Stochastic fragments: A framework for the exact reduction of the stochastic semantics of rule-based models

In this paper, we propose an abstract interpretation-based framework for reducing the state space of stochastic semantics for protein protein interaction networks. Our approach consists in quotienting the state space of networks. Yet interestingly, we do not apply the widelyused strong lumpability criterion which imposes that two equivalent states behave similarly with respect to the quotient, but a weak version of it. More precisely, our framework detects and proves some invariants about the dynamics of the system: indeed the quotient of the state space is such that the probability of being in a given state knowing that this state is in a given equivalence class, is an invariant of the semantics. Then we introduce an individual-based stochastic semantics (where each agent is identified by a unique identifier) for the programs of a rulebased language (namely Kappa) and we use our abstraction framework for deriving a sound population-based semantics and a sound fragments based semantics, which give the distribution of the traces respectively for the number of instances of molecular species and for the number of instances of partially defined molecular species. These partially defined species are chosen automatically thanks to a dependency analysis which is also described in the paper.