JKAU: Eng. Sci., Vol.19 No.2, pp: 41-49 (2008A.D. / 1429 A.H.) 41 [Short Note] On Limitations of Using Scalar Equations for Analyzing Synchronous Boolean Networks Ali M. Ali Rushdi and Sultan O. Saad Al-Otaibi Department of Electrical and Computer Engineering King Abdulaziz University, Jeddah, Saudi Arabia arushdi@kau.edu.sa and sootaibi@hotmail.com Abstract. A total description of a synchronous Boolean network is typically achieved by a matrix recurrence equation. A simpler description is possible when such a matrix equation is replaced by a scalar equation or a reduced scalar equation. Unfortunately, the scalar- equation technique suffers from several shortcomings and limitations, both in procedure and results. We discuss these shortcomings and support our arguments with two illustrative examples. 1. Introduction A Synchronous Boolean network is a set of n nodes, each of which is either in state 1 (on) or state 0 (off) at any given time t. Each node is updated at time t+1 by inputs from any fixed subset of the set of nodes according to any desired logical rule. Since the total number of possible network states is finite (2 n ) and the network changes states sequentially in discrete time steps, the network must necessarily return to a previously occupied state in a finite time (at most 2 n time points). This means that all possible trajectories of the network consist of either cycles (loops) of any length from size one (a fixed point) to a maximum of 2 n , or transient states leading eventually to a cycle. An ideal total description of the network (in which one accounts for all 2 n states) can be realized only for small n, and would be unfeasible for most networks of interest that