View source for Variety (cybernetics)

{{Short description|Number of states of a cybernetic system}}
In [[Cybernetics|cybernetics]], the term '''variety''' denotes the total number of distinguishable elements of a [[Set (mathematics)|set]], most often the set of states, inputs, or outputs of a [[Finite-state machine|finite-state machine]] or [[Transformation (function)|transformation]], or the binary logarithm of the same quantity.<ref name="Ashby 1956">{{cite book
 |last= Ashby
 |first= William Ross
 |date= 1956
 |title= An Introduction to Cybernetics
}}</ref> Variety is used in cybernetics as an [[Information theory|information theory]] that is easily related to deterministic finite automata, and less formally as a conceptual tool for thinking about organization, regulation, and stability. It is an early theory of [[Complexity|complexity]] in [[Automata theory|automata]], complex systems,<ref name="Ashby 1956"/>{{rp|6}} and [[Operations research|operations research]].<ref name="Ashby 1958">{{cite journal
| last1      = Ashby
| first1     = William Ross
| date       = 1958
| title      = Requisite Variety and its Implications for the Control of Complex Systems
| url        = http://134.184.131.111/Books/AshbyReqVar.pdf
| journal    = Cybernetica
| volume     = 1
| issue      = 2
}}</ref> 

== Overview ==
The term "variety" was introduced by [[Biography:W. Ross Ashby|W. Ross Ashby]] to extend his analysis of machines to their set of possible behaviors.<ref>{{harvnb|Ashby|1956|p=121}}: "In Part I we considered the main properties of the machine, usually with the assumption that we had before us the actual thing ... To progress in cybernetics, however, we shall have to extend our range of consideration. The fundamental questions in regulation and control can be answered only when we are able to consider the broader set of what it ''might'' do..."</ref>{{rp|121}} Ashby says:<ref name="Ashby 1956"/>{{rp|126}}
<blockquote>
The word <em>variety</em>, in relation to a set of distinguishable elements, will be used to mean either (i) the number of distinct elements, or (ii) the logarithm to the base 2 of the number, the context indicating the sense used. <!-- emphasis in original -->
</blockquote>

In the second case, variety is measured in [[Bit|bits]]. For example, a machine with states <math>\{a,b,c,d\}</math> has a variety of four states or two bits. The variety of a [[Sequence|sequence]] or [[Multiset|multiset]] is the number of distinct symbols in it. For example, the sequence <math>a,b,c,c,c,d</math> has a variety of four. As a measure of uncertainty, variety is directly related to information: <math>\text{Uncertainty} = - \text{Information}</math>.<ref name="Pask 1961">{{cite book
 |last= Pask
 |first= Gordon
 |date= 1961
 |title= An Approach to Cybernetics
}}</ref>{{rp|26}}

Since the number of distinguishable elements depends on both the observer and the set, "the observer and his powers of discrimination may have to be specified if the variety is to be well defined".<ref name="Ashby 1956"/>{{rp|125}} Gordon Pask distinguished between the variety of the chosen reference frame and the variety of the system the observer builds up within the reference frame. The reference frame consists of a state space and the set of measurements available to the observer, which have total variety <math>\log_2(n)</math>, where <math>n</math> is the number of states in the state space. The system the observer builds up begins with the full variety <math>\log_2(n)</math>, which is reduced as the observer loses uncertainty about the state by learning to predict the system. If the observer can perceive the system as a deterministic machine in the given reference frame, observation may reduce the variety to zero as the machine becomes completely predictable.<ref name="Pask 1961"/>{{rp|27}}

Laws of nature constrain the variety of phenomena by disallowing certain behavior.<ref name="Ashby 1956"/>{{rp|130}} Ashby made two observations he considered laws of nature, the law of experience and the law of requisite variety. The law of experience holds that machines under input tend to lose information about their original state, and the law of requisite variety states a necessary, though not sufficient, condition for a regulator to exert anticipatory control by responding to its current input (rather than the previous output as in [[Negative feedback#Error-controlled regulation|error-controlled regulation]]).

==Law of experience==
The ''law of experience'' refers to the observation that the variety of states exhibited by a deterministic machine in isolation cannot increase, and a set of identical machines fed the same inputs cannot exhibit increasing variety of states, and tend to synchronize instead.<ref>
{{harvnb|Ashby|1956|p=138}}: "It is easy to see, therefore, that, provided the same change is made to all, change of parameter-value to the whole set cannot increase the set's variety ... change of parameter value makes possible a fall to a new, and low, minimum ... Since this will often happen we can make the looser, but more vivid statement that a uniform change at the inputs of a set of transducers tends to drive the set's variety down."</ref>
<blockquote>
Some name is necessary by which this phenomenon can be referred to. I shall call it the <em>law of Experience.</em> It can be described more vividly by the statement that information put in by change at a parameter tends to destroy and replace information about the system's initial state.<ref name="Ashby 1956"/>{{rp|139}}<!--emphasis in original-->
</blockquote>

This is a consequence of the ''decay of variety'': a deterministic transformation cannot increase the variety of a set. As a result, an observer's uncertainty about the state of the machine either remains constant or decreases with time. Ashby shows that this holds for machines with inputs as well. Under any constant input <math>P_1</math> the machines' states move toward any [[Attractor|attractors]] that exist in the corresponding transformation and some may synchronize at these points. If the input changes to some other input <math>P_2</math> and the machines' behavior enacts a different transformation, more than one of these attractors may sit in the same basin of attraction under <math>P_2</math>. States which arrived and possibly synchronized at those attractors under <math>P_1</math> then synchronize further under <math>P_2</math>. "In other words," Ashby says, "changes at the input of a transducer tend to make the system's state (at a given moment) less dependent on the transducer's individual initial state and more dependent on the particular sequence of parameter-values used as input."<ref name="Ashby 1956"/>{{rp|136-138}}

While there is a law of non-increase, there is only a tendency to decrease, since the variety can hold steady without decreasing if the set undergoes a [[Injective function|one-to-one transformation]], or if the states have synchronized into a subset for which this is the case. In the [[Formal language|formal language]] analysis of finite machines, an input sequence that synchronizes identical machines (no matter the variety of their initial states) is called a [[Synchronizing word|synchronizing word]].

==Law of requisite variety==
[[File:LRV game.svg|thumb|D emits disturbances, to which R emits responses. The table T describes the interaction between D and R's output, and the outcome of this interaction is expressed in E.<ref name="Ashby 1956"/>{{rp|210}} ]]
Ashby used variety to analyze the problem of [[Finance:Regulation|regulation]] by considering a two-player [[Game theory|game]], where one player, <math>D</math>, supplies disturbances which another player, <math>R</math>, must regulate to ensure acceptable outcomes. <math>D</math> and <math>R</math> each have a set of available moves, which choose the outcome from a table with as many rows as <math>D</math> has moves and as many columns as <math>R</math> has moves. <math>R</math> is allowed full knowledge of <math>D</math>'s move, and must pick moves in response so that the outcome is acceptable.<ref name="Ashby 1956"/>{{rp|202}} 

Since many games pose no difficulty for <math>R</math>, the table is chosen so that no outcome is repeated in any column, which ensures that in the corresponding game any change in <math>D</math>'s move means a change in outcome, unless <math>R</math> has a move to keep the outcome from changing. With this restriction, if <math>R</math> never changes moves, the outcome fully depends on <math>D</math>'s choice, while if multiple moves are available to <math>R</math> it can reduce the variety of outcomes, if the table allows it, dividing by as much as its own variety of moves.<ref name="Ashby 1956"/>{{rp|204}} 

<math>
\begin{array} {c c | c}
& & R \\
& & \begin{array} { c c c }
\alpha & \beta & \gamma
\end{array}
\\ \hline
D & \begin{array}{ c | c c c }
    1 \\ 2 \\ 3  \\ 4 \\ 5 \\ 6
    \end{array}
& \begin{array}{c c c }
  \mathbf{a} & f & d \\
  \mathbf{b} & e & c \\
  c & d & \mathbf{b} \\
  d & c & \mathbf{a} \\
  e & \mathbf{b} & f \\
  f & \mathbf{a} & e \\
  \end{array}
\end{array}
</math>

The ''law of requisite variety'' is that a deterministic strategy for <math>R</math> can at best limit the variety in outcomes to <math>\tfrac{D\text{'s variety}}{R\text{'s variety}}</math>, and only adding variety in <math>R</math>'s moves can reduce the variety of outcomes: "<em>only variety can destroy variety</em>".<ref name="Ashby 1956"/>{{rp|207}}<!-- emphasis in original--> For example, in the table above, <math>R</math> has a strategy (shown in bold) to reduce the variety in outcomes to <math>|\{a,b\}| = 2 = \tfrac{6}{3}</math>, which is <math>\tfrac{D\text{'s variety}}{R\text{'s variety}}</math> in this case. Ashby considered this a fundamental observation to the theory of regulation.

It is not possible for <math>R</math> to reduce the outcomes any further and still respond to all potential moves from <math>D</math>, but it is possible that another table of the same shape would not allow <math>R</math> to do so well. Requisite variety is necessary, but not sufficient to control the outcomes. If <math>R</math> and <math>D</math> are machines, they cannot possibly choose more moves than they have states. Thus, a perfect regulator must have at least as many distinguishable states as the phenomenon it is intended to regulate (the table must be square, or wider). 

Stated in bits, the law is <math>V_O \ge V_D - V_R</math>. In Shannon's information theory, <math>D</math>, <math>R</math>, and <math>E</math> are information sources. The condition that if <math>R</math> never changes moves, the uncertainty in outcomes is no less than the uncertainty in <math>D</math>'s move is expressed as <math>H(E|R) \ge H(D|R)</math>, and since <math>R</math>'s strategy is a deterministic function of <math>D</math> set <math>H(R|D) = 0</math>. With the rules of the game expressed this way, it can be shown that <math>H(E) \ge H(D) - H(R)</math>.<ref name="Ashby 1956"/>{{rp|207–208}} Ashby described the law of requisite variety as related to the tenth theorem in Shannon's Mathematical Theory of Communication (1948):<ref>W. R. Ashby (1960), [https://archive.org/details/designforbrainor00ashb/ "Design for a Brain"], p. 229.</ref>
<blockquote>
This law (of which Shannon's theorem 10 relating to the suppression of noise is a special case) says that if a certain quantity of disturbance is prevented by a regulator from reaching some essential variables, then that regulator must be capable of exerting at least that quantity of selection.
</blockquote>

Ashby also postulated that the law of requisite variety allows for the measurement of regulation, namely that the requirement for a well-functioning regulation is that the regulator or regulators in place are designed to account for all the possible states in which the variable or variables to be regulated may fall within, so as to ensure that the outcome is always within acceptable range.<ref name="Ashby 1956"/>{{rp|209}} Ashby saw this law as relevant to problems in biology such as [[Homeostasis|homeostasis]], and a "wealth of possible applications". Later, in 1970, Conant working with Ashby produced the [[Good regulator|good regulator]] theorem<ref>Conant 1970</ref> which required autonomous systems to acquire an internal model of their environment to persist and achieve stability (e.g. [[Nyquist stability criterion]]) or [[Physics:Dynamic equilibrium|dynamic equilibrium]]. 

Boisot and McKelvey updated this law to the '''"law of requisite complexity"''', that holds that, in order to be efficaciously adaptive, the internal [[Complexity|complexity]] of a system must match the external complexity it confronts. A further practical application of this law is the view that information systems (IS) alignment is a continuous coevolutionary process that reconciles top-down ‘rational designs’ and bottom-up ‘emergent processes’ of consciously and coherently interrelating all components of the Business/IS relationships in order to contribute to an organization’s performance over time. 
<ref>{{Cite journal |last1=Benbya |first1=H. |last2=McKelvey |first2=B. |date=2006 |title=Using coevolutionary and complexity theories to improve IS alignment: a multi-level approach |journal=Journal of Information Technology |language=en |volume=21 |issue=4 |pages=284–298 |doi=10.1057/palgrave.jit.2000080 |s2cid=15214275}}</ref><ref>{{Cite journal |last1=Boisot |first1=M. |last2=McKelvey |first2=B. |date=2011 |title=Complexity and organization-environment relations: revisiting Ashby's law of requisite variety |journal=P. Allen, the Sage Handbook of Complexity and Management |language=en |pages=279–298}}</ref>

The application in project management of the law of requisite complexity is the model of [[Project complexity|positive, appropriate and negative complexity]] proposed by Stefan Morcov.

== Applications ==

Applications to organization and management were immediately apparent to Ashby. One implication is that individuals have a finite capacity for processing information, and beyond this limit what matters is the organization between individuals.<ref name="Ashby 1958"/>
<blockquote>
Thus the limitation which holds over a team of ''n'' men may be much higher, perhaps ''n'' times as high, as the limitation holding over the individual man. To make use of the higher limit, however, the team must be efficiently organized; and until recently our understanding of organization has been pitifully small.
</blockquote>
Stafford Beer took up this analysis in his writings on [[Management cybernetics|management cybernetics]]. Beer defines variety as "the total number of ''possible'' states of a system, or of an element of a system".<ref name="Beer 1981">Beer (1981)</ref> Beer restates the Law of Requisite Variety as "Variety absorbs variety."<ref>Beer (1979) p286</ref> Stated more simply, the logarithmic measure of variety represents the minimum number of choices (by binary chop) needed to resolve [[Physics:Uncertainty|uncertainty]]. Beer used this to allocate the management resources necessary to maintain process viability.

The cybernetician Frank George discussed the variety of teams competing in games like football or rugby to produce goals or tries. A winning chess player might be said to have more variety than his losing opponent. Here a simple [[Order (group theory)|ordering]] is implied. The [[Physics:Attenuation|attenuation]] and [[Engineering:Amplifier|amplification]] of variety were major themes in Stafford Beer's work in management <ref name="Beer 1981"/> (the profession of control, as he called it).  The number of staff needed to answer telephones, control crowds or tend to patients are clear examples.

The application of natural and analogue signals to variety analysis require an estimate of Ashby's "powers of discrimination" (see above quote). Given the [[Butterfly effect|butterfly effect]] of dynamical systems care must be taken before quantitative measures can be produced. Small quantities, which might be overlooked, can have big effects. In his ''Designing Freedom'' Stafford Beer discusses the patient in a hospital with a temperature denoting fever.<ref>Beer (1974)</ref> Action must be taken immediately to isolate the patient. Here no amount of variety recording the ''patients' average temperature'' would detect this small signal which might have a big effect. Monitoring is required on individuals thus amplifying variety (see ''Algedonic alerts'' in the [[Viable system model|viable system model]] or VSM). Beer's work in management cybernetics and VSM is largely based on variety engineering.

Further applications involving Ashby's view of state counting include the analysis of digital [[Bandwidth (computing)|bandwidth]] requirements, [[Redundant code|redundancy]] and [[Software bloat|software bloat]], the bit representation of data types and [[B-tree|indexes]], [[Analog-to-digital converter|analogue to digital conversion]], the bounds on finite state machines and [[Data compression|data compression]]. See also, e.g., [[Physics:Excited state|Excited state]], [[State (computer science)]], [[State pattern]], State (controls) and [[Cellular automaton]]. Requisite Variety can be seen in Chaitin's [[Algorithmic information theory]] where a longer, higher variety program or finite state machine produces incompressible output with more variety or information content.

In general a description of the required inputs and outputs is established then encoded with the minimum variety necessary. The mapping of input bits to output bits can then produce an estimate of the minimum hardware or software components necessary to produce the desired [[Control system|control]] behaviour; for example, in a piece of computer software or [[Computer hardware|computer hardware]].

Variety is one of nine requisites that are required by an [[Ethical regulator|ethical regulator]].<ref>M. Ashby, [http://ashby.de/Ethical%20Regulators.pdf "Ethical Regulators and Super-Ethical Systems"], 2017</ref>

== See also ==
* [[Cardinality]]
* [[Complexity]]
* [[Chemistry:Degrees of freedom (physics and chemistry)|Degrees of freedom]]
* [[Power set]]
* Practopoiesis
* [[Waterbed theory]]
* [[Good regulator]]
* [[Ethical regulator]]
* [[State (computer science)|State (Computer Science)]]
* Myhill-Nerode Theorem
* [[Space complexity]]
* [[Project management#Project complexity|Project Complexity]]

== References ==
<references/>

== Further reading ==
* Ashby, W. R. 1956, An Introduction to Cybernetics, Chapman & Hall, 1956, {{ISBN|0-416-68300-2}} (also available in [http://pespmc1.vub.ac.be/ASHBBOOK.html electronic form as a PDF] from ''Principia Cybernetica'')
* Ashby, W. R. 1958, [https://web.archive.org/web/20181016052643/http://pespmc1.vub.ac.be/books/AshbyReqVar.pdf Requisite Variety and its implications for the control of complex systems], Cybernetica (Namur) Vol. 1, No. 2, 1958.
* Ashby, W. R. 1960, Design for a brain; the origin of adaptive behavior, 2nd ed. ([https://archive.org/details/designforbrainor00ashb/ Electronic versions on Internet Archive]).
* Beer, S. 1974, Designing Freedom, CBC Learning Systems, Toronto, 1974; and John Wiley, London and New York, 1975. Translated into Spanish and Japanese.
* Beer, S. 1975, Platform for Change, John Wiley, London and New York. Reprinted with corrections 1978.
* Beer, S. 1979, The Heart of Enterprise, John Wiley, London and New York. Reprinted with corrections 1988.
* Beer, S. 1981, Brain of the Firm; Second Edition (much extended), John Wiley, London and New York. Reprinted 1986, 1988. Translated into Russian.
* Beer, S. 1985, Diagnosing the System for Organisations; John Wiley, London and New York. Translated into Italian and Japanese. Reprinted 1988, 1990, 1991.
* Conant, R. 1981, Mechanisms of Intelligence: Ross Ashby's papers and writings, Intersystems Publications, {{ISBN|1-127-19770-3}}.

==External links==

*[http://pespmc1.vub.ac.be/REQVAR.html The Law of Requisite Variety] in the Principia Cybernetica Web, 2001.
*[http://www.isss.org/911/leonard.htm Systems concepts and 9/11] Allenna Leonard on Requisite Variety
*[http://www.ashby.de/rossashby.info/journal/index/thumbs/requisite.html All references to The Law of Requisite Variety] in Ross Ashby's journal 1953–1961.
*[https://www.youtube.com/watch?v=uu9yMq4z2cA&feature=channel Management Cybernetics: The Law of Requisite Variety] Livas short introductory videos on [[Company:YouTube|YouTube]]
*[http://www.danko-nikolic.com/practopoiesis/ Practopoiesis]: How biological systems get their variety
*[https://www.cbc.ca/radio/ideas/the-1973-cbc-massey-lectures-designing-freedom-1.2946819 The 1973 CBC Massey Lectures, "Designing Freedom"]

{{DEFAULTSORT:Variety (Cybernetics)}}
[[Category:Cybernetics]]

{{Sourceattribution|Variety (cybernetics)}}