Rate-Based Stochastic Fusion Calculus and Angelo Troina Continuous - - PowerPoint PPT Presentation

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains

Gabriel Ciobanu, Angelo Troina ICTCS’12, September 21, 2012, Varese, Italy

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Features of Fusion Calculus

A global fusion relation modelling global shared state.1

π-Calculus Fusion Calculus Communication Effect Local Local or global Binding Input and Restriction Only one binding operator Operators Input always binds Neither Input or output Output sometimes binds are binding Bisimulation Early Only one Congruence Late congruence: Open Hyperequivalence Which one is “best”?

• 1J. Parrow, B. Victor.

The Fusion Calculus: Expressiveness and Symmetry in Mobile Processes.

• Proc. LICS, IEEE Computer Society, 176-185, 1998.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Fusion in Action

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Syntax

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Semantics

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Delayed Input

In π extended with delayed input, u(x) : P stands for a non stopped process P which can perform input on channel u at any time. In fusion calculus, delayed input is given by the process: (x)(ax | P) (x)(crx | Pcell | [x = s]PActPath)

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Continuous Time Markov Chains

A Continuous Time Markov Chain (CTMC) is a triple S, T, s0, where:

◮ S is the set of states, ◮ T : S × S → I

R≥0 is the transition function,

◮ s0 ∈ S is the initial state.

The system passes from a state s to a state s′ by consuming an exponentially distributed quantity of time, in which the parameter of the exponential distribution is T(s, s′). The cumulative distribution is T(s, C) =

s′∈C T(s, s′)

Many analysis techniques and tools are available for CTMCs, e.g. the PRISM model checker.2

• 2M. Kwiatkowska, G. Norman, D. Parker.

Probabilistic Symbolic Model Checking with PRISM: A Hybrid Approach.

• Int. J. on Soft. Tools for Technology Transfer, vol.6, 128142, 2004.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMC: Simulation

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMC: Simulation

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMC: Simulation

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMC: Simulation

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMCs: Race Condition

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMCs: Race Condition

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMCs: Race Condition

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMCs: Race Condition

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

CTMCs: Race Condition

with duration ti=min{t1, t2, t3}.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Markovian Bisimulation

Given a CTMC S, T, s0, an equivalence relation R on S is a strong Markovian bisimulation if and only if for all (s1, s2) ∈ R and for all equivalence classes C ∈ S/R, T(s1, C) = T(s2, C). Two states s1 and s2 are strong Markovian bisimilar, written s1 ∼M s2, if and only if there exists a Markovian bisimulation R on S with (s1, s2) ∈ R. Two CTMCs MC1 = S1, T1, s01 and MC2 = S2, T2, s02, are strong Markovian bisimilar, written MC1 ∼M MC2, if s01 ∼M s02.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Rate-Based Stochastic Fusion Calculus

We use the generic notation: µ = (α, r) for actions, where α can be either an input u x, an output u x, or a fusion ϕ, and r is the rate of the exponential distribution modelling the action duration.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Rate-Based Transitions

We define the semantics of SFC via a transition relation: P

α

− → ρ associating to a given process P and a transition action α a next state function (NSF):3 ρ : SFC → I R≥0. where ρ(Q) returns the rate of reaching process Q from process P through action α.

• 3R. De Nicola, D. Latella, M. Loreti, M. Massink.

Rate-based Transition Systems for Stochastic Process Calculi.

• Proc. ICALP, LNCS vol.5556, 435-446, 2009.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Rate-Based Transitions: NSF construction

Base case: If P = (α, r).Q then: P

α

− → ρ where: ρ(Q) = r and ρ(Q′) = 0 ∀Q′ = Q. Adding Operators: If P = P1 + P2 with P1

α

− → ρ1 and P2

α

− → ρ2, then: P

α

− → ρ where: ρ(Q) = ρ1(Q) + ρ2(Q).

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Stochastic Semantics

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Stochastic Bisimulation

Definition

A stochastic bisimulation is an equivalence relation R over SFC s.t. for each pair (P, Q) ∈ R, for all actions α, and for all equivalence classes S ∈ SFC/R, we have that: γα(P, S) = γα(Q, S). where γα(R, S) = {ρ(R′) | R

α

− → ρ, R′ ∈ S}. Two processes P and Q are stochastic bisimilar, written P

.

∼SB Q, if they are related by a stochastic bisimulation.

Theorem

Stochastic bisimilarity is NOT a congruence.

(y, ry) | (z, rz)

.

∼SB (y, ry).(z, rz) + (z, rz).(y, ry) {y = z}.((y, ry) | (z, rz))

.

≁SB {y = z}.((y, ry).(z, rz) + (z, rz).(y, ry))

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Stochastic Hyperequivalence

We look for the largest congruence included in the stochastic bisimilarity closing the definition of stochastic bisimulation under arbitrary substitutions.

Definition

A stochastic hyperbisimulation is an equivalence relation R

• ver SFC satisfying the following properties:

i) R is closed under any substitution σ, i.e., PRQ implies PσRQσ for any σ; ii) for each pair (P, Q) ∈ R, for all actions α, and for all equivalence classes S ∈ SFC/R, we have γα(P, S) = γα(Q, S). Two processes P and Q are stochastic hyperequivalent if they are related by a stochastic hyperbisimulation (written P ∼SH Q).

Theorem

Stochastic hyperequivalance is a congruence.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

From SFC Processes to CTMCs

The semantics of a SFC process P, given a global fusion relation ϕ is transformed into a CTMC by considering pairs (P, ϕ) as states of the CTMC. T((P, ϕ), (P′, ϕ′)) is the sum of the rates of all reactions from P to P′ updating the fusion ϕ to ϕ′: T((P, ϕ), (P′, ϕ′)) =

α{ρ(P′) | P α

− → ρ} where the action α updates the fusion ϕ to ϕ′ (or leaves it unchanged, i.e. when ϕ = ϕ′). We call [|P|]MC the CTMC obtained as the semantics of the SFC process P.

Theorem

Stochastic hyperequivalence preserves strong Markovian bisimulation, i.e., for P, Q ∈ SFC, P ∼SH Q implies [|P|]MC ∼M [|Q|]MC.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

The Arbuscular Mycorrhizal Symbiosis

The arbuscular mycorrhizal (AM) symbiosis is an example of association between fungi and the roots of most land plants.

◮ AM fungi are obligate symbiont which supply the host

with essential nutrients such as phosphate, nitrate and

• ther minerals from the soil. In return, AM fungi receive

carbohydrates derived from photosynthesis in the host.

◮ The symbiosis also confers resistance to the plant

against pathogens and environmental stresses.

◮ The fungus develops an extensive network of hyphae

which explores and exploits soil microhabitats for nutrient acquisition. Despite the central importance of AM symbiosis in both agriculture and natural ecosystems, the mechanisms for the formation of a functional symbiosis between plants and AM fungi are largely unknown.

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Arbuscular Micorrhiza

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

AM Communication

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Arbuscular Mycorrhizal (AM) Symbiosis

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Activated Cells: Calcium Spiking

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

SFC Model

Initial System: PlantRoot | Soil PlantRoot ::= (us, rs) | 1000 × EpiCell Soil ::= (ux, 1) | 10 × Spore Spore ::= x = s?Hypha : Spore Hypha ::= (vm, rmB).Hypha + (vm, rmD).0 EpiCell ::= (y)((vy, 1).0 | IntEpiCell) IntEpiCell ::= y = m?Active : IntEpiCell

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

From SFC to CTMCs: PRISM Model

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Simulation Results

Mean from a 1000 sample of stochastic simulations:

Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus

Features Syntax and Semantics

Continuous Time Markov Chains

Definition Simulating CTMCs Markovian Bisimulation

Stochastic Fusion Calculus

Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs

The AM Symbiosis

Biological System Formal Modelling Simulation Results

Conclusions

Conclusions

Stochastic Fusion Calculus constitutes a successful tool for the analysis of concurrent distributed systems combining aspects of:

◮ compositionality (typical of process algebra), ◮ delayed input (non binding input), ◮ one to many communication (global fusion relation), ◮ probability distributions (race conditions), ◮ action durations (exponential distribution), ◮ model checking (translation to CTMCs).

Possible future investigations:

◮ explore other features of communications in fusion

calculus,

◮ develop a tool for automatic translation from SFC to

CTMCs.