Redefined observability matrix for Boolean networks and distinguishable partitions of state space

This paper redefines the observability matrix and defines the observability index for Boolean networks (BNs). In the new definition, a BN is observable if and only if the observability matrix is of full rank. The observability matrix is calculated by a simple algorithm and is applied to the problem of distinguishability of partitions. A partition of the state space is said to be distinguishable if any two distinct subsets are distinguishable, and the finest distinguishable partition (FDP) is finer than any other distinguishable partition. A necessary and sufficient condition is proposed for checking the distinguishability of any given partition, and an algorithm is proposed for calculating the FDP. The proposed results are illustrated by examples.