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Based on the discrete definition of biological regulatory networks developed by René Thomas, we provide a computer science formal approach to treat temporal properties of biological regulatory networks, expressed in computational tree logic. It is then possible to build all the models satisfying a set of given temporal properties. Our approach is(More)
Given a Boolean function F : {0, 1} n → {0, 1} n , and a point x in {0, 1} n , we represent the discrete Jacobian matrix of F at point x by a signed directed graph G F (x). We then focus on the following open problem: Is the absence of a negative circuit in G F (x) for every x in {0, 1} n a sufficient condition for F to have at least one fixed point? As(More)
Boolean networks are discrete dynamical systems extensively used to model biological regulatory networks. The dynamical analysis of these networks suffers from the combinatorial explosion of the state space, which grows exponentially with the number n of components. To face this problem, a classical approach consists in deducing from the interaction graph(More)
R. Thomas conjectured, twenty years ago, that the presence of a positive circuit in the interaction graph of a dynamical system is a necessary condition for the presence of several stable states. Recently, E. Remy et al. stated and proved the conjecture for Boolean dynamical systems. Using a similar approach, we generalize the result to discrete dynamical(More)
We consider a product X of n finite intervals of integers, a map F from X to itself, the asynchronous state transition graph Γ(F) on X that Thomas proposed as a model for the dynamics of a network of n genes, and the interaction graph G(F) that describes the topology of the system in terms of positive and negative interactions between its n components.(More)
In this paper, we are interested in the number of fixed points of functions f : A n → A n over a finite alphabet A defined on a given signed digraph D. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on D. We then discover relationships between the number of fixed points of f and(More)
We consider a class of Boolean networks called and-nets, and we address the question of whether the absence of negative cycle in local interaction graphs implies the existence of a fixed point. By defining correspondences with the notion of kernel in directed graphs, we prove a particular case of this question, and at the same time, we prove new theorems in(More)
In this paper, we propose a refinement of the modelling of biological regulatory networks based on the discrete approach of Rene Thomas. We refine and automatise the use of delays of activation/inhibition in order to specify which variable is more quickly affected by a change of its regulators. The formalism of linear hybrid automata is well suited to allow(More)
Linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations towards their destinations. The solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding. The guessing number approach converts this(More)