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Representation of gene regulation networks by hypothesis logic-based Boolean systems

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Abstract

Boolean Dynamical Systems (BDSs) are networks described by Boolean variables. A new representation of BDSs is presented in this article by using modal non-monotonic logic (\({\mathcal {H}}\)). This approach allows Boolean Networks to be represented by a set of modal formulas and therefore can be used to describe and learn their properties. The study of a BDS focuses in particular on the search of stable configurations, limit cycles and unstable cycles, which help to characterize a large type of Gene Networks. In this article is presented the identification of such asymptotic properties by introduction of a new concept, ghost extensions. Using ghost extensions, it is possible to translate BDSs in propositional calculus and consequently to use SAT algorithms.

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Notes

  1. The full definition of \(\mathcal {H}\) further states that any formula of first-order logic is in \(\mathscr {L}(\mathcal {H}{})\), and that, whenever f and g are in \(\mathscr {L}(\mathcal {H}{})\), \(\lnot f\), \((f \wedge g)\), \((f \vee g)\), \((f \rightarrow g)\), are in \(\mathscr {L}(\mathcal {H}{})\) too.

  2. Uncountable because we can apply the Cantor’s diagonal argument on the set of deterministic updating modes which are basically defined as infinite sequences of subsets of nodes of the network.

  3. Function k may appear naive, because \(x_1 \wedge \lnot x_1 \wedge x_2 = \bot\) (\(\top\) is the logic formula True and \(\bot\) is False), which gives an equivalent translation \(TR (h) = \{\mathrm {H}2 \rightarrow \text{L} \lnot 1, \mathrm {H}\lnot 2 \rightarrow \text{L} 1, \mathrm {H}\bot \rightarrow \text{L} 2, \mathrm {H}\top \rightarrow \text{L} \lnot 2\}\). However, one of the aims of this study is also to show that we can deal with functions of any kind, without the need of a pre-processing. The formalism of \(\mathcal {H}\) implicitly makes the expected simplifications.

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Authors and Affiliations

  1. Aix Marseille Univ., CNRS, LIS, Marseille, France

    Pierre Siegel, Vincent Risch & Sylvain Sené

  2. CNRS, LAAS, French West Indies University-LAMIA, Université de Toulouse, 7 Avenue du Colonel Roche, 31400, Toulouse, France

    Andrei Doncescu

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  1. Pierre Siegel
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Correspondence to Andrei Doncescu.

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Siegel, P., Doncescu, A., Risch, V. et al. Representation of gene regulation networks by hypothesis logic-based Boolean systems. J Supercomput 79, 4556–4581 (2023). https://doi.org/10.1007/s11227-022-04809-5

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