Observability and dead-beat observers for Boolean networks modeled as polynomial discrete-time systems

Boolean networks are considered in this paper and are represented through polynomial discrete-time systems over the Galois field F₂. The local and the global observability properties of a Boolean network are then expressed in terms of the polynomial ideal generated by the measured output and its repeated time-shifts (the observability ideal). It is shown that any globally observable Boolean network can be immersed into a polynomial system that is linearizable by output injection, whence a dead-beat observer can be designed in such a case. In case of globally observable Boolean networks, the left inverse of the observability mapping can always be computed in closed-form through a Gröbner basis of the observability ideal. More in general, the computation of a Gröbner basis of the observability ideal is useful also for the partial inversion of the observability mapping in case of Boolean networks that are not globally observable, as it is shown for a genetic system, the lac operon.