Most Permissive Semantics of Boolean Networks

As shown in (http://dx.doi.org/10.1101/2020.03.22.998377), the usual update modes of Boolean networks (BNs), including synchronous and (generalized) asynchronous, fail to capture behaviors introduced by multivalued refinements. Thus, update modes do not allow a correct abstract reasoning on dynamics of biological systems, as they may lead to reject valid BN models. This technical report lists the main definitions and properties of the most permissive semantics of BNs introduced in http://dx.doi.org/10.1101/2020.03.22.998377. This semantics meets with a correct abstraction of any multivalued refinements, with any update mode. It subsumes all the usual updating modes, while enabling new behaviors achievable by more concrete models. Moreover, it appears that classical dynamical analyzes of reachability and attractors have a simpler computational complexity: - reachability can be assessed in a polynomial number of iterations. The computation of iterations is in NP in the very general case, and is linear when local functions are monotonic, or with some usual representations of functions of BNs (binary decision diagrams, Petri nets, automata networks, etc.). Thus, reachability is in P with locally-monotonic BNs, and P$^{\text{NP}}$ otherwise (instead of being PSPACE-complete with update modes); - deciding wherever a configuration belongs to an attractor is in coNP with locally-monotonic BNs, and coNP$^{\text{coNP}}$ otherwise (instead of PSPACE-complete with update modes). Furthermore, we demonstrate that the semantics completely captures any behavior achievable with any multilevel or ODE refinement of the BN; and the semantics is minimal with respect to this model refinement criteria: to any most permissive trajectory, there exists a multilevel refinement of the BN which can reproduce it. In brief, the most permissive semantics of BNs enables a correct abstract reasoning on dynamics of BNs, with a greater tractability than previously introduced update modes.