The Idea of Systems in Ashby's "Design for a Brain"

Introduction

It is hard to imagine that anyone reading W. Ross Ashby’s “Design for a Brain” could fail to grasp his notion of system. However, it appears to be the case that there are those who indeed fail to grasp his concept of system. I will try to give a brief summary of his ideas using some slightly different wording to emphasise certain points but I wholly recommend reading the book itself.

Ashby was a medical doctor and a psychiatrists primarily concerned with the working of the brain. His approach to studying systems as studying systems behaviour rather than structure is aimed at dealing with the enormous complexity of the brain. In particular he wanted to know what, if any, principles might hold for regulatory and adaptive systems i.e. regulatory systems that adapt the regulatory strategy to cope with environmental changes. His definition of system and behaviour are chosen with that goal in mind. The definition he uses has many similarities with that of Ludwig von Bertalanffy and Jay Forester, but also has significance differences. For Ashby a system is a closed system where feedback can be quantified in terms of information. For Forrester again systems are closed system but the quantities are physical representing the flows of materials and energy. For Bertalanffy the quantities are again physical quantities but systems are open, that is, in particular they are subject to inputs which are not part of the system. Each author has chosen a notion of system and systems behaviour suited to their particular purpose of study. There is no point debating which is correct, only which is useful for what purpose.

Ashby said there are two ways that we may understand the behaviour of a system (a) by examining its parts and deriving how it must behave and (b) by the process of tracing out its behaviours by starting the system from each possible initial starting point and tracing its behaviour from there. He was well aware that the real process of understanding a system involves both routes. A system may be too complicated to understand exactly how it’s parts generate the behaviour but we may be able to sample its behaviours and plot out an approximation. We may then be able to utilise some knowledge of how it is constructed to tell us how the samples plots related to its overall behaviour. For example we may determined from its construction that in a realm of interested that it is approximately (mathematically) linear (i.e. solutions are additive). This may then allow us to fill in the curves in its behaviour between the sampled points.

His book “Design for a Brain” was written to introduce his ideas about adaptive systems to the biological, medical and cybernetic community of his day. The main text of the book describes his ideas, avoiding the use of mathematics, providing extensive examples and plain English explanations. The appendices recapitulate much of text much more succinctly using the theory of dynamical systems. His later book, “An Introduction to Cybernetics”, was a tutorial text intended to be a self-teaching guide to the ideas behind cybernetics and their use in analysing systems. It explains the essential mathematical ideas behind his analysis of systems using simple examples and has many exercises to help the reader understand the content of each chapter. Most of the exercises are not framed in mathematical terms. It’s mathematical level rarely move beyond English maths ‘O’ level (or the equivalent) and when it does it is only for optional exercises. The greatest mistake a reader can make with this book is not doing the exercises (in this respect it is very like a maths book). After writing “an Introduction to Cybernetics” Ashby produced a second edition of “Design for A Brain”. This new edition brings the two books into better alignment and allowed Ashby to improve his explanation of ideas around adaptive systems building on the improved understanding he had developed in the intervening years. Ashby says it is the latter two thirds of the book which has been altered most significantly. That said, it is presentation rather than content which has altered and the first third of the book is where Ashby introduces his notion of system that we are concerned with in this essay. So whichever work you have read or are reading this should be relevant to you.

Ashby’s Notion of System

Before setting out Ashby’s definition of system it is first advisable to consider the meaning of certain terms. A variable is a named, measurable, quantity, that may vary over time, associated with some given phenomenon. The measured quantity for the variable at a given time is called the value of the variable at that time. So if water is flowing into a tank then we may name a variable <height of water in tank> and measure the height of the water in the tank at some given time, say today at 5pm, and we find it was 6cm, and we may say that the value of the variable <height of water in tank> at 5pm today was 6cm.

If we wish to think about several variables simultaneously we can think of each variable being an axes creating a coordinate system and the values of the variables as defining a point in the space defined by the coordinates. Fixing the order in which we take the coordinates fixes our viewpoint on the set of variables. Given some phenomenon and a selection of variables if we measure each variable over time and plot these as points in the coordinate system we will get a line. This line Ashby calls a line of behaviour.

In the following we will assume we are dealing with some fixed phenomenon and some fixed selection of variables. Clearly the order we take the coordinates in does not alter what we are looking at and so for convenience we will assume some fixed order is given throughout in what follows.

Talking about the set of variables and the corresponding set of values of those variables is slightly cumbersome, so we will call the set of variables, given in the order corresponding to fixed coordinate system the state vector, and the the values, given in the same order, the state.

Now consider all lines of behaviour that can be created by setting the state vector to a particular state where each state is tried many times. The plot of all possible lines of behaviour is called the field of the state vector. The field so generated is called the behaviour of the phenomenon.

The way the state changes with time when the state vector is started at some state is constrained by whatever mechanism underlies the particular phenomenon and links the variables.

If there is only one line of behaviour for each state, the field is said to be state determined and we say the phenomenon has state determined behaviour.

A mechanism which gives rise to a state determined behaviour is a deterministic system (aka state determined system, determinant system; Ashby used the term determinant system).

If there are some states from which there are multiple lines of behaviour, so that starting at such a state the line of behaviour followed is a matter of chance, then the field is said to be non-deterministic and the phenomenon is said to have non-deterministic behaviour. A mechanism which give rise to non-deterministic behaviour is said to be a non-deterministic system.

Ashby’s general objective is to find a set of variables that render a deterministic system. In much of his writing the term unqualified term system means deterministic system.

Note that a system is the mechanism that connects the variables that constitute the state vector associated with some phenomenon. In this sense when talking about systems, Ashby is talking about concrete things rather than descriptions of things. That is, for Ashby, systems are distinct from system descriptions which are how a system might be described in some language.

Clearly the process outlined above cannot be carried out in practise for most systems. We cannot simply start the state vector in every possible state and watch the evolution of the state vector from that point. The process is an idealisation designed to introduce the notion of a behaviour.

Systems and Systems Descriptions

Finally it is perhaps worth clearing up the matter of system versus system description. At least in relation to Ashby’s systems. Here we need a little mathematics. To simplify things we will consider only discrete time systems. A discrete time deterministic system is a function that maps a state vector to a new state vector. Because time is discrete there is a well defined notion of the next state following a given state and we get from the given state to the next state by applying the function (often called the transition function). If we think of a simple example of such a thing we can consider a counter which increments by 1 on each application of the function. So we can start at any state, say 4,? and keep applying the function e.g. f(f(f(4))) to move the state to 7. Now assume we can have a system with the state being integer values and the transion function being any function from integers to integers. Trivially there are uncountable many such functions which are distinct but there are only countable many descriptions of functions. So there are fewer descriptions of systems than there are systems, at least in the Ashby sense of system.

Of course, you may only deal with systems with finite descriptions. Indeed, if you only ever construct systems rather than study existing systems this might be adequate. Similarly you may only ever study finite systems. So why not adopt such limits? Well, perhaps that’s a topic for another day.