# The Number of Fixed Points of AND-OR Networks with Chain Topology

## Author(s)

Alan Veliz-Cuba, Ph.D.

Mathematics

4-1-2019

Honors Thesis

## Abstract

Boolean networks are sets of Boolean functions, which are functions that contain Boolean variables and the logical operators AND, OR, and NOT. In the simple case, the variables can be in one of two states—either 1 or 0, which can be interpreted in different ways such as ON or OFF, or TRUE or FALSE, depending on the application. Arranging model systems into Boolean functions, we can study steady states of these networks. This refers to the overall state of the dynamical system given an initial condition and another theoretical condition such as a subsequent point in time. Boolean networks have many applications, such as those in mathematics and computer science, and they can be used to study biological systems, especially to model gene networks.

The wide range of applications for Boolean networks brings us to two important questions: how do we compute steady states, and how do we find the number of fixed points? Computing the number of fixed points is very difficult. One way to simplify the computation is to focus on certain classes of networks. Another way to simplify our scope is by focusing on certain network topologies. We focus on AND-OR networks with chain topology.

AND-OR networks are Boolean networks where each coordinate function is either the AND or OR logical operator. We study the number of fixed points of these Boolean networks in the case that they have a wiring diagram with chain topology. We find closed formulas for subclasses of these networks and recursive formulas in the general case. Our results allow for an effective computation of the number of fixed points of AND-OR networks with chain topology. We further explore how our approach could be used in “fractal” chains.

## Permission Statement

This item is protected by copyright law (Title 17, U.S. Code) and may only be used for noncommercial, educational, and scholarly purposes

Mathematics

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