# Parameter Space Abstraction and Unfolding Semantics of Discrete Regulatory Networks

## TL;DR

This paper introduces an abstraction method for the parameter space of discrete regulatory networks, enabling more efficient analysis of their behaviors while ensuring conservative reachability results.

## Contribution

It presents a novel convex sublattice abstraction of the parameter space and an unfolding semantics that exploits concurrency for compact state reachability analysis.

## Key findings

- The abstraction is proven to be optimal and conservative.
- The method is effective on Boolean and multi-valued networks.
- Prototype implementation demonstrates scalability to large networks.

## Abstract

The modelling of discrete regulatory networks combines a graph specifying the pairwise influences between the variables of the system, and a parametrisation from which can be derived a discrete transition system. Given the influence graph only, the exploration of admissible parametrisations and the behaviours they enable is computationally demanding due to the combinatorial explosions of both parametrisation and reachable state space. This article introduces an abstraction of the parametrisation space and its refinement to account for the existence of given transitions, and for constraints on the sign and observability of influences. The abstraction uses a convex sublattice containing the concrete parametrisation space specified by its infimum and supremum parametrisations. It is shown that the computed abstractions are optimal, i.e., no smaller convex sublattice exists. Although the abstraction may introduce over-approximation, it has been proven to be conservative with respect to reachability of states. Then, an unfolding semantics for Parametric Regulatory Networks is defined, taking advantage of concurrency between transitions to provide a compact representation of reachable transitions. A prototype implementation is provided: it has been applied to several examples of Boolean and multi-valued networks, showing its tractability for networks with numerous components.