Towards a Boolean dynamical system representation in a nonmonotonic modal logic
Résumé
Boolean dynamical systems (BDSs) represent the evolution
of interactions inside a finite network of entities taking
Boolean states over discrete time. These networks are classically
used to model interactions of biological networks. In
this context, a genetic network can be represented by both
a Transition Graph (TG) and an Interaction Graph (IG). The
precise relationship between IG and TG has been studied for
many years in dynamical systems theory while still an open
question. The global purpose of this article is to further study
this relationship via a logical representation of BDSs into
a nonmonotonic modal logic called Hypothesis Logic (H).
While the dynamics of a BDS are characterized by a function
f, an important part of the studies focused on the analysis
of both stable configurations (i.e. fixed points of f), and
stable/unstable cycles of f. For the representation of some
genetic networks with no negative feedback circuits, results
were previously obtained with some well known nonmonotonic
formalisms. So far however, BDSs representation by
most of these formalisms does not permit to capture cyclic
dynamical behaviors. Notably, the equivalent of a negative
circuit has no extension in default logic (DL). This is embarrassing
because these cycles may represent real interactions
in living organisms like the cell cycle. This possible
lack of extensions in DL was studied in H, for which theories
always have extensions while some of these, called ghost
extensions, are actually not extensions of the corresponding
theories in DL. This paper addresses to the question of a
first representation of the dynamics of BDSs with H, and
ghost extensions appear to be a powerful tool in this respect.
As we are especially concerned with cycles, it provides us
with hints of simple algorithms for computing exhaustively
both stable/unstable cycles and fixed points: distinguishing
between stable/unstable as well as enumerating all the solutions
in practice would be a major advance that would lead to
apprehend better inner fundamental aspects in biology.