  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.
