Towards a Boolean dynamical system representation in a nonmonotonic modal logic

International audienceBoolean dynamical systems (BDSs) represent the evolutionof interactions inside a finite network of entities takingBoolean states over discrete time. These networks are classicallyused to model interactions of biological networks. Inthis context, a genetic network can be represented by botha Transition Graph (TG) and an Interaction Graph (IG). Theprecise relationship between IG and TG has been studied formany years in dynamical systems theory while still an openquestion. The global purpose of this article is to further studythis relationship via a logical representation of BDSs intoa nonmonotonic modal logic called Hypothesis Logic (H).While the dynamics of a BDS are characterized by a functionf, an important part of the studies focused on the analysisof both stable configurations (i.e. fixed points of f), andstable/unstable cycles of f. For the representation of somegenetic networks with no negative feedback circuits, resultswere previously obtained with some well known nonmonotonicformalisms. So far however, BDSs representation bymost of these formalisms does not permit to capture cyclicdynamical behaviors. Notably, the equivalent of a negativecircuit has no extension in default logic (DL). This is embarrassingbecause these cycles may represent real interactionsin living organisms like the cell cycle. This possiblelack of extensions in DL was studied in H, for which theoriesalways have extensions while some of these, called ghostextensions, are actually not extensions of the correspondingtheories in DL. This paper addresses to the question of afirst representation of the dynamics of BDSs with H, andghost extensions appear to be a powerful tool in this respect.As we are especially concerned with cycles, it provides uswith hints of simple algorithms for computing exhaustivelyboth stable/unstable cycles and fixed points: distinguishingbetween stable/unstable as well as enumerating all the solutionsin practice would be a major advance that would lead toapprehend better inner fundamental aspects in biology