Hennessy-Milner Logic 1.

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1 Hennessy-Milner Logic 1. Colloquium in honor of Robin Milner. Crlos Olrte. Pontifici Universidd Jverin 28 April Bsed on the tlks: [1,2,3]

2 Prof. Robin Milner (R.I.P). LIX, Ecole Polytechnique.

3 Motivtion How to Verify the Correctness of Concurrent System? By Using Equivlences impl spec is n equivlence (e.g., bisimultion) The specifiction nd the implementtion re written in the sme lnguge, e.g., CCS. Spec provides the full specifiction of the intended behvior.

4 Motivtion How to Verify the Correctness of Concurrent System? Model Checking Approch impl = Property = is the stisfction reltion. Property is prtil specifiction of the intended behvior. Property is formul in logic.

5 Motivtion Specifiction of Properties Henessy-Milner Logic : Modl logic to express properties of rective systems. Modlities: Necessity nd Possibility. The ction cnnot hppen. After n ction, the systems cn perform n ction b. After n ction, the system never exhibits b ction. More Exmples: A coffee is given fter coin is inserted. After coin is inserted, either coffee or te re dispensed.

6 Bckgroud Lbelled Trnsition Systems (LTS) A LTS is triple S, A, T where S is set of sttes. A is set of ctions (e.g.,,, b, c, τ,...). T S L S is the trnsition reltion: s 1 s2 mens (s 1,, s 2 ) T p r b,c s d d e q

7 Bckgroud Bisimultion Given two LTSs, reltion R S S is clled bisimultion whenever: If (p, q) R nd p p then there exists q s.t. q q nd (p, q ) R. If (q, p) R nd q q then there exists p s.t. p p nd (q, p ) R. q p r t u Bisimilrity is the finest resonble equivlence, where resonble mens tht we cn observe only the behvior nd not the stte-spce.

8 Two systems tht re not bisimilr p q c b r 1 r 2 S1 =.(b + c) p q 1 q 2 b c r 1 r 2 S2 =.b +.c S1 S2 Notice tht in S 1 : p q b r 1 while in S 2 : p q 2 b

9 Henessy-Milner Logic Syntx Let A be set of ctions. Formule in HM Logic re build from: HM Logic Syntx Φ ::= tt ff Φ 1 Φ 2 Φ 1 Φ 2 [A]Φ A Φ tt: The constnt true formul. ff: The constnt flse formul. Φ 1 Φ 2 : Conjunction. Φ 1 Φ 2 : Disjunction. [A]Φ: Red s box A Φ. For ll A-derivtion, Φ holds. A Φ: Red s dimond A Φ. There exists n A-derivtion s.t. Φ holds.

10 Henessy-Milner Logic Semntics (Intuition) The formul tt is stisfied for ll process. No process stisfies ff. The opernds nd re interpreted s usul in logic. [A]Φ mens, ll -sucessor ( A) stisfies Φ. A Φ mens, there exists n -sucessor ( A) tht stisfies Φ.

11 Henessy-Milner Logic Semntics Let (Proc, L, { L}) be n LTS. Vlidity of P = Φ P = tt for ech P Proc P = ff P = Φ Θ iff P = Φ nd P = Θ P = Φ Θ iff P = Φ or P = Θ P = [A]Φ iff P = Φ for ll P Proc, A s.t. P P P = A Φ iff P P for some P Proc, A s.t. P = Φ We sy tht formul Φ is: Stisfible : if there exists P s.t., P = Φ. Unstisfible : if no process stisfies it. Vlid if ll processes stisfy it.

12 Exmples P = tick tt : P cn do n tick. P = tick tock tt : P cn do tick nd then tock. P = {tick, tock} tt: P cn do tick or tock. P = [tick]ff: P cnnot do tick. P = tick [tock]ff: P performs tick nd goes to stte from which there re no tock trnsitions. P = tick ff: This is lwys flse. P = [tick]tt: This is lwys true.

13 Exmples Continution Let L be the set of ctions, A L nd A the complement of A. P = [L]ff: P is dedlock (it cnnot perform ny ction). P = L tt: P cn perform some ction. P = L tt [{}]ff: must hppen next. P = L tt [L]Φ: Φ holds fter one step.

14 Negtion nd De Morgn lws P = Φ iff P = Φ tt = ff. ff = tt. (Φ Θ) = Φ Θ. (Φ Θ) = Φ Θ. [A]Φ = A Φ. A Φ = [A] Φ. With the subsets {,, ff, }, {[ ],, tt, } or {, [ ],,, tt, ff} one gets the full logic.

15 More Exmples Formule Distinguishing Systems s 1 s 1 s 2 C =.C C = []ff D =.D +.nil D = []ff

16 More Exmples Formule Distinguishing Systems p q c b r 1 r 2 = ( b tt c tt) p q 1 q 2 b c r 1 r 2 = ( b tt c tt)

17 More Exmples Formule Distinguishing Systems p q c b r 1 r 2 = ( b tt) p q 1 q 2 c b c r 1 r 1 = ( b tt) r 2

18 HM Login nd Strong Bilimilrity Imge-Finite System The LTS (Proc, L, { L}) is imge-finite if for every P Proc nd every A the set {P Proc P P } is finite. Theorem (Henessy-Milner) Let (Proc, L, { L}) be imge-finite LTS nd P, Q Proc. The following sentences re equivlent: 1 P Q (P nd Q re strongly bisimilr). 2 For every HM formul Φ, P = Φ Q = Φ.

19 I mke no clim tht everything cn be done by lgebr... It is perhps eqully true tht not everything cn be done by logic; thus one of the outstnding chllenges in concurrency is to find the right mrrige between logic nd behviorl pproches Robin Milner.

20 Sources 1 Pwel Sobocinski. Bisimultion, Gmes nd Hennessy Milner logic. Lecture 1 of Modelli Mtemtici dei Processi Concorrenti. 2 Mrtin Wirsing nd Axel Ruschmyer. Prozesslgebr: Hennessy-Milner Logic. Bsierend uf Lecture Notes von Rocco De Nicol. 3 Modl Logic.

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