Solving Equations on Discrete Dynamical Systems

Alberto Dennunzio, Enrico Formenti, Luciano Margara, Valentin Montmirail, Sara Riva

Boolean Automata Networks ; Discrete Dynamical Systems ; Decidability

Boolean automata networks, genetic regulation networks, and metabolic net- works 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 exper- imental data or inducing the model under uncertainties in the data. Equipping finite discrete dynamical systems with an algebraic structure of commutative semiring pro- vides a suitable context for hypothesis verification on the dynamics of DDS. Indeed, hypothesis on the systems can be translated into polynomial equations over DDS. Solu- tions to these equations provide the validation to the initial hypothesis. Unfortunately, finding solutions to general equations over DDS is undecidable. In this article, the authors want to push the envelop further by proposing a practical approach for some decidable cases in a suitable configuration that the authors call the Hypothesis Checking. the authors 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.