/pdf/an-algorithmic-pipeline-for-solving-equations-over-discrete-3vq9oh9q.pdf

An algorithmic pipeline for solving equations over discrete dynamical systems modelling hypothesis on real phenomena

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.

https://hal.archives-ouvertes.fr/hal-03380303/document

MDDs Boost Equation Solving on Discrete Dynamical Systems

Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing the division and computing $k$-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$. These results are some of the important final steps for solving more general polynomial equations on FDDS. 

pdf/roots-in-the-semiring-of-finite-deterministic-dynamical-149xgrnzlg.pdf

Roots in the semiring of finite deterministic dynamical systems

Asynchronous, finite dynamical systems

In conventional discrete dynamical systems, the new configuration depends solely on the configuration at the preceding time step. This contribution considers an extension to the standard framework of dynamical systems by taking into consideration past history in a simple way: the mapping defining the transition rule of the system remains unaltered, but it is applied to a certain summary of past states. This kind of embedded memory implementation, of straightforward computer codification, allows for an easy systematic study of the effect of memory in discrete dynamical systems, and may inspire some useful ideas in using discrete systems with memory (DSM) as a tool for modeling non-markovian phenomena. Besides their potential applications, DSM have an aesthetic and mathematical interest on their own, as will be briefly over viewed. The contribution focuses on the study of systems discrete par excellence, i.e., with space, time and state variable being discrete. These discrete universes are known as cellular automata (CA) in their more structured forms, and Boolean networks (BN) in a more general way. Thus, the mappings which define the rules of CA (or BN) are not formally altered when implementing embedded memory, but they are applied to cells (or nodes) that exhibit trait states computed as a function of their own previous states. So to say, cells (or nodes) — canalize — memory to the mapping. Automata on networks and on proximity graphs, together with structurally dynamic cellular automata, will be also studied with memory. If time permits, systems that remain discrete in space and time, but not in the state variable (e.g., maps and spatial games), will be also scrutinized with memory. A list of references on DSM may be found in http://uncomp.uwe.ac.uk/alonso-sanz.

Cellular Automata and other Discrete Dynamical Systems with Memory.

Discrete dynamical systems (DDS) are a useful tool for modelling the dynamical behavior of many phenomena occurring in a huge variety of scientific domains. Boolean automata networks, genetic regulation networks, and metabolic networks are just a few examples of DDS used in Bioinformatics. Equations over DDS have been introduced as a formal tool to check the model against experimental data. Solving generic equations over DDS has been proved undecidable. In this paper we propose to solve a decidable abstraction which consists in equations having a constant part. The abstraction we focus on consists in restricting the solutions to equations involving only the periodic behavior of DDS. We provide a fast and scalable method to solve such abstractions.

/pdf/solving-equations-on-discrete-dynamical-systems-uljtxm473j.pdf

Solving Equations on Discrete Dynamical Systems

Nineteen elementary cellular automata rules (and their conjugation, reflection, reflected-conjugation) have been proven not maximum sensitive to synchronism, i.e. they do not have a different dynamics for each (non-equivalent) block-sequential update schedule (defined as ordered partitions of cell positions). In this work we present exact measurements of the sensitivity to synchronism for these rules, as functions of the size. These exhibit a surprising variety of values and associated proof methods, such as the isomer pairs of rule 128, and the connection to the bisection of Lucas numbers of rule 8. 

Non-maximal sensitivity to synchronism in elementary cellular automata: Exact asymptotic measures

Minimal trap spaces (MTSs) capture subspaces in which the Boolean dynamics is trapped, whatever the update mode. They correspond to the attractors of the most permissive mode. Due to their versatility, the computation of MTSs has recently gained traction, essentially by focusing on their enumeration. In this paper, we address the logical reasoning on universal properties of MTSs in the scope of two problems: the reprogramming of Boolean networks for identifying the permanent freeze of Boolean variables that enforce a given property on all the MTSs, and the synthesis of Boolean networks from universal properties on their MTSs. Both problems reduce to solving the satisfiability of quantified propositional logic formula with 3 levels of quantifiers ($\exists\forall\exists$). In this paper, we introduce a Counter-Example Guided Refinement Abstraction (CEGAR) to efficiently solve these problems by coupling the resolution of two simpler formulas. We provide a prototype relying on Answer-Set Programming for each formula and show its tractability on a wide range of Boolean models of biological networks.

pdf/tackling-universal-properties-of-minimal-trap-spaces-of-2zirscre.pdf

Tackling Universal Properties of Minimal Trap Spaces of Boolean Networks

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