Modal hypothesis logic, Boolean dynamical systems and genetic networks

Genetic networks are biological systems that represent how genes or proteins interact in a cell. They are especially studied by means of automata networks and Boolean dynamical systems (BDSs). This article studies the representation of BDSs, using a modal hypothesis logic, namely H. In the BDS formalism, a genetic network can be represented either by an interaction graph (IG) or by a transition graph (TG). Each of these representations stress distinct characteristics. The dynamics of a BDS is characterized by a function f and an updating mode μ that organizes the entities updates over time. An important part of the studies done on BDSs focused on the analysis of both the stable configurations, or fixed points, of pf, μq, and stable/unstable cycles of pf, μq. The representation of a BDS by whatever default logic, ASP, or other nonmonotonic formalisms, enables to find stable configurations. However, these representations are not suitable to capture cyclic dynamical behaviors. We introduce representations for both IGs and asynchronous TGs in H, which leads to new formal results. They aim at making possible to discriminate between stable configurations, limit cycles and unstable cycles. While a previous work has studied in detail IGs, the present paper focuses mainly on TGs, for which ghost extensions, defined in H, play a key role.