A Correct Abstract Machine for the Stochastic Pi-calculus

Machine for Stochastic Pi-Calculus (SPiM) ➢ Formalise how the simulator works (program specification). ➢ Prove properties about the simulator (program verification). ➢ Give greater confidence in the simulation results. ➢ Improve on existing simulators (BioSpi). Andrew Phillips BioConcur 2004 19 Machine Data Structures ➢ General Machine Term: νn1 νn2 ...νnN (Σ1 ::Σ2 :: ... ::ΣM :: []) ➢ Syntax Definition: V,U ::= νn V Restriction p A List A,B ::= [] Empty p Σ::A Summation Andrew Phillips BioConcur 2004 20 Machine Encoding ➢ Encoding (P ): (P ) , P ◦ [] ➢ Construction (P ◦ V ): n 6∈ fn(P ) ⇒ P ◦ (νn V ) , νn (P ◦ V ) 0 ◦A , A (P | Q) ◦A , P ◦Q ◦A n 6∈ fn(P ◦A) ⇒ (νm P ) ◦A , νn (P{n/m} ◦A) !π.P ◦A , (π.(P | !π.P ) + 0) ◦A (π.P + Σ) ◦A , (π.P + Σ)::A Andrew Phillips BioConcur 2004 21 Machine Execution ➢ Reduction (V r −→ V ′): V r −→ V ′ ⇒ νx V r −→ νx V ′ ∣∣∣∣∣ x = Next(A) ∧A Â (x(m).P + Σ)::A′ ∧A′ Â (x〈n〉.Q + Σ′) ::A′′ ⇒ A rate(x) −→ P{n/m} ◦Q ◦A′′ ➢ Selection (A Â B): A Â A A Â Σ′ ::A′ ⇒ Σ::A Â Σ′ ::Σ::A′ Σ::A Â (π′.P ′ + Σ′) ::A ⇒ (π.P + Σ) :: A Â (π′.P ′ + π.P + Σ′) ::A Andrew Phillips BioConcur 2004 22 Channel Activity ➢ Choose next channel x = Next(A) by stochastic algorithm [Gillespie, 1977] ➢ Gillespie chooses next channel based on activity : activity = number of possible interactions on a channel ➢ In SPiM, activity of channel x in term A: Actx(A) = (Inx(A) ∗Outx(A))−Mixx(A) ❑ Inx(A) = number of unguarded inputs on channel x in A. ❑ Outx(A) = number of unguarded outputs on channel x in A. ❑ Mixx(A) = sum of Inx(Σi)×Outx(Σi) for each summation Σi in A. Andrew Phillips BioConcur 2004 23 Gillespie: Choosing the Next Reaction Next(A) 1. For all x ∈ fn(A) calculate ax = Actx(A) ∗ rate(x) 2. Store non-zero values of ax in a list (xμ, aμ), where μ ∈ 1...M . 3. Calculate a0 = ∑M ν=0 aν 4. Generate two random numbers n1,n2 ∈ [0, 1] and calculate τ, μ such that: τ = (1/a0) ln(1/n1) μ−1 ∑ ν=1 aν < n2a0 ≤ μ ∑ ν=1 aν 5. Next(A) = xμ and Delay(A) = τ . Andrew Phillips BioConcur 2004 24 Correctness of the Machine ➢ Safety : no runtime errors (no crashes) Lemma 1. ∀V.V ∈ SPiM ∧ V r −→ V ′ ⇒ V ′ ∈ SPiM ➢ Soundness: machine only performs valid executions steps (behaves well) Theorem 1. ∀V.V ∈ SPiM ∧ V r −→ V ′ ⇒ [V ] r −→ [V ′] ➢ Completeness: machine can perform all execution steps of the calculus Theorem 2. ∀P.P ∈ SPi ∧ P r −→ P ′ ⇒ (P ) r −→≡ (P ′). ➢ Termination: machine does not loop forever unnecessarily Theorem 3. ∀P.P ∈ SPi ∧ P 6−→ ⇒ (P ) 6−→ Andrew Phillips BioConcur 2004 25 Correctness of the Machine ➢ Duration: reactions in machine and calculus have same average duration ❑ Gillespie algorithm proved correct for selecting next reaction channel. ❑ Also need to ensure that reaction has correct duration ❑ E.g. reduction in P1 is twice as fast as reduction in P2: P1 , xr〈n〉.P + xr〈n〉.P | x(m).Q P2 , xr〈n〉.P | x(m).Q ❑ Sufficient to ensure that machine and calculus have same channel activity. Proposition 1. ∀V ∈ SPiM.Actx(V ) = Actx([V ]) Proposition 2. ∀P ∈ SPi.Actx(P ) = Actx((P )) Andrew Phillips BioConcur 2004 26 Implementation ➢ Abstract Machine maps almost directly to program code ➢ Implemented a polymorphic type system and type checker ➢ Correctness of the machine gives greater confidence in the simulation results ➢ Demo... Andrew Phillips BioConcur 2004 27 Conclusion➢ Presented a graphical representation for pi-calculus:❑ Precise, compositional, executable descriptions.❑ Used to model regulatory systems at the molecular level.➢ Presented an abstract machine for the stochastic pi-calculus:❑ Correctness proof: safety, soundness, completeness, termination, duration.❑ Maps readily to program code.❑ Could be used as a basis for implementing new calculi.➢ Built a simulator based on the machine.❑ Plan to incorporate a graphical editor as a front-end.Andrew Phillips BioConcur 200428 References[Gillespie, 1977] Gillespie, D. T. (1977). Exact stochastic simulation of coupledchemical reactions. J. Phys. Chem., 81(25):2340–2361.[Priami et al., 2001] Priami, C., Regev, A., Shapiro, E., and Silverman, W.(2001). Application of a stochastic name-passing calculus to representationand simulation of molecular processes. Information Processing Letters. in press. Andrew Phillips BioConcur 200429 Link Encoding➢ Encoding uses restriction, replication, parallel composition and communication.➢ A linked node → a replicated input on a fresh channel x, in parallel with anoutput on x➢ A link to the node → an output on x.➢ E.g.:ΣΣππ’’=ΣΣππ’’new x!x() x<>x<> Andrew Phillips BioConcur 200430 Safety ProofLemma 2. ∀V.V ∈ SPiM ∧ Vr−→ V ′ ⇒ V ′ ∈ SPiMProof. By Lemma 3, Lemma 4 and by induction on reduction in SPiM. 2Lemma 3. ∀A ∈ SPiM.A Â B ⇒ B ∈ SPiMProof. By induction on selection in SPiM. 2Lemma 4. ∀V.∀P.V ∈ SPiM ∧ P ∈ SPi ⇒ P ◦ V ∈ SPiMProof. By induction on construction in SPiM. 2 Andrew Phillips BioConcur 200431 Soundness ProofTheorem 4. ∀V.V ∈ SPiM ∧ Vr−→ V ′ ⇒ [V ] r−→ [V ′]Proof. By Lemma 5, Lemma 6 and by induction on reduction in SPiM. 2Lemma 5. ∀A.A ∈ SPiM ∧A Â B ⇒ [A] ≡ [B]Proof. By induction on selection in SPiM. 2Lemma 6. ∀V.∀P.V ∈ SPiM ∧ P ∈ SPi ⇒ [P ◦ V ] ≡ P | [V ]Proof. By induction on construction in SPiM. 2[νn V ] , νn [V ][[]] , 0[Σ::A] , Σ | [A]Andrew Phillips BioConcur 200432 Completeness ProofTheorem 5. ∀P.P ∈ SPi ∧ Pr−→ P ′ ⇒ (P ) r−→≡ (P ′).Proof. By Lemma 7 and by induction on reduction in SPi, where the rule forparallel composition is expanded over the remaining rules. 2Lemma 7. P ≡ Q ⇒ (P ) ≡ (Q)Proof. By induction on structural congruence in SPi. 2Lemma 8. ∀V.V ∈ SPiM∧U ≡ V ∧V r−→ V ′ ⇒ ∃U ′.U r−→ U ′∧U ′ ≡V ′Proof. By induction on structural congruence in SPiM. 2 Andrew Phillips BioConcur 200433 Structural CongruenceV≡α U ⇒ V ≡ Ux 6∈ fn(V ) ⇒ νx V ≡ Vνx νy V ≡ νy νx VΣ::Σ′ ::A ≡ Σ′ ::Σ::AA ≡ A′ ⇒ Σ::A ≡ Σ::A′(π.P + π′.P ′ + Σ)::A ≡ (π′.P ′ + π.P + Σ)::AΣ::A ≡ Σ′ ::A ⇒ (π.P + Σ)::A ≡ (π.P + Σ′) ::AAndrew Phillips BioConcur 200434 Termination ProofTheorem 6. ∀P.P ∈ SPi ∧ P 6−→ ⇒ (P ) 6−→Proof. By Theorem 4 and by basic relationships between encoding and decod-ing. 2 Andrew Phillips BioConcur 200435 Graphical SemanticsxPx(m)P’’ xPx(m)P’’xPx(m)P’xPx(m)P’ΣΣ!Σ!!Σ! ➢ Requires some imagination: for substituting names and for cloning graphs.Andrew Phillips BioConcur 200436