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Unifying Theories of Concurrency: CCSandCSP He Jifeng and Tony Hoare BCTCSApril 6, 2006

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Why? just for the sake of it –as a scientific achievement to explain differences between theories –and what they are good for to integrate more general toolsets –for coherence and consistency –in system design, implementation,...

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A Transition System a set P of processes: nil, p, q, Lp,… a set A of observations: a, b, … –communications: x, y,... –hidden events: , ,... –meaningful barbs: ref(X), δ … a relation T P × A × P a {(p,q) | (p,a,q) T}

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a b a c ref(X) x b

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Traces p q p = q p s r q. p a q & q s r p s _ q. p s q traces(p) { s | p s _ }

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(Strong) Simulation ≤ is the weakest x P×P such that a:A, x ; a a ; x –describes efficient model checking algorithm ≡ ≤ ∩ ≥ Theorem: ≤ and ≡ are pre-orders – Id and ≤ ; ≤ satisfy the defining equation

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Refinement ⊑ is the weakest x P×P such that s:A*, x ; s s ; U Theorem: ≤ ⊑ –one defining equation implies the other Theorem: p ⊑ q iff traces(q) traces(p)

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L : P → P is a link if it maps all processes of its source theory to all processes of its target theory. ≤ L L ; ≤ ; L –i.e.,p ≤ L qiffLp ≤ Lq ⊑ L L ; ⊑ ; L Theorem: ≤ L, ⊑ L are preorders – L ; L = Id

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L is monotonic ≤ ≤ L or equivalently: – p ≤ q Lp ≤ Lq, all p, q – ≤ ; L L ; ≤ consequently: –all order-theorems of source theory are valid in the target theory

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L is idempotent L ; L ; ≤ = L ; ≤ or equivalently: –L(Lp) ≡ Lp,all p consequently: –≤ L =≤ (restricted to target theory) –Lp ≡ p iff p is in target theory

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L is decreasing L ≤ or equivalently: –Lp ≤ p,for all p – ≤ L ; ≤ consequently: –the target theory is more abstract –Lp is the closest abstraction of p within the target theory.

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L is efficient L ; ≤= ≤ L or equivalently: –Lp ≤ qiffLp ≤ Lq,all p, q consequently: –to test : spec ≤ L imp, model-check : L(spec) ≤ imp, –(as is done in FDR)

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L is a retraction iff it is decreasing ≤ L ; ≤ it is idempotentL ; L ; ≤ L ; ≤ it is monotonic ≤ ; L L ; ≤ Theorem: L is a retraction iffL is efficient iffL ; ≤ is a preorder

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quarter of the proof L is a retraction (L ; ≤) is a preorder –Id (≤) (L ; ≤) {L dec} –(L ; ≤ ; L ; ≤) (L ; L ; ≤ ; ≤) {L mon} L ; ≤ {L idem}

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Weak Simulation p =a=> q ----------------------- Wp Wq where = => * and =a=> * * for a and * … Theorem: W is a retraction

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The original graph a b

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W only adds transitions so it is decreasing a b W W W W a a a W

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W W adds no more so it is idempotent a b W WW a a a W

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(W; ≤ ) is weak simulation Theorem: it is the weakest solution of the defining equations –x ; * * ; x, for a – x ; * ; x CCS/weak simulation is a retract (by W) of CCS/strong simulation

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After p / sis the most general behaviour of p after performing all of trace s p s _ ----------------------- p/s a p/(s )

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The original graph b c a a p

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The effect of _ /a b b c c a a p/a p/ac p p/ab

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Trace refinement _ & p/a = q p a _ & p/a = q ----------------------------- Tp a Tq Theorem: T is a retraction and (T ; ≤ ) = ⊑

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The original graph b b c c a a p/a p/ac p p/ab

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The effect of T b b c c a a T(p/a) T(p/ac) TpTp T(p/ab) a

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CSP is a retract of CCS Theorem: (W;T) is a retraction and (W; T; ≤ ) is CSP trace refinement Conclusion: CSP/trace refinement is a retract of CCS/weak simulation.

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ref(X) is a refusal where X is a set of communications x X { } p x _ p x q -------------------- --------------- Rp ref(X) Rp Rp x Rq Theorem: (R ; ≤ ; R ) is ⅔ simulation

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Divergences p p' p'' … forever ------------------------------------------ Dp δ Dr & Dp a Dr p a q --------------- Dp a Dq Theorem: D is a retraction

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CSP/FDR = L(CCS / ≤ ) where L = D ; R ; W ; T is a retraction –with respect to ≤ D;R L is defined by SOS transition rules. CSP healthiness conditions are expressed p ≡ L(p) CSP refinement coincides with simulation variations of CSP and CCS defined by selection from: T, D, R, W,…

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CCS is more general –applies to all edge-labelled graphs has less laws –the minimum reasonable set is less expressive –uses equivalence rather than ordering

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CSP describes distributed computing –graphs restricted by healthiness conditions has more laws –for optimisation and reasoning –the maximum reasonable set respecting deadlock and divergence is more expressive –ordering represents correctness –and refinement of system from specification

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