# Categorization Problem on Controllability of Boolean Control Networks

## TL;DR

This paper introduces an algebraic graph theoretic method to classify pairs of states in Boolean control networks based on whether their reachable and unreachable time step sets are finite or infinite.

## Contribution

It develops a novel algebraic graph approach to categorize state pairs in BCNs by their reachability properties, addressing the controllability categorization problem.

## Key findings

- Method can classify all state pairs into four categories.
- Provides a systematic way to analyze controllability in BCNs.
- Applicable to various types of Boolean control networks.

## Abstract

A Boolean control network (BCN) is a discrete-time dynamical system whose variables take values from a binary set $\{0,1\}$. At each time step, each variable of the BCN updates its value simultaneously according to a Boolean function which takes the state and control of the previous time step as its input. Given an ordered pair of states of a BCN, we define the set of reachable time steps as the set of positive integer $k$'s where there exists a control sequence such that the BCN can be steered from one state to the other in exactly $k$ time steps; and the set of unreachable time steps as the set of $k$'s where there does not exist any control sequences such that the BCN can be steered from one state to the other in exactly $k$ time steps. We consider in this paper the so-called categorization problem of a BCN, i.e., we develop a method, via algebraic graph theoretic approach, to determine whether the set of reachable time steps and the set of unreachable time steps, associated with the given pair of states, are finite or infinite. Our results can be applied to classify all ordered pairs of states into four categories, depending on whether the set of reachable (unreachable) time steps is finite or not.