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Synchronous Parallel Composition in Process Calculus for Ecological Models Anna Philippou Department of Computer Science University of Cyprus Joint work with Mauricio Toro Department of Comp. Sc. EAFIT University Spyros Sfenthourakis Christina Kassara Department of Biology University of Patras Department of Biology University of Cyprus Motivation and Background • Population Ecology – Population dynamics – Conservation schemes, species reintroduction • Individual-based modeling vs. population-based modeling • Formal methods for ecological systems – Process calculi, P-systems, cellular automata 2/25 Motivation and Background • Previous work – PALPS (Process Algebra with Locations for Population Systems) • a discrete-time, probabilistic process calculus with locations – Translation from PALPS to model checker PRISM – Application of the framework for studying population dynamics through model checking • Limitations – State space explosion – Interleaving semantics in process calculi vs. phase-based execution considered by ecologists 3/25 Illustration by example • Consider a species that cycles through dispersal and reproduction phases: 𝑆 = 𝑚𝑜𝑣𝑒. 𝑟𝑒𝑝. √. 𝑆 • State space of a single individual move rep √ … 4/25 Illustration by example 𝑆 = 𝑚𝑜𝑣𝑒. 𝑟𝑒𝑝. √. 𝑆 • System with 2 individuals: S | S • State space of 2 individuals (first time unit): √ = … = move rep 5/25 Illustration by example 𝑆 = 𝑚𝑜𝑣𝑒. 𝑟𝑒𝑝. √. 𝑆 • System with 3 individuals: S | S | S • State space of 3 individuals (first time unit): = √ … = move rep 6/25 Illustration by example • Intention: individuals execute their dispersal and reproduction phases simultaneously. • Proposed solution – Syntax Group individuals into populations – Semantics Synchronous parallel composition • E.g. – System with k individuals: Sk – state space for k individuals movek repk 7/25 S-PALPS • An extension of PALPS: a discrete-time, probabilistic process algebra with locations • World consisting of a set of locations each featuring its own environmental characteristics and inhabited by populations. World implemented as a graph: 8/25 S-PALPS • Basic entities – Channels: a, b, … • can be used in input position, a, b, or in output position a, b – Locations: l1, l2, … • And a set of logical expressions, e, referring to properties of locations – Species: s1, s2, … • Individuals – – – – Modelled as processes Possess a species and a location Locations may change dynamically Engage in activities (e.g. reproduction, dispersal, predation, death) 9/25 Syntax (1) • The Individual level 𝑃 ∷= 0 | 𝜂. 𝑃 | 𝑖∈𝐼 𝑝𝑖 : 𝑃𝑖 | 𝛾? (𝑃1 , 𝑃2 ) | 𝑃1 | 𝑃2 | 𝐶 • Actions 𝜂 ∷= 𝑎 | 𝑎 | 𝑔𝑜 𝑙 | √ Inactive Individual Action prefix Probabilistic choice Conditional choice Parallel composition Constant Input action Output action Dispersal action Tick action 10/25 Syntax (2) • The system level 𝑆 ∷= 0 | 𝑃: 𝑠, 𝑙, 𝑞 | 𝑆1 | 𝑆2 | 𝑆\L Inactive system Located Population Parallel Composition Restriction 11/25 S-PALPS Example • The individual level: Individuals live on an nn lattice of sites and they disperse and reproduce in two discrete phases. • The system level: A system comprised of 2 individuals at location ℓ and one individual at location ℓ’: 12/25 S-PALPS Semantics • Two transition relations: – Probabilistic transition relation 𝑆 𝑤 𝑝 𝑆′ – Nondeterministic transition relation 𝑆 𝛼 𝑛 𝑆′ • Key concepts – All components must synchronize on a tick-action – Probabilistic transition take precedence over nondeterministic transitions – All components able to execute a nondeterministic action must proceed synchronously 13/25 Selected Rules – Population Level • (ACT) • (MOVE) 14/25 Selected Rules – System Level • (TICK) • (SYNCH) 15/25 Model checking S-PALPS models • S-PALPS models where probabilistic and nondeterministic behavior co-exists • PRISM model checker for probabilistic systems (Markov Decision Processes, Discrete Markov Chains, Continuous MCs) • Prism language – State-based – Guarded commands – Set of modules that communicate with each other on shared actions (CSP-style communication) 16/25 Encoding PALPS into PRISM • To translate SPALPS into the PRISM language – Each state of a located population is a module – A module has a variable that counts the number of individuals at the specific state – the execution flow is facilitated by a local variable – all modules synchronize via one of the following actions: • synch – executed when a nondeterministic transition takes place • tick – executed when a timed action takes place • prob – executed when a probabilistic transition takes place 17/25 Encoding S-PALPS into PRISM – A Snapshot global sp:[0,max] init 5 global sQ:[0,max] init 0 System = 𝑃: 𝑠, 𝑙, 5 𝑃 = 𝑎. 𝑄 𝑄 =… module P st:[0..5] init 1; nP:[0..max] [] (st=1)&(sP>0) (tact’=0)& (st’=2)&(np’=sP); [] (st=1)&(sP=0) (st’=2); [synch] (st=2) (st’=3); [](st=3)&(pact=0)(sP’=sp– np) &(sQ’=sQ+nP) [prob] (st=3)&(pact=1) … module Q … 18/25 Correctness of the encoding Theorem. For any S-PALPS process 𝑆 and its PRISM translation 𝑆 we have 1. If 𝑆 2. If 𝑆 3. If 𝑆 √ 𝑛 𝑆′ then 𝑤 𝑝 𝑆′ then 𝛼 𝑛 𝑆′ then 𝑆 𝑆 𝑆 𝑠𝑦𝑛𝑐ℎ 𝑡𝑖𝑐𝑘 𝑆′ . 𝑠𝑦𝑛𝑐ℎ 𝑝𝑟𝑜𝑏,𝑤 𝑠𝑦𝑛𝑐ℎ 𝛼 𝑆′ . 𝑆′ . 19/25 Case Study – The Eleonora Falcon • Migrant species that breeds in the Mediterranean sea • Local conservation importance • Three types of individuals – Juveniles – Young adults (sexual maturity at the age of 4) – Mature adults (more likely to choose a protected nest) • Parameters – Probability of an individual to return to the island to breed (mortality) – Availability of less-exposed nests – Probabilities of offspring survival in exposed/less-exposed nests 20/25 The life cycle Newborns Dispersal Newborns Newborns Juveniles of Age 4 Mortality Surviving Adults Dispersal First-year breeders Experienced breeders Reproduction in exposed nest Reproduction in less exposed nest Adults Adults DIspersal DIspersal Surviving Adults Mortality Surviving Adults Mortality 21/25 The model in S-PALPS 22/25 Analysis in PRISM • Verification of probabilistic temporal PCTL properties, e.g. – Probability of extinction of the population in the next 10 years is less than a certain threshold pe – The total average number of individuals of species s at time unit t • Semantics of PRISM model checking – Defined over Markov Decision Processes: Computes minimum and maximum probabilities – Approximation defined over Discrete-Time Markov Chains: Computes reward-based properties, steady state and reachability • Simulation – Explore random paths of execution – Search for deadlocks using Prism simulation – Perform model-checking by simulation (statistical model checking) 23/25 Representative results • Impact of changing the offspring survival rates on the size of the population • Results indicate a fair degree of stability in the evolution of the species and a relative insensitivity to small changes in local conditions 24/25 Conclusions and future work • S-PALPS – Synchronous parallel composition – more faithful models – Reduction of the state space of systems – Case study – promising results • Current/future work: – Translation from S-PALPS to PRISM has been automated – Improve the translation: generality vs. efficiency – Further calibrate our model and take into account more aspects of interest to biologists – Apply to other case studies – Mean field semantics à la CCS within a spatially-explicit framework 25/25
