MDDs Boost Equation Solving on Discrete Dynamical Systems

Abstract

Discrete dynamical systems (DDS) are a model to represent complex phenomena appearing in many different domains. In the finite case, they can be identified with a particular class of graphs called dynamics graphs. In [9] polynomial equations over dynamics graphs have been introduced. A polynomial equation represents a hypothesis on the fine structure of the system. Finding the solutions of such equations validate or invalidate the hypothesis. This paper proposes new algorithms that enumerate all the solutions of polynomial equations with constant right-hand term outperforming the current state-of-art methods [10]. The boost in performance of our algorithms comes essentially from a clever usage of Multi-valued decision diagrams. These results are an important step forward in the analysis of complex dynamics graphs as those appearing, for instance, in biological regulatory networks or in systems biology.

An Algorithmic Pipeline for Solving Equations over Discrete Dynamical Systems Modelling Hypothesis on Real Phenomena

“…Equation ( 6) is still hard to solve in this form. We can further simplify it by performing a contraction step which consists in rewriting it in an equivalent way as union of systems of the following type, one for each vector (n 11 1 , . .…”

“…Thus, D p,q = {2}. Indeed, the following situations occur Experiments show how this method can achieve interesting performances in time and memory [11].…”

“…The first level, corresponding to the monomial C 1 4 ⊙ X 1 , only contains the node root. According to the necessary equations defined p = 4 and q = 4, the first monomial is able by itself to form n 11 2 cycles of length 4 where n 11 2 ∈ {1, 2, 3, 4}. Regarding the second monomial, it is able by itself to form either n 21 2 = 2 or n 21 2 = 4 cycles of length 4.…”

“…Indeed, the result of the above mentioned algorithm depends on the order by which the SB-Cartesian MDDs are considered when the intersection is performed. In [11], a new algorithm performing the intersection has been proposed in such a way that it properly works independently of that order. Let us recall its underlying idea.…”