Generalized Communicating P Systems Working in Fair Sequential Model

In this article we consider a new derivation mode for generalized communicating P systems (GCPS) corresponding to the functioning of population protocols (PP) and based on the sequential derivation mode and a fairness condition. We show that PP can be seen as a particular variant of GCPS. We also consider a particular stochastic evolution satisfying the fairness condition and obtain that it corresponds to the run of a Gillespie’s SSA. This permits to further describe the dynamics of GCPS by a system of ODEs when the population size goes to the infinity.

💡 Research Summary

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The paper introduces a new derivation mode for Generalized Communicating P Systems (GCPS), called the fair sequential mode (fs‑mode), and establishes a tight correspondence with Population Protocols (PP). Traditional GCPS operate under a maximally parallel derivation strategy, where as many rules as possible fire simultaneously. In contrast, the fs‑mode restricts each transition step to a single rule application while enforcing a fairness condition: any configuration that appears infinitely often must also generate all its possible successors infinitely often. This mirrors the fairness requirement of PP, where agents interact pairwise in a sequential fashion and the overall execution must be fair.

The authors show how any PP can be encoded as a one‑symbol GCPS. Each state q of the PP becomes a membrane (cell) labelled q, and a token • placed in that membrane represents an agent in state q. Every PP interaction (q₁,q₂) → (q₀₁,q₀₂) is translated into a GCPS rule (•,q₁)(•,q₂) → (•,q₀₁)(•,q₀₂). The initial multiset of agents is reflected by the initial number of tokens in each membrane. Under the fs‑mode, the GCPS reproduces exactly the set of fair executions of the original PP. Conversely, any GCPS that does not use the environment cell can be simulated by a PP, highlighting the structural equivalence of the two models.

Beyond the deterministic mapping, the paper proposes a stochastic execution strategy for GCPS that satisfies the fairness condition. Each rule μ is assigned a stochastic rate constant c_μ, and the propensity a_S^μ = c_μ·h_S^μ is computed from the current configuration S, where h_S^μ counts the number of distinct pairs of objects that can fire μ. Selecting the next rule and the time interval according to these propensities reproduces Gillespie’s Stochastic Simulation Algorithm (SSA). Consequently, a GCPS running under this stochastic fs‑mode behaves as a continuous‑time Markov chain identical to the one used in biochemical reaction network simulations.

Taking the limit as the population size N → ∞, the authors normalize token counts to concentrations x_i = |z_i|/N. Assuming mass‑action kinetics, the propensities scale as a_S^μ/N → k_μ·Π_i x_i^{α_i}, where α_i denotes the number of tokens of type i consumed by rule μ. In this limit the stochastic dynamics converge to a deterministic system of ordinary differential equations (ODEs). The paper further provides sufficient conditions under which a given ODE system can be represented by a GCPS with concentration‑dependent stochastic rules, thus establishing a two‑way bridge between membrane computing and classical chemical kinetics.

To illustrate the practical relevance, two case studies are presented. The first models the Lotka‑Volterra predator‑prey dynamics using a small set of GCPS rules; stochastic simulations match the trajectories obtained from the corresponding ODEs, confirming the correctness of the Gillespie correspondence. The second demonstrates how GCPS can approximate algebraic numbers (e.g., √2) by designing a rule set whose fair sequential execution converges to the desired value, showcasing the computational power of the model beyond traditional number generation.

In summary, the contributions are: (1) defining the fair sequential derivation mode for GCPS; (2) proving that PP is a special case of GCPS under this mode; (3) linking stochastic GCPS executions to Gillespie’s SSA and, via the thermodynamic limit, to ODEs; (4) providing reverse construction criteria from ODEs to GCPS; and (5) validating the framework with biologically inspired and number‑theoretic examples. This work opens new avenues for using membrane‑based formalisms in the analysis of distributed algorithms, biochemical networks, and hybrid discrete‑continuous systems.