Quotients of probabilistic Boolean networks

TL;DR

This paper introduces a method to analyze quotients of probabilistic Boolean networks (PBNs), enabling simplified control synthesis and applications in biological systems by leveraging quotient transition systems.

Contribution

It develops a systematic approach to obtain quotient PBNs from equivalence relations, facilitating easier controller synthesis and analysis.

Findings

Quotient PBNs can be constructed to capture behavior of original networks.

Control laws can be designed on quotient systems and lifted back to original networks.

Application demonstrated with a biological example.

Abstract

A probabilistic Boolean network (PBN) is a discrete-time system composed of a collection of Boolean networks between which the PBN switches in a stochastic manner. This paper focuses on the study of quotients of PBNs. Given a PBN and an equivalence relation on its state set, we consider a probabilistic transition system that is generated by the PBN; the resulting quotient transition system then automatically captures the quotient behavior of this PBN. We therefore describe a method for obtaining a probabilistic Boolean system that generates the transitions of the quotient transition system. Applications of this quotient description are discussed, and it is shown that for PBNs, controller synthesis can be performed easily by first controlling a quotient system and then lifting the control law back to the original network. A biological example is given to show the usefulness of the developed results.