to the lphbet Σ. If Σ = Σ 1 nd π = P P1... is n infinite sequence of trnsitions, then π = A if the utomton ccepts the word ; lterntively, Σ = Σ 2 nd π = A if it ccepts Prop(P 0 ) Prop(P 1 )... where Prop(P ) is the subset of Prop tht is true t P. Exercise Let Prop = {p}, S = {s, t}, δ(s, {p}) = {t}, δ(s, ) = {s}, δ(t, {p}) = {t} nd δ(t, ) = {t}, s 0 = s nd F = {t}. Wht property of n infinite run of LTS does this Büchi utomton express? When ech formul of logic is equivlent to n utomton, stisfibility checking reduces to the non-emptiness problem for those utomt: whether n utomton ccepts some word (pth or whtever). This my hve lgorithmic benefits in reducing n pprently complex stisfibility question into simple grph-theoretic procedures: Büchi utomton, for instnce, is non-empty if there is pth s 0 * s F nd cycle s * s (equivlent to n eventully cyclic model). Indeed the introduction of Büchi nd Rbin utomt ws for showing decidbility of mondic second-order theories by reducing them to utomt, see the tutoril text [GTW02] for detils. The ide of recognizing bounded brnching trees extends the definition of A to ccept n-brnching infinite trees. With word utomton, stte s belonged to δ(s, ); now it is tuples (s 1,..., s n) tht belong to δ(s, ). A tree utomton trverses the tree, descending from node to ll n-child nodes, so the utomton splits itself into n copies, nd proceeds independently. A run of the utomton is then n n-brnching infinite tree lbelled with sttes of the utomton. A run is ccepting if every pth through this tree stisfies the cceptnce condition F. In the cse of Rbin cceptnce F = {(G 1, R 1 ),..., (G k, R k )} where ech G i, R i S nd π obeys F if there is j such tht inf (π) G j nd inf (π) R j =. A vrint definition is prity cceptnce where F mps ech stte s of the utomton to priority F (s) N. We sy tht pth stisfies F if the lest priority seen infinitely often is even. It is not hrd to see tht prity condition is specil cse of Rbin condition; it is lso true, though somewht trickier, tht Rbin utomton cn be trnslted to n equivlent prity utomton. Such utomt cn recognize bounded brnching unrvellings of LTSs. Exercise Tree utomt chrcterize rooted n-brnching infinite tree LTS models for µm formuls. Such model L = A if A ccepts the behviour tree tht replces ech stte P Pr with Prop(P ). Let Prop = {p}, S = {s, t}, δ(s, {p}) = {(s, s)}, δ(s, ) = {(t, t)}, δ(t, {p}) = {(s, s)} nd δ(t, ) = {(t, t)} nd s 0 = s. This utomton A hs prity cceptnce condition F (s) = 1 nd

23 1.5 Mondic second-order logic nd bisimultion invrince 23 F (t) = 2. Wht µm formul is equivlent to A over infinite binry-tree models? (Hint: wht fixpoints re coded by sttes s nd t?) There is slight mismtch between (the unrvellings of) LTSs nd bounded brnching trees becuse of the fixed brnching degree nd the explicit indexed successors; for instnce, see the unrvelled LTS of Figure 1.4. Wht is wnted is n utomton tht cn directly recognize LTS nd which preserves the virtue of simple locl definition of trnsition function. We shll define vrint of lternting prity utomt which is due to Wlukiewicz (lso see [KVW00]). The rnge of trnsition function of n utomton A will be locl formul. For word utomton, if δ(s, ) = {s 1,..., s m } then it is the formul s 1... s m. For n-brnching tree utomton if δ(s, ) = {(s 1 1,..., s1 n),..., (s m 1,..., sm n )} then it is ((1, s 1 1 )... (n, s1 n))... ((1, s m 1 )... (n, sm n )): here the element (i, s ) mens crete n ith-child with lbel s. A word or tree is ccepted if there exists n ccepting run for tht word or tree; hence, the disjuncts. However, for tree, every pth through it must be ccepting; hence the conjuncts. In lternting word utomt, the trnsition function is given s n rbitrry boolen expression over sttes: for instnce, δ(s, ) = s 1 (s 2 s 3 ). In lternting tree utomt it is boolen expression over directions nd sttes: for instnce, ((1, s 1 ) (1, s 2 )) (2, s 3 ). Now the definition of run becomes tree in which, successor trnsitions obey the boolen formul. In prticulr, even for n lternting utomton on words, run is tree, nd not just word. The cceptnce criterion is s before, tht every pth of the run must be ccepting. An lternting utomton is just two plyer gme too where one plyer is responsible for choices nd the other plyer for choices. The trnsition function for n utomton A tht recognises LTSs hs the form δ : S Σ 2 Φ(Σ 1, S) where Φ(X, Y ) is set of formuls over X nd Y. One ide is tht this formul could be simple modl formul. For instnce, if s is the current utomton stte t P Pr nd δ(s, Prop(P )) = s 1 [c]s 2 nd P P 1, P b P 2, P c Q i, for ll i 0 then the utomton moves to P 1 with stte s 1 nd to ech Q i with stte s 2. As with tree utomt, run of A on LTS is lbelled tree of rbitrry degree. Such modl utomt when the cceptnce condition for infinite brnches is the prity condition hve the sme expressive power s µm. However, to prove Proposition Jnin nd Wlukiewiciz use FOL formuls. The ide for tomic predictes is to replce pirs (i, s) of tree utomton with elements of U = {(, s) Σ 1 nd s S}. Now, for ech s S nd W Prop, δ(s, W ) is formul of the form (*) x 1... x k. (u 1 (x 1 )... u k (x k )) x. (u 1 (x)... u k (x))

24 24 1 Bisimultion nd Logic where ech u i U. An exmple, is φ = x 1. x 2. (, s)(x 1 ) (b, s )(x 2 ) x. (, s)(x) (b, s )(x). If t lbels the stte P of the LTS nd W = Prop(P ) nd δ(t, W ) = φ nd P P i, P b Q j, i, j > 0 then the utomton t the next step would spwn copy t ech P i with stte s nd ech Q j with stte s. Notice tht such formul is quite similr to the components MOD (, P ) of chrcteristic formul described in Proposition Every µm formul is equivlent to such n utomton; the different kinds of fixpoint re ctered for in the prity cceptnce condition. Let dis(x 1,..., x n ) be the FOL formul 1 i<j n x i x j. There is very similr chrcteriztion of MSO formuls over trees. where now ech δ(s, W ) hs the form (**) x 1... x k. (D u 1 (x 1 )... u k (x k )) x. D (u 1 (x)... u k (x)) where D = dis(x 1,..., x k ) nd D = dis(x, x 1,..., x k ). Now the result follows: if φ(x) is n MSO formul tht is bisimultion invrint then it is true on ny n-unrvelled model nd so (*) nd (**) will be equivlent for n k.

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