Towards a Boolean dynamical system representation in a nonmonotonic modal logic

Abstract : Boolean dynamical systems (BDSs) represent the evolution of interactions inside a finite network of entities taking Boolean states over discrete time. These networks are classically used to model interactions of biological networks. In this context, a genetic network can be represented by both a Transition Graph (TG) and an Interaction Graph (IG). The precise relationship between IG and TG has been studied for many years in dynamical systems theory while still an open question. The global purpose of this article is to further study this relationship via a logical representation of BDSs into a nonmonotonic modal logic called Hypothesis Logic (H). While the dynamics of a BDS are characterized by a function f, an important part of the studies focused on the analysis of both stable configurations (i.e. fixed points of f), and stable/unstable cycles of f. For the representation of some genetic networks with no negative feedback circuits, results were previously obtained with some well known nonmonotonic formalisms. So far however, BDSs representation by most of these formalisms does not permit to capture cyclic dynamical behaviors. Notably, the equivalent of a negative circuit has no extension in default logic (DL). This is embarrassing because these cycles may represent real interactions in living organisms like the cell cycle. This possible lack of extensions in DL was studied in H, for which theories always have extensions while some of these, called ghost extensions, are actually not extensions of the corresponding theories in DL. This paper addresses to the question of a first representation of the dynamics of BDSs with H, and ghost extensions appear to be a powerful tool in this respect. As we are especially concerned with cycles, it provides us with hints of simple algorithms for computing exhaustively both stable/unstable cycles and fixed points: distinguishing between stable/unstable as well as enumerating all the solutions in practice would be a major advance that would lead to apprehend better inner fundamental aspects in biology.
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Submitted on : Tuesday, April 9, 2019 - 1:20:21 PM
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  • HAL Id : hal-02093398, version 1

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Pierre Siegel, Andrei Doncescu, Vincent Risch, Sylvain Sené. Towards a Boolean dynamical system representation in a nonmonotonic modal logic. International Workshop on Non-Monotonic Reasoning (NMR'18), Oct 2018, Tempe, United States. pp.53--62. ⟨hal-02093398⟩

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