# Solving Equations on Discrete Dynamical Systems (Extended version)

## TL;DR

This paper introduces a practical method for solving certain decidable polynomial equations over discrete dynamical systems, enabling hypothesis verification in biological models with promising scalability.

## Contribution

It proposes a novel approach to solve specific decidable equations in discrete dynamical systems, facilitating hypothesis testing in biological modeling.

## Key findings

- Approach effectively reduces complex equations to simpler forms.
- Experimental results show good scalability of the method.
- Method enables enumeration of all solutions for hypothesis verification.

## Abstract

Boolean automata networks, genetic regulation networks, and metabolic networks are just a few examples of biological modelling by discrete dynamical systems (DDS). A major issue in modelling is the verification of the model against the experimental data or inducing the model under uncertainties in the data. Equipping finite discrete dynamical systems with an algebraic structure of commutative semiring provides a suitable context for hypothesis verification on the dynamics of DDS. Indeed, hypothesis on the systems can be translated into polynomial equations over DDS. Solutions to these equations provide the validation to the initial hypothesis. Unfortunately, finding solutions to general equations over DDS is undecidable. In this article, we want to push the envelope further by proposing a practical approach for some decidable cases in a suitable configuration that we call the Hypothesis Checking. We demonstrate that for many decidable equations all boils down to a "simpler" equation. However, the problem is not to decide if the simple equation has a solution, but to enumerate all the solutions in order to verify the hypothesis on the real and undecidable systems. We evaluate experimentally our approach and show that it has good scalability properties.