Pask Gordon an Approach to Cybernetics

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An ApproachtoCybernetics

Gordon Pask


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An approach toCyberneticsGordon PaskWITH A PREFACE BY WARREN S. McCULLOCHMASSACHUSETTS INSTITUTE OF TECH NOLOGY

A RADIUS BOOK / HUTCHINSON


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H UTC H INSON & CO (Publishers) LTD3 Fitzroy Sqflare. London WiLondon Melbourne Sydney AucklandWellington Johannesburg Cape Town

and agencies throughout the worldFirst pliblishl'd 1961

This edition March 1968Second inrprusion October 1968Third impression July 1971

To Elizabeth

10 Gordon Pask 1961Printed in Great Britain by liIho on smooth wove paper

by Anchor Press, and bound by Wm. Brendan,both 0/ Tiptr,.,.. Essex

ISBN 0 09 086810 2 (cased)o 09 08681l 0 (paper)

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ContentsACKNOWLEDGMENTS

PREFACE

I TH E BA C KGROUN D OF CYBERNETICSIntroduction - Origins - Definitions of cyberneticsCommon misconceptions - Summary

2 LEARNING, OBSERVATION AND PREDI C TIONObservers - The consequences of uncertainty - The typeof uncertainty - The source of uncertainty - Definitionof a system - Phase space - Procedure of an observerMeasuremen t of uncertainty and of information conveyed.

3 TH E STATE DE T ERMINED BEHAVIOUREquilibrium behaviour - Working models and relationbetween systems - Object language and metalanguagePartitioning systems - Coupling systems - Alternativeprocedure - Statisticaldeterminacy- Markovian systems- Stochastic models - Non-Stationary systems - These lf-organizing system

4 CONTROL SYST EMSRegulation an d requisite variety - Automatic controllers - Distinction between perfectly and imperfectlyinformed controllers - Predictive controllers _ Conditional probability machines - Imitative controllersThe adaptive control system - Adaptive controllers inindustry - Adaptive controllers able to deal with a lesstidy environment - A real life artifact


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CONTEN T S5 BIOLOGICAL CONTROLLERS

Survival - Adaptation - The regulation of breathing-The brain/ike artifact - Problem solving - Recogn itionand abstraction of form s - Learning to recognize forms

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6 TEACHING MA CH I NES S8Adapti \"c teachers - Descriptive mode l- Add listing -Card punching - Aptitude testing

7 THE EVOLUTION AND RE P RO D UCTION O F MACHINESThe self-organizing systems - Abstract approach -Chemical c o m p u t ~ r ]008 II"DUS T R I AL CYBERNETICS

Its impact - The Slruclure of industry - DecisionmakingGLOSSARY

REFERENCES

APPENDICES

I NDEX

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114117J21127

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PlatesI (i) A pupil-teacher system

(ii) Adaptive reorganizing automatonfacin

I I Automatic keyboard instructor and maintenancetra ining teaching machine

III Chemical computersIV Act ivity surge in 2-d imensional cell array


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AcknowledgmentsI WI SH fO ack nowledge the help afforded by Alex: Andrew, W.Ross Ashby, Stafford Beer. and Christopher Bailey. who readthe draft manuscript, and Brian N. Lewis, whose commentshave been freely incorporated. lowe especial thanks to WarrenMcCulloch for days of constructive criticism and. in particular,to Heinz Von Foerster who undertook a comprehensive surveyf Ihe book and suggested many sorely needed improvements.My wife, Elizabeth, has typed several versions of the text andMr Tom Dalby. Hutchinsons and Harpers have shown patienced offered valuable advice.

GORDON PASK

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PrefaceT H [S book is not for the engineer content with hardware,the biologist uneasy outside his specialty; for it depicts thcegenation of Art and Science which begets inan imate objcbehave like living systems. They regulate themselvesand sThey adapt and they compute: They invent. They co-operthey compete. Naturally they evolve rapidly,Pure ma thematics, being mere tau tology, an d pure pbeing mere fact, could not have engendered them; for crto live, must sense the useful and the good; and enginesmust have energy available as work : and both, to endureregulate themselves. So it is to Thermodynamics andbrotherIp iogp, called Information T heory, that we look dis.tinctions between work and energy and betwcen signnOise.Fo r like cause we look to reflexology and its brotheback, christened Multiple Closed Loop Servo Theory, foranical explanation of Entelechy in Homeostasis and in appeThis is that governance, whether in living creatures ansocieties or in ou r lively artifacts. that is now called CyberBut under that title Norbert Wiener necessarily su bsumcomputation that, from afferent signals, forecasts succonducts in a changing world.To embody logic in proper hardware explains the lathought and consequently stems from psychology. Fo r nuthe digital ar t is as old as the abacus, but it came alive onlyTuring made the next operation of his machine hingevalue of the operand, whence its ability to compute anyputable number.

Fo r Aristotelian logic, the followers of Ramon Lull, incLeibnitz, have frequently made machines for three, andtimes four, classifications. The first of these to be live ly comcontingent probabilitiesWith this ability to make or select proper filters on its

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10 PREFACEs u ~ h a device e x p l ~ j n s the central problem of experimentalepistemology. The nddles of stimulus equivalence or of localcircuit action in thc brain remain only as parochial problems.This that ~ p a n d i n g world of beings, man-made or begotten,

c o n c e ~ 1 1 I n g which Ross Ashby asked, 'How can such systems?rgamze t?emselves?' His answer is, in one sense, too general andIts embodiment, too special to satisfy him, his friends or his followers.

This book describes their present toil to put his ideas to workso as to come to grips with his question.20th December, 1960. WARREN S. MCCULLOCH

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I The Background of CyberneticsIntroductionCYBERNETICS is a young discipline which, like applied matics, cuts across the entrenched departments of natural sthe sky, the earth, the animals and the plants. It s interdiscicharacteremerges when it considers economy no t as an econbiology no t as a biologist, engines no t as an engineer. In eaits theme remains the same, namely, how systems regulateselves, reproduce themselves, evolve and learn. Its highthe qucstion of how they organize themselves.A cybernetic laboratory has a varied worksheet - concemation in organized groups, teaching machines, brain mand chemical computers for use in a cybernetic factory. Ascientists we are concerned with brain-like artifacts, withtion, growth and development; with the process of thinkigetting to know about the world. Wearing the ha t of ascience, we aim to create what Boulanger,' in his presiaddress to the International Association of Cybernetics,the instruments of a new industrial revolution - control misms that lay their own plans.The crux of organization is stability, for 'that which iscan be described; either as the organization itself, orcharacteristic which the organization preserves intact.which is stable' may be a dog, a population, an aeroplanJones', Jim Jones's body temperature, the speed of a shindeed, a host of other things.In chemistry, for example, Le ChatelIier's Principle is ament that the equilibrial concentration of reactants in a vessel is stable, for it asserts that the assembly will react snullify thermal or chemical disturbances. But the equiliwhich is always implied by the word stability, is rarelysimple kind. Jim Jones is in dynamic equilibrium wenvironment. He is no t energetically isolated and his consmaterial is being continually built up and broken dow

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12 AN APPROACH TO CYBERNET ICSinterchanged. When we say 'Jim Jones is stable', we mean theform, the organization that we recognize as Jim Jones, is invariant.Again, if Jim Jones drives his motor car his behaviour is (statistically speaking) stable, and (in the sense that a destination is reachedand no collision occurs) J im Jones and his aulomobile arc inequilibrium with their world.

Origins of CybemelicsA great deal of cybernetics is concerncd with how stability ismaintained with 'control mechanisms'. One of the first of theseto be treated explicitly was Watfs invcntion of the governor (atheoretical analysis was offered by Maxwell in 1865). The deviceillustrates a principle called lIegmil'eJeedback. A signal, indicatingthe speed of a steam engine, is conveyed to a power amplifyingdevice (in this case, a steam throttle) in such a way that whenthe engine accelerates the steam supply is reduced. Hence, thespeed is kept stable. The signalling arrangement is independentof energetic considerations, and it is legitimate to envisage thegovernor as a device which feeds back information in order toeffect speed control.PhYSiological SourcesPerhaps the earliest cybernetic thinking comes within the compassof physiology, where the key notions of information feedbackand control appear as the ideas of reflex and homeostasis. In1817 Magendie defined a reflex as an activity produced by adisturbance of some part of the body which travelled (over thedorsal nerve roots) to the central nervous system, and was reflected(through the ventral nerve roots) to the point of origin where itmodified, slopped or reversed the original disturbance. The basicidea of signalling and directed activity is apparent (the commonmisinterpretation of a reflex as a mere relay action should beavoided). The elaboration of this idea in the early part of thepresent century, and the experiemental study of reflexes up toand beyond Pa vlov, is well known.Whereas rcflexis preserves the organism against the flux o fits environment, homeostasis counters the internally generatedchanges which are prone to disrupt the proper structure anddisposition of parts in the organism. Homeostatic mechanismsmaintain the milieu intemale of Claude Bernard, the proper

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THE BACKGROUND OF CYBERNETICSvalues of acidity, water balance and metabolites - a bodyature which the cells of the body ca n tolerate. Th e firsthensive study was published by Cannon in 1932 t and tvast amount of recent work (to cite a few representativeStanford Goldman; tTealiog blood sugar control as a fmechanism, T. H. Benzinger for a discussion of theregulator in the hypothalamus, an d Magoun, Peterson, Land M c C u l l o c h ~ for a study of feedback in postural treIn much, though not all, physiological control thech ief controller, and in effcctingcontrol, chief rccogn izer,izer and arbiter. Hence cybernetic thinking stems alpsychology and io turn makes comment. Studying themeet a feature common to most cybernetic investigatioassembly is so large that its details always, and its generasometimes, remain necessarily obscure. Hcre the mathmodels of our science are particularly valuable. One model is a network of formal neurones (a formal neurconstruct, depicting the least set of properties whichneurone, a constituent active cell of the brain, could possess). McCulloch, who p i o n e ~ r e d this field has renumber of conclusions. In particular he and Pitts showyears ago 70 that plausible networks of these formal nwere automata capable of many gambits, such as learnelaboration of gestalten and the embodiment of unHence, the corresponding modes of mentality are surprising nor adventitious when they appear in the felaborate real brain.Finally there is the question of 'purpose'. All the homand reflexive mechanisms are goal-directed and self.regThere is no magic about this and, whilst we can discern tno mystery either. But when. as often happens, a goal iby several interacting mechanisms, or several goals apbe sought by one, we might apply the term 'purposiveresulting behaviour. There is no suggestion of a vital fothough we rightly marvel at the organization, there is nointroduce teleological concepts). In particular we aret id purposive behaviour in assemblies like brains, wlarge and incompletely observed. But I do not wish toimpression tha t the generdtion ofpurposive or any other beis enlodged within a particular assembly. In cybernetic

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14 AN APPROACH TO CYBERNETICSthinking of an organization. Citing McCulloch's 1946 lecture,'Finality and Form' " . . some re-entrant paths lie within thecentral nervous system, others pass through remote parts of thebody and still others, leaving the body by effectors, returning byreceptors, traverse the external world. The functions of the firstare, at present, ill defined, the second constitute the majority ofour reflexes, the last our appetites and purposes . . .' The irtotality is the organism we study in cybernetics.Olher SourcesIn zoology and in embryology there used to be a problem equivalent to the teleolog ica l dilemma of purposive behaviour. Heft! ittook the name equifinality. Driesch, for example, was led tobelieve in a vital force, because the development of sea urchinembryos seemed to be pre-determined 'from outsidc' since theyreached the same final form even though crassly mutilated . Bythe ea rly 1910's biologists were thinking in terms of organiza tion(there is a classic paper of Paul Weiss," which bears this out)and it became obvious that in a wholly pedestrian manner thewhole of an organization is more than the sum of its parts, Themystique behind cquifinality (which lay there because, from aci rcumscribed point of view, the parts should ad d up to thewhole) evaporated like the apparent magic of purposiveness.Von Bcrtalanffy's thinking in this direction exerted considerableinfluence, no t only in biology bu t also in the social sciences, andhe gave the name system to the organization which is recognizedand studied (we speculate abou t the system which is the organization of a leopard and not about the leopard itself). Further, vonIkrtalanffy realized that when we look at systems (which cyberneticians always do) many apparently dissimilar assemblies andprocesses show features in common.' He called the search forunifying principles which re late different systems, GeneralSystems Theory.

General Systems Th eo ry found little acceptance in engineeringand had little relatior. to the physiological developments untilthe mid-194Os. About then, engineers had to make computingand control devices elaborate enough to exhibit the troublesomekinds of purposiveness already familiar in biology. Also it wasin the 1940 's that Julian Bigelow, then Rosenblueth and Wienerrealized the significance of the organizational viewpoint, and had

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THE BACKGROUND OF CYBER.NETIC'Sthe insight to wed together the developments we have dand the rigorous mathematics of communication enginDefinitions of CybeflleticsThus, cybernetics was born. Since then it has been vdefined. At one elltrerne, there is the or isinal d'the science of control and communication in the animamachine; advanced by Norbert Wiener ' when he adoword- in 1948 in the book Cybernetics which is the first cstatement of the discipline (a paper" anticipates a paar gument). At the other extreme is Lou is Couffignal'slo ppu t forward as an expansion in 1956, 'L a Cybernctrart d'assurer l'efficacite de raction.' The gap betweenand an is filled by a continuum of interpretations. Thus,Beerll looks upon cybernetics as the science of properwithin any assembly that is treated as an organic windustry, for example, this cou ld be the science of manaAlso he regards Operational Research, in its widest sensprincipal experimental method of cybernetics, the scienAshby," on the other hand, gives emphasis to abstrcontrolfable system from the flux of a real world (for abis a prerequisite of talk about contro!), and he is concerthe entirely general synthetic operations which can be peupon the ab stract image. He points ou t that cybernetmore restricted to the control of observable assembliesabstract systems that correspond with them, than seorestricted to describing figures in the Euclidean spacmodels ou r environment.

For my own part,n I subscribe to both Ashby's anview, finding them compatible. Their definitions arc bothby Wiener's global dictum.Th e cybernetician has a well spec ified , though giganticinterest. His object of study is a system, either construct

abstracted from a physical assembly, that it exhibits inbetween the parts, whereby one controls another, unclothe physical character of the pans themselves. He manand modifies his systems often using mathematical tecbut, because in practical affairs cybernetics is most

The world 'Cybernetics' was first used by Ampere as thesociologica l study, I t is derived from the Greek word for s


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16 AN .... PPROACH TO CYBERNETICSapplied to a wry large system, he may also build mechanicalarlifacts to model them. Simply because the particulars areirrelevant, he can legitimately examine such diverse .assembliesas genes in a chromosome, the contents of books in a library(with respect to information storage), ideas in brains, governmentand computing machines (with respect to the learning process).Commoll MisconceptiollsIt is easy to misinterpret the whole idea and conclude thatcybernetics is a trivial or even meaningless pursuit. We haveto answer the kind of criticism offered by Buck" - thatanything whatever can be a system - according to most cybernetic definitions of the word. But I believe an answer can begiven, providing we do not confuse the strict identity of principlebetween the workings of several assemblies, which the cybernetician tries to embody in his abstract system, with mere facileanalogy. The confusion does occur when people over-simplifythe supposed activities of a cybernetician, perhaps, for a popularaccount of them, by expressing these activities in tenns of asingle experiment.Let us suppose, for example, that Mr X is building a cyberneticmodel of some region of the brain. Mr X is approached by MrY who asks his profession. 'Cybernetician,' says Mr X. 'Suchnonsense,' says Y, 'I've never heard of it, bUI,' he adds, 'I cansee youre making a mode l of the brain .Be sensible and tell mewhether you are a psychologist, or an electronic engineer: If MrX insists that he is neilher, but a cybernetician, Y will make someprivate reservations and humour the man, pressing Mr X todescribe his activity 'a s though he were a psychologist' or 'a sthough he were an electronic engineer', because he can 'understand Ihal sort of language'. Fo r Y is convinced that X is makingsome electrical imitation of the brain. But if the device is a cybernetic model, then it is almost certainly a lery poor imitation. Inconsonance with Beer' Tsubmit that the workings ofa cyberneticmodel are identical with some feature in the work.ings of a brainwhich is relevant to the control within a brain. Most likely, thisfeature is not readily describable in terms of psychology orelectronics. So, having missed the point, Y is ap t to depart underthe impression that X is bad at psychology and bad at electronicsand a little demented.

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THE BACKGROUND OF CYBERNETICSIt is easy to cite brain models which are merely im

most well-behaved robots, most of the tidy automata thaa naughts and crosses player, nearly all of the mazemachines (though there are some, like Deutsch's Rat, S wused explicitly to illustrate an organizational principlthan to imitate a response). There are not so many cymodels to choose from, but one of them, made by Ashcalled the Homcostat, admirably illustrates the distinis made up of four interacting regulators and an indeswitching mechanism which changes the interconnectionsthese elements until a stable arrangemenUs reached. rt cthe viewpoint of psychology and engineering respectidubbed a 'brain-like analogue and a 'device for solvingtial equations', for it does, rather imperfectly, display like behaviour and it will, rather eccentrically, solve difequations. I ts imperfections as an equation solver (whichmeant to be) are obvious from its construction and have a good deal of heavy-handed criticism. Its imperfectiobrain-like analogue (which, once again, it is not meanoccur because at the level of functional analogy the orgaof a homeostat is not particularly brainlike. It is only wcome to the level intended in the cybernetic abstractionself-regulation in a homeostat is idl'lIlical with the self-rein a brain, and with referc:,.ce to this feature the homeocybernetic model of all brains.SummaryTo summarize, a cybernetician adopts, so far as posattitude which lays emphasis upon those characteristphysical assembly which are common to each diseip'abstracts' them into his 'system.This is no t a prudent methodology, for it runs theseeming to be impertinent. It is justified in so far as it dto effective control procedures, efficient predictions, andable unifying theories (and whilst this is true of 011)' scisanctions are rightly enough weighted against a Jactrades). But the risk, on balance, is worth while, for the cyapproach call achieve generality and yield rigorous coupon organization.

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2 Learni ng, Observation and PredictionOBSERVERS are men, animals, or machines able to learn abouttheir environment and impelled to reduce their uncertainty aboutthe events which occur in it, by dint of learning. In this chapterwe shall ellamine human observers who, because we have aninside understand ing of their observational process, belong to aspecial category. Fo r the moment, we shall no t bother withHOW an observer learns, but will concentrate upon WHAT helearns about, i.e. what becomes more certain.As observers we expect the environment to change an d try todescribe those features that remain unchanged with the passageof time. An unchanging form of events due to the activity withinan assembly is called a behOl'iollr. The behaviour of a steamengine is a recurrent cycle of steam injection and piston movements that remains invariant. The behaviour ofa ca t is made upof performances like eating and sleeping and, once again, it is aninvariant fo rm selected from the multitude of things a cat mightpossibly do. The behaviour of a statue is a special case, for thestatue is immobile, or to usc an equivalent fo rmalism, it changesal each in stant of time into itself. We sha ll neglect the specialcase entirely. An 'assembly' is the dynamic part of an observer'senvironment, a piece of the real world, which is freely suppliedwith energy. Although the energetics do not immediately concernus, the assembly embodies one or many more or less regularmodes of dissipating the energy - steam expansion or metabolism- as a result of which it produces an unlimited supply of observable events.The Consequences of UncertaintyWhen we say that ou r uncertainty about the environment hasbeen reduced we mcan that a larger number of the behavioura lpredictions we make are turning out to be right. But J take as

(17) is a comprehensive textbook dealing with scientific observation.18

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LEARNING, OBSERVATION AN D PRIiDICTIONan axiom that our uncertainty about the environmenbe entirely removed. Any ObSCTI'Olioll of the real worldand occupies a definite interval Lll.On the other hand , predictiolls are always dogmaticthe dogma ca n be modified in the light of further evidencommon usage ' I predict event A with probability 08 aB with probability 0-2', is no exception. This statea shorthand version of '1 predict (with certainty) that tof a variable called the probability of A,namelyp(A)equaithe value of a correspo nding variable for B. namely pC0'2', In other words, we are not predicting evecertain abstract entities called the probabilities of evencan be variously interpreted, for example, in the presas an assertion that if either A or B (but no other e\-able to occur upon many occasions, 80 per cent of theoccurence would be A, and 20 per cent of the time it woThus, it follows from ou r axiom, that we do not maktions about a piece of the real world, an 'assembly'which is unknowable in detail. Rather, we make prabout some simplified abstraction from the rea lsome incomplete image - of which we can become(the probability model is, of course, an abstractionkind). Subject to some important qualifications, whappear in the discussion, this simplified abstractisystem'.71u! Type of Uncertainty\Vhat is an observer uncertain about? Tn the first observer, absurd as it sounds, may be uncertain aobjective, that is, about the kind of predictions he wmake. This is rarely the case so far as a scientific obconcerned. A scientist usually knows whether he wantsclinically useful observations, commercially useful tions, or observations compatible with the hypothetico-dstructure of physics. On the other hand, there are casestante observation, where the objective is not obvious atset and only becomes so when some tentative knowlbeen gained. This situation is not readily analysed, foonly speak about a source of uncertainty relativeobjective or other , i.e. clinical, commercial or physical p


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20 AN APPROACH TO C YBERNETI CSmaking and for the moment we shall deal exclusively with thosecases wh ere thc objective is s ~ c i f i e d .Secondly, an observer with an objective has a structuraluncertainly about the kind of assembly he is dealing with andthe measurements that are relevant. Take, for example, a brainand thc objective of investigating the auditory mechanism. Th eobserver is uncertain about the anatomical regions that performvarious computations and even about the validity of dividingthe auditory mechanism into functional parts. In lurn, he isuncertain of the inquiries to make about a brain; where, for

example. to place the recording e!eclrodcs.Structural uncertainty about metabolism entails ignorance ofthe h ierarchical arrangement of the enzymes which catalyse thereact ion; or, at a deeper level, about whether enzymes are theactive catalysts. Structural uncertainty about an industry isignorance of the flow diagram to represent the interchange ofenergy, goods or information.Finally, supposing the observer has some strU,cture and thussome set of relevant measurements in mind, he is liable to metricaluncertainty about the values of these measurements. (SecAppendix /. )As a case in point, there is a mod erately good picture of wha thappens when a nerve impu lse travels along a fibre, but physiolo-gists would like to know more ab out the effect which is exertedwhen the impulse reaches a synaptic connection between thefibre and the cell body of another neurone. Our structuralnotions of impulse transmission suggest measuring the depolari-zation of the cell membrane in the synaptic region and it ispossible to obtain a ve ry accurate measurement of the electricalpotential of a microelectrode inserted into the region concerned.But this, of course, is only an index of the measurement required.The potential itself depends upon a number of unknown quan-tities and although the observer is sure enough concerning themeasurement he ought to make (membrane depolarization) andsure about the value of the index which is technically available(micro..clectrode potential) he remains uncertain about the valueof the relevant measurement. Indeed, according to ou r init ia lalliom, an Obsen 'er is bound to accept some minimum uncer-tainty from one source or another, st ructural uncertainty ormetrical uncertainty or both . We shall rationalize the axiom in

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LEARNING, OBSERVATION AN D PREOICTIOa rough and ready fashion by noting that the more dobserver's structural knowledge Ih e more difficultmeasurements he is impelled to makeThe Source of UncertaintyUncertainty stems from ourselves and ou r contactworld. A real obse rver is able to recognize some, bupossible forms of behaviour. These recognizable formpercepts and there is a finite set of them. We have all eXthe sensation ' j can't pu t my finger on anything'. Of cmean that there is no form that we are able to recothat there is no form to be recognized. Ou r ideas of chfrom percepts we have available, which, from ou r po inare not chaotic, or, alternatively, from conventionhave been accepted. From the whole gamut of orders thin the world we can recognize only a few and these weassimilate at a limited rate, through observations at AtWhilst the ultimate restriction is imposed by ou r obilities, we are commonly up against other and artificities. Because of these the object of the study appearsclosed in a container, the so called 'Black Box', to whobservers, have incomplete access. A 'Black Box' l f situarise to either structural or metrical uncer ta in ty or bosimplest case, the assembly, a piece of electrical equipexample, is literally enclosed in a black box with input aconnections.Tests applied at the input and output yield some inabout the equipment, but w ill not specify its conditibiguously. Further tests wou ld involve opening the blacthis is disallowed either by a capricious rule or becausement must be tested whilst it is functioning (the equipbe a running dynamo which cannot be stopped for tbusiness efficiency expert allo ...Cd to see some, bu t noclient's books is in a somewhat analogous position.

We take it, as a matter of belief, that the world is sueh such Ihal we see some order in Ihe world. As RashevskyIh is much must be admitted in order to make science possibfSuch as the convention that a sct of unifo rmly distributis more chaotic than a configured set of particles. Wh ilstusefu l convention, there is nothing sacrosanct about this,pointed OUI."


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22 AN APPROA C H TO CYBERNETICSecologist who, in order to study the interactions within ananimal commun ity. is bou nd to interfere with the ecologicalbalance.Individuals circumvent their imperfections by forming a simplified abstraction of the real world, through learning andconcept formation (as a result of which. amongst other th ings,they learn to recogn ize new percepts). This abstraction, of course,is a private image, but it allows them to deal with an d decideabout their environment. On the other hand, just because of ou rhuman limitations there is advantage to be gained if a group ofobservers, an",ious to make the same so rt of predictions, communicate with one another and in place of many private images,build up one common ly understood abstraction (such as thehypothetico-deductive structure of science). This will be a publicimage of the world within which all observations are assimilableand in terms of which behavioural predictions are made. Anobserver who subscribes to the plan, must limit himself to observations that are mutually inte lligible and which can be assimilated.Again, the rules of deduction which appl y in the abstract structure(and on the basis of which these predictions are made) must berules which have me t wi th public approval.Definition o f a SystemWe are now in a pos it ion to discuss a system of which thesimplified abstraction we have e:'lamined is a particular case. Inthe firs t place, a system entails an a priori struc/llre whichspecifies the logical possibili ties an observer can talk about. Wesha ll call it a 'universe of d iscourse' and will denote it as V.Sometimes U is a loosely related collection of names for objectsor events. At the other extreme U is an elaborate mathematicalmodel wherein names are related by manipulable calcul i, sothat gillen one rela tion many others are deducible. In either case,its nalllC3 and relations and its deductive content (the 'logicallytrue' statements possible in U) exist in the observer's mindindependently of any assembly whatever. V does depend uponthe observe r's previous c:'Iperience, his objective an d his hunchbout a use ful form of description.Second ly, a system entails an identification L between theames in V and those attributes of the assembly which thebserver regards as relevant to his objective. Hence L specifies

LEARNING, OBSERVATION AND PREDICTIONthe set of possible observations. At one c:'Itreme L is dea sta tement like ' I am looking ou t of an aeroplane wincloud shadows fleeting over the ground (I recognized istinguished by the categorical attributes "angular", "band so on)'. In this case the 'system' is no more than aof the cloud configurations, for the attributes are notcommunicable. A t the other e:'ltreme L is the precise specof a reproducible e:'lperiment that a potential is mcasurenearest millivolt at point x, a pressure at po int) ' and sothis case the 'system' is a public abstraction since the atpo tential and pressure are commonly understood. As a rthe identification the logically true statements in Uplausible hypotheses about the relevant and observable atof the assembly and we shall call the pair V,L, a reference(See Appendix 2.)

The reference frame itself is a system. It satisfies a dproposed by Colin Cherry" that a systcm is an 'enseattributes' . But it has no predictive value. In order to shit becomes of predictive value we shall first introduce at io n for rep resenting U, L, called a 'phase space'. Seconsha ll credit the observer with a special objective vI> namake predictions about any beh41'iour in V, L. In other wdiscover all he can about a given way of looking at the asAlthough 'special' VI is sha red by nearly al1 'scientific obsPerhaps it is also tru e that we aTe impelled to adopt v t byin the underlying regularity of the world, and that this rewill be apparent in the reference frame we have chosen.Pllase SpaceSuppose the observer can unambiguously describe his attIf he can, his senses can be replaced by instruments whivert events from the assembly inlo numerically va lued avariables (including, possibly, two valued variables whichI, if an attribute is present and equals 0 if it is absent), Xl> x 2, x .. and displayed in a common modality (perdials or melers). In the simplest case, the observe r knolittle about the assembly. It is a black bo:'l with /I Iunrefated outputs. By the usual convention , we represenoutputs, the values of the x variables, as independent co-orin a phase space. If /II = 2 the phase space will be a p

.."." ... " , , , = RIt a" aSia, II"


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AN APPROACH TO CYBERNETICSFigure I, if II I = 3 a cube, and if II I = 4 a four-dimensional

The phase space is U. The chosen set of m instruments deter-L. We now define the state of the system at any instant t as

(I ) = Xl (r), X: (I), . . . Xm (I), that is, as an instantaneous ob of all relevant attributes. Now X(t) is a point in Umarking of f observed values of the attribute variablesthe co -ordinates of U. Observations can be made no more

than each Lfl. Since no absolute value is assigned to LIt,may as well say LIt = I. In this case a behaviour of theF ~ " " 5 E SPA.CE

11 , , 5 , 7 8,

PH.o..$E SPACE

1

(i )

(ii)

1 Z 3 4.iXiXiXi, . ,. , ., , 7 ,STA.TE ( 'MPH

hlly C Q n ~ e c t Stot . Grapllfo r 8 States

l(i) and \(ii). Each cell in a quantised phase space is representeda single point in the equivalent state graph. Points are connected bywith arrows showing possible transitions. Since the state need notfrom each point and return. These linesare omitted, for clarity.

-' _ ' " = oiL " , - " ,,,.. " " - " .

,Ii

LEARNING, OBSERVATION AN D PREDICTIONsystem is a sequence of states X (0), X (I ) . . . , observabl0, 1, . . _

Because of the observer's metrical uncertainty, obsecannot be exact. Hence the dials may as well be marked ian intermediate reading counting as the nearest markedIn this case, the phase space is q u a n l j ~ c d into unit cellsreal observation can locate the state point within a cell, bno greater precision . Given these modifications, the statetion graph of Figure I (i ) is equivalent to the phase spaceNotice, some structure has been introduced with ouspace. It was tacitly assumed that the number '2 ' on a diaa greater value than the number ' I ' , that '3 ' is greater and '4 ' than '3'. As a result, some transitions are prohibthe state graph (compare it with Figure 1 (ii)). Maydetermines too structured a U (it might, if the attributcribed cloud shadows). In this case, the observer couldto a set of two valued variables, which merely indicaexistence of an attribute. Fo r the same number of vthere are, of course, fewer states, but as in Figure 1 (ii) antransition is possible. On the other hand, if the observersomething about the structure of the assembly beforehandchoose a more structured U, for example, he may knowpossible behaviours carry a state point along a line or ba pair of lines and if so, he can restrict his system to thisof U, L.Finally, as a point of nomenclature, when we do adstate graph picture it seems more natural to talk aboutransitions, or state selections, occurring in discrete jumpthan behaviours leading the state point along a given patProcedure of an Observer with Objective VIA system of predictive value is constructed in U, L, throempirical confirmation or denial of hypotheses. Each hypwhich tallies with an observation is tentatively 'proven', emin U, and its deductive consequences worked ou t tofurther hypotheses for testing. (From this point efforts arto disprMe tentatively accepted hypotheses.)

The observer is mostly concerned with predictive hypabout behaviour, that have the form, 'given the locus ofA, the locus of XCt + l) is B'. Such behavioural predicti

, - .. , ....... . .=.


14/68

2. AN APPROACH TO CYBERNETI CSadvanced by the observer whenever Ihe events in the assemblymove the state point in U, and they arc tested by observing thesubsequent behaviour of the Slale point. (fhis is the effort todisprove current hypothesis.) But when a prediction is consistcntly confirmed and never denied, it acquires the status ofan empirical truth on a pa r with logically tfue statements in U(such as 'an attribute cannot have two values at once' or 'to getfrom Xl = 1 to XI = 5, you must pass through values Xl = 2,Xl = 3, Xl = 4'), In this case the behaviour is regarded asentirely predictable and ilean beembodied in a rule. or behavioural

~ q u a t i o n (or alternatively it can be described by a behaviouralpalh in U). Any entirely predictable behaviour is called statedetermined, and, by definition, an observer with objective VItries to specify as many state determined behaviours in U, L aspossible. Strictly speaking, an inductive procedure like this cannever lead to certainty, for, though a single negative case deniesan hypothesis, no number of positive cases entirely confirm it.ftThus, we assume that at some point the observer becomesconfident that some of his predictions, which have never beforebeen denied, never will be denied.Measurement of Uncertainty and of Information Com'eyedGiven a well-defined sct of elements, it is possib le to measure theamount of uncertainty with rderence to this set. The referenceframe provides a set of states, hence a measur.: of uncertainty ispossible and is c.dled the variety of the sct. The simplest caseis the system in Figure I (ii), where, at any instant, eachstate is equally likely to occur. Since there are n states, anobserver is initially uncertain about 'which of n', or conversely,the appearance of one particular Slale removes this uncertaintyand conveys an 'amount of information', se lecting one of npossibilities. Information an d uncerta inty, if expressed in anadditive form as logarithmic measures , are very simply relatedindeed, Uncertainty = - InformationBecause of this, observation can either be thought of as 'removinguncertainty' about a set of possibilities, or selections from theset of possibilities can be thought of as a 'source of information'.We thus define the variety as +Log.tII or the information

i

i,I,,1Iii,II

LE AR N ING , OBSERVA.TION A.ND PREDICTIONinitially conveyed per observation as - L o g . ~ I / . As the obusing Vh learns and as his system becomes of predictive vainformation conveyed by the appearance of an event is rehe can predict what will occur. If the system becomes epredictable, and all behaviours state determined, when tno uncertainty about it, the information is reduced to O.must be careful to distinguish:(I) The variety of the chosen reference frame U, L,remains for n unrestricted states always Log. ," per otion. (The variety in Figure I (i) is less due to the restrictionphase space .)(2) T he variety of the system which the observer buildthis reference frame (or the variety measured with referethe observer), which is initia!ly Log'2ll, but which is ras the system becomes ofpredict iye value. If you like, the nof possibilities contemplated by the observer = /I " are rand the system variety = Log.,n".There is no measurable variety of Ihe assembly, or of thof the assembly, for in neither case is th ere a well-definedpossibilities. In order to have any measurable variety thebe an agreed reference frame.


15/68

3 The State Determined BehaviourA BEHAVIOUR is state determined if an observer, knowing thestate at I, is able to predict the state at 1 + I with certainty.Rephrased; a behaviour is state determined if X (I + I) dependsin a unique fashion upon X(t) and, in the phase space, thismeans that the path describing a state determined behaviourdoes not bifurcate.We describe the path by a behavioural equation: X (t + I) ""X ( l) ' E. Where E is the transformation in co-ordinates x (themathematical instruction forchanging point X (I) into X (I + I .If the behaviour described by this equation is state detenninedE is a closed, single "allied transformation, that is, the next stateis always one of the states in the phase space and the next stateis always uniquely specified.For the state transition graph, the behavioural equation isexpressed in an equivalent but slightly different form. The statesare labelled I, 2... n. If the behaviour is stat e determined onestate is unambiguously defined at each instant, hence the state ofthe graph is specified, at an instant I, by a binary number I (I).having n entries indexed by the state labels. Of these entriesn - I are always 0 and one entry, with index corresponding tothe current state. is I. I f n = 4. for example, and the secondstate is current at t = 0, the number1(0) = 0, 1.0,0. A behaviouris a sequcnce of binary numbers:

1(0) ) / (1) .......... 1 (r )such as 0, . O , O . ~ O . O , I , O , . . . . . . . 1,0,0,0.So the state transition at each step, if the system is stale determined, will be a closed, sillgle vaillcd, selec/ive operalion F uponI (I) written asJ(1 + I) = I ( I) ' F.Since each entry J,(l ) in the binary number I (t) is in one to

Much of this chapter reflects the views of Ashby and his detailedargument should be consulted." n Detailed references will not be given.28

,

,,

TH E STATE DETERMINED BEHAVIOURone correspondence. by indexing, with a state X, it is not dto see that the two forms of thc behavioural equation are elent.I t is more convenient to express a state determined behas powers of a transformation than as a sequence of soperations. Thus. in the phase space. the state at t = 2, XX(I) ' E= X(O) E. E = X(O). 2, or in general, at I ='.X (0)' E. Similarly. in the state graph,J(r ) z> J(O)' P. WhF' are the roth powers of the transformation E, F, an d reconcisely that the operation has been repeated upon r sucoccasions.Equilibrium BehaviourA moment's consideration will convince you that (since thmust be unique) a state determined behaviour must eithverge, as in Figure 2, to a fixed state called the 'equilPOint'. or enter a behavioural cycle' as in Figure 3. Eitheof behaviour is called a stable equilibrium because. unlesis some disturbance which moves the state point (or altsubsequent transformation). its behaviour remains invari

Fig. 2. Stable point in a phase space - an-o"'"5 convergNOTE: We use the convention of showing a few repsentative behaviours in the phase space. by single linIn fact. there are indefinitely many lines.

Mathematically this is due to a property of the powE and F. namely that for some r = 1. 2, ... and for some2, . . . with n>1. E = E'+! and F' = P +'.Thus. i f I = I. we have the equilibrium 2 and if I> 1 wthe equilibrium 3 represented by the sequences:X(r) = X(r + I) = .. . or/(r) = 1(r + I) = .. . f


16/68

JO AN APPROACH TO CY8ERNET I CSand by

X(r)-+X(r + ) . . . X(r + l) = X(r) orJ ( r )#J(r + 1) .. .J(r + /) J(r)withX(r);eX(r + I) and J(r) J(r + 0- for 3.

Fig. 3. A cycleSince this is true of any slate determined behaviour and sincea state determined system is made up of state determinedbehaviours, we define a state determined system as a collection 0/L identified state determined behaviours which tonverge to a stableequilibrium in a g i ~ ' e l l U (the system may be all of these or onlysome) and it is demarked as a stable region in the phase space,as shown in Figure 4.

Fig. 4. Siable region enclosed by dotted lineNot all equilibria are stable. A ball balanced on a pin, shownabstractly in Figure 5, is in unstable equili brium because theslightest disturbance will displace it irreversibly. On the otherhand, a ball resting in a hollow is in stable equilibrium providing

that the disturbances able to push it around are not large enoughto move it over the edge of the ho llow. ([nstability is associated

For any equilibrial state the selective operation F is a permutationof the position or the ' \ ' in J(t) . This includes the identity permutationthat leaves T in the same position and corresponds to the equilibriumpoint.

... - , .

,I,

,I,,i

T H STATE DETERMINED BEHAVIOURwith the uncontrolled dissipation of energy: stability with ament of an energy minimum, and cyclic activity with condissipation. It is helpful to think in this way, providing tkeep in mind that the behaviour in a phase space is an accobservable events and makes no direc t comment upon th~ aspects of the assembly.)

Fig. 5. Unstable point - arrows diverge.Except in the 'pure' case, where the system is wholly i

and there are no disturbances the distinction betweenand unstable equilibria is one of degree rather than kinthese are useful concepts and their imperfections netrouble us too much for we shall rarely encounter thecase of an isolated and state determined system. Themajo rity of systems have many equilibria. Displacementstate point from one equilibria may lead (i ) to ano ther,to some condition, true enough an equilibrium bu t onethe observer cannot discern for it is outside V. L. This,like, is real instability for nothing can be said about it.

."""" -- - -- ._ -- -- - ---_ .. _--, ,SOUII.CE OJ'"

DISTURBANCE

FILTEA ,DEFINING CONSTRAINTS ' ,___ ___ ______ _ __ __ __ .. . . _ . __ _ ,

TH E ASSEMBLYFig. 6. To an observer the assembly enclosed by a dotted line lothe simulated model sho\\TI inside.Working Models on d Relations between SystemsA reference frame is chosen an d imposed upon the assemthe observer an d from his point of view the assembly

= R " ...... 14W .'OS"'.1 ." . . . . . .


17/68

AN APPROACH TO CYBERNETICSbe replaced by a lileraJly constructed 'black box'

some device for producing the events whichmotions of the state points in U and somemechanism which selects the events of admissibleof all possible events). I have shownobserver's eye view in Figure 6 and it is essential to noticethe filtering mechanism summarizes only those constraintsthe assembly which act upon the relevant attributes (not aUexist). Because of this any system forms theor simulation of some fa cets of manymadeCommonly, for example, working models are made usinglogic is identical withobserver's eye view of Figure 6. The box of constraints, themechanism, is some arrangement of parts in the comwhich, physically speaking, has equilibria that COllt:SpOndthe abstract equilibria. and behaviours that correspond tobehaviours. The model is set in motion to generate allan auxiliary mechanism which feedsdisturbances into the constraint box. In our abstractof course, these correspond with displacements of theBut precisely the same arrangement of parts in therepresent the spread of an epidemic, the spread ofin a community." the development of rust on a piece ofin a semi-conductor.It is natural to ask how models and systems are related. In theof models the answer is easy. for we have explicitly neglectedchoice of L. I f two models, such as the 'epidemic' and themodel are mathemically identical. we say they are]f they differ only with respect to detail, for example,each cycle in the first corresponds with an equilibrium in thewe say that the second model is a homomorph of theif the second is a mapping of the first which preof the stale transfonnation - herecation). Now the second, homomorphic model, isthe observer's eye view ofan observer who had thrown awayof the available information (in a carefully calculatedimage is less detailed than but consistentoriginal). So, in this sense, we can say th at two systems

e isomorphic or homomorphic. But, is this useful? On these

- " . s .. "

IIII

,.

TE ..CHEl'.SIMUlJ\TOR

CONTROLCONSOLE

PUPIL~ I U L " , I ' O R

roPLATE I (i) Simulating a pupil-teacher system. S o l a r t ~ o n fUC(see page 67).(ii) Murray Babcock's adaptive reorganising automatOn. Networktions are made by plugged leads between 'neurone' and 'synapsc'State is displayed on a neon tube matrix (sec page 67).(iii) A practical evolutionary system. Learning machine is ma rkedthread structures are developed in dishes marked S. This demowas set up by the aUlhof at the Symposium on the Mechanisation ofProcesses, held at the National Physical Laboratory, No\'embcr, 1

,., "' , ,

.. . . ......

a E", Eo " " ad ' " "


18/68

(Abon').

(see pageMain

in theown(see page

,

,

IiijI

ii

TH E STATE DETERMINED BEHAVIOURgrounds a system representing the motion of a roundisomorphic with a circular argument. True, they both eidea of going round, but that is the content of the isomand I am not entirely certain what it means. Fo r thethe roundabout are not only different from the stateargument, they are described in a different and, at the mincomparable language. 1 am disinclined to accept the umathematical relations between such states or the corressystems. On the other hand, I am prepared to say that therepresenting the 'epidemic', 'rumour, 'rust" and 'semi-coassemblies are isomorphic because, although the stadifferent, we can talk about them in the same langucompare the L determined measurements we make.According to this view, a pair of systems are compathe L of their reference frames are comparable . In pasystems in the same reference frame must be comparabthis fact allows us to give a rigorous expression to the bof the black box. Any state determined system is themorph of some more detailed system which is also statmined. Ultimately, if we believe in the underlying regulariassembly, there is a state determined system of immenswhich, due to ou r imperfections, we cannot directly obseObject Lang uage and MetalanguageFo r the rest of the discussion we shall adopt an omattitude and look externally upon the observer and hibox. We are now talking about the observer rather thanthe world through his eyes, and, of course, we talk in dterms. Since we shall use this gambit and others like it deal, I shall call the observer's language an object lan(with words that refer to states in his reference frame) alanguage (in tenns of which we talk about an observmetalanguage. I am introducing the distinction at thibecause it will be convenient if we can look inside the obblack box and know in greater detail than he does whof assembly there is. To keep something tangible in mindpose that the assembly is actually a town, with the road F igure 7, and the attribute variables are actually meteread the number of motor vehicles residing at a given upon the labelled intersections in Figure 7. The obsener i


19/68

34 AN APPROACH T O CYBER NETICSto make sense of what we call 'traffic flow', and , in practice, whenthe box is not completely black, he may be morc or less awarethat this is his job in life. Now , in this case, when we are talkingabou t 'an observe r' , both the metalanguage and the objectlanguage are well determined.

There is a second innovation. So far we have thought of5041'-':6 o fActivity

(Vehicles)

Tho.or.. ' < ' O " 9 ~ "';">10 . )(i,.."x &lo


20/68

AN APPROACH TO CYBE RNETICSwill remark there are two substantiallysubsystems (namely, IX - x .. x . x , and fJ = x x.).

other words, he partitions the variables into two subsets onewe know refers to A and the other to B. Partitioning ise important way to reduce the elaboration of a gigantic systemvast numbers of equilibria. The gambit works whenever

are structural co nstraints such as the components in adifferent processes in a factory. different tissues in anor different traffic streams in a town.

The phrase 's ubsystem ' is natural enough if we happen tothat the subset of va riables refcr to streams of traffic. Butsubset of variables is closely related also to ourof a 'machine' (not necessarily a collection of physical

but any entity which does a specific job). The relation is ofSuppose the participant observer could change Xl and X,will (these variables being called the 'input' to the 'machine' ex)which is called the 'output' of the 'machine' ex wouldway. Commonly we say the output is a mathof the input and in electronic machines it isfunction: X3 = f, (Xl> x,) which, givenstringent conditions assumed a moment ago,

to = Xl + Xt. In this case we know that ex is achanging the distribution of motor vehicles', and itto say ex is 'the road layout'. But this would be wrong.is what the road layout actually does, specified by fl . Annttd know nothing about motor vehicles and still seesame machine, only he might call it 'a machine for addingx, and x,'.But a participant observer may do more than 'stimulate'. His

of actions is likely to include such things as C =the traffic signal connection with sensing element atand stop lamps at b. Obviously, this altersf, into some otherf" plausibly enough into:

X. = I, (Xit xJ = Xz + x,O - z) with z a positive, fractionalconstant* The term 'machine' corresponds with the current usage in thisA state determined subsystem is equivalent to the most [ e m e ~ t a r yparadigm a 'Turing Machine'" which has one binary mputone binary output determined by its input and the state of thewhen the input is appl ied.

,TH E STATE DETERMINED BEHAVIOUR

since congestion will occur at the traffic signal stop lampsnumber or motor veh icles proportiona l to Xl will filter D which has become, for them, a most direct route. We cor anything that changes J. a 'parameter of the subsystemachine.

I t must be admitted that the distinction between an 'and a 'parameter' is a little arbitrary. When XI and x. inin the rush hour fwill be changed. If x. is given a posit i\'CI will probably change and the observer is at liberty eithdefine a new system, including XI or to regard XI as a paraof X. Then the whole concept of a subsystem is 'arbitrary',sense that it depends not only upon the -regularities' iassembly which, from omniscience, we know 10 exisl buupon those the observer chooses to recognize.Coupling SystemsApart from the actions of a participant observer, a subscan be affected by the other subsystems. Thus ex can be afby p, in which case we say that ex is COl/pIed to p. As a resa coupling the integrity of the subsystems is partly lost. ever, it is still useful to distinguish between them if the mof coupling is specified by some function, say g to distingurrom f. It may be, for example that g (fJ) which relates exinvolves only some of the variables of P or only some ostates of P (coupling is significant only if there is a partidistribution of the traffic). In common with 'actions theofpmay act as either 'inputs' (as stimuli) or as parameter chaWhilst admitting that the distinction is tenuous it is stillvenient 10 represent these possibilities separately. Hence, (ou r definition of a subsystem, as a relatively isolated functentity) we show subsystems as boxes and distinguish:

Output state of (3 acts as input to a.Fig. 8

, ,


21/68

38 AN APPROACH TO CYBERNETICS

(3 I .l

O ~ t p u t ~ t Q t e of (3 (.hongu parom.ters of 0:::Fig. 9

remembering in each case that the box does no t necessarilyimply a collection of physical par ts. Of these, Figure 8 may entailcoupling the traffic light linkage d, c, of Figure 7 which stopstraffic flow along E when XI increases beyond a limit, also to lamph, whereas Figure 9 may entail a device which renders the linkagea, b operative if and only if X$ exceeds this limit.

If the coupling is two-way. so that P ffects a and IX affects p,we say that ex and P re interacting. When the interaction is veryseverely restricted there is some point in talking about feedback:as we did in Chapter I, and analysing the system in terms offeedback: theory. But most of the systems that concern us are soelaborate that the techniques of feedback theory are inapplicable.Interaction by feedback makes the sub-systems very hazy and,as mentioned in Chapter I, gives rise to apparently purposiveforms of behaviour.Of course, from ou r om n iscient viewpoint, the black boxand the observer are merely a pa ir of subsystems; subsystems inou r metalanguage, however! In the upper picture, Figure 10(0I have tried to show what goes on in these terms when anobsen.er aims for 1)1 (to obtain a participant observer connectchannel 5 ; to make yourself a plain observer disconnectchannel 5). The lowe r pict ure re fers to the next part of ou rdiscussion.Alternaljre ProcedureThere is no guarantee that an observer, using 1)1 will achieve astale determined system . Some of the behaviours in his phasespace may remain ambiguous, like Figure II , where A goessometimes to B and sometimes to C.

TH E STATE DETERMINED BEHAVIOUR

(NVIIl,O,. ...U n

NVlROIfMtIH

,,

II.II

I

II----..-I

II 0 8 ~ [ R V E R.II.I

DIM"RlI, . , 'c .r i,i.,.., .,--".,". COh'll'.o.1lt' _ . . _ J

OBSERHR.

,, .........-! QMPAR(I.,......_-- ... f - ~ _____ ........... "' .............S H OFTRIA .

ACTIONS II

Fig. 10. Different kinds or observation and experimentation viewesystems

Bc

AFig. I !

:. -


22/68

40 AN APPROACH TO CYBERNETICSIn this case the observer may either:(i) Examine a system of greater detail and diversity, so thaiA becomes a pair of states, 0 , which always leads 10 B, and aswhich always leads 10 C as Figure 12. B", ,-,, ,,, ,a ( ) 'Z :> , CFit 12(ii) Resort to statistical observation.First ofall, let us look at (i). We and possibly the observer know

that motor vehicles are being counted. They are discrete entitiesand, unless the instruments are misfunctioning, they cannot becounted more accurately! So it is only possible to improve themeasuremelllS by reducing L11 and counting 'more often'. Evenhere a useful lower limit is set by the maximum speed of the motorvehicles and we may as well assume that L1t is within the limit.So the observer must look for a greater diversity of data, forexample,he must investigate more of the intersections of Figure 7,since XI> x" . . , x .. are only a subset of the possible measuresx, , x" ... n > m, which are potentially available. Thisdoes, of course, entail changing L an d possibly also U (sincemore, as well as differen t, variables may be nceded to describe astate detennined system). Hence, the objective is no longer II,.Instead, the observer is looking for a state detennined system,in allY reference frame available (and we suppose that this searchis permitted). Unless some restriction is imposed, the search willbe haphazard. Thus, we assume that the observer wishes to dis-cover a state determined system sufficient to make some specifiedkind of prediction, for example, sufficient to control the traffic.Any such objective will be called VI and the procedure adoptedby an observer will have the fonn 'Choose a reference frameVI> L, an d test for a state detennined system in VI> L" but ifthis is no t achieved after a certain arbitrary effort, choose afurther reference frame VI,L I and if necessary another V3,L 3,andanother V., L. and so on'. (Figure 10 (ii).) Whilst the procedure

,I,I

,,,,

I

,

,,,I

THE STATE DETERMINED BEHAVIOURfor VI was essentially a matter of chance trial, liZ is likely tothe elements of insight and invention. This becomes

w h ~ n we consider the L an observer may choosc. If thattnbute number of motor vehicles' provides insufficienlhe may take the make and model of the motor vehicaccount or , for that matter, Ihc drivers' occupations tthat are running in town, or the day of the weck. 'Statistical DeterminacySuppose that the observer is not allowed Ihis latitude. H i

~ e n t s are given and. he ":lust stick to the method of F igu~ m c e he. cannot split A mt o Q, and 0 " he may have to10 despair. On the other hand, it may be possible to negleof the detailed state changes and make consistent stassertions. But, this possibility depends very much upassembly, and an observer can in no way guarantee succ

If he looks long enough for many, say 100, transittake place from state A in Figure 12 an observer may beconclude:(I) That A always went either to B or to C.(Ii) That it went into B 80 limes, and C 20 times. ou t ofTo summarize the i n f o n ~ ~ l t i o n writes proportionS!land 11 c . . 02. These empIrical estlmatesof the transition

bilities from stale A to state B, and state A to state Cobtained by 'time averaging ' the results. There is, however, a basically different way to glean std ~ t a . Suppose there are many, say 100, observers loodifferent, but macroscopically similar, assemblies in threference fra?"Le. If 80 of them report simultaneously, p

t h ~ t A goes into Band 20 of them report that A goes th iSknowledge may also besummarized by proportionSjl.


23/68

42 AN APPROACH TO CYBERNETICSused in psychology (SO subjects passed a test, 20 failed a lest)and in any case we need the concept for ou r later discussion.

Returning 10 the single observer: if on repealed inspection thevalues of 1 A /I and" . C" do not change he will become convincedthat there is an underlying statistical constraint because of whicf:these proportions exist. In other words, he uses the consistenc)of '1.0. and '1 .. c as empirical evidence in favour of an h y p o t h e s i ~that there is regularity in the world, and infers the existence of astatistical structure (which determines the detailed behavioursomewhere within the black box). Suppose, that for each statei = 1,2, . . . nand j = I, 2, . . . n, it is true that empiricalestimates 'lu ar c unchanging, an observer may legitimately infera set of related statistical constraints that determine a statisticalsystem. Because the estimates are invariant the statistical systemis said to be a slationary syslem. One important consequence ofstationarity is that for long enough or large enough samples'1'J = J1 P'J > o. Since some transition occur at each instant (possibly the transformation of ainto itself) the sum of the probabilities associated with armoving away from a state (including the arrow which maway and returns) must equal I.By analogy with Fwe can construct a probabilistic transfotion of the binary number which represents the state of the syat t = 0, by summarizing the P I} in a transit ion probability rL'!.: P. (See Appendix 3).However, the transformation no longer leads to a unique but to a probability distribution or in other words, a statefor each of he n states of their probabilities ofoccurrence at tgiven the state specified at t = O. We call this distribution.

p, (I ) = P;l (I), p" (I ) , . . . p;n (I )and writep, (I ) = J (0)'P, or since J (I) is a special case of p (I) witentries I or 0, pil) = P, (0)' P.

We continue, as with the state determined system but ob taifurther distributionsp, (2) = p/ (I)'P = p,(O)'F, or for 1 _ rp,er) = p , (r-l) 'P = p, (o )r

,


24/68

44 AN APPROACH TO CYBERNETICSA distribution p, (t) is the state of the Markovian system an d asequence of distributions is a behaviour of the Markovian system,conditional upon the chosen initial state;. Instead of choosing aparticular initial state we could have chosen a probabilitydistribution - in particular - if we had chosen the distribution

I, I . . . 1 p; (0) = - - so that each state IS equally likely, the resultingn n ndistributions would be the unconditional states of the Markoviansystem.We can also construe the statistical transformation as aninstruction to take a four-sided, or in general, an n-sided diceand to bias it according to the entries in the row of P whichcorresponds with ou r chosen initial state. The dice is thrown andthe outcome determines Ihe state at t = 1, of a hypothetical,determinate system (let us call it a representative system) whichis one of a statistical ensemble. The row of P selected by thisoutcome is used to bias the dice for a second throw, the outcomeof which selects the state of the represelltative system at t _ 2, andso on. In the phase space the sequence of states generated bydice-throwing delineates the behaviour of a single r e p r e s e n t a t i ~ esystem.

Consider a large number of dice thrown simultaneously, manyfrom each different initial state and each according to theseinstructions . Each one determines a representative system and isassigned to a point in the phase space (the whole set of statepoints forming an ensemble). A sequence of throws generates abehaviour of each representative system and the points move.If the number of representative systems, and hence of statepoints is very large, we can neglect their individual behaviour andconsider only the density of points, that is, the behaviour of theensemble. The behaviour of Ihe ensemble is the behaviour of theMarkovian system.Stochastic ModelsSince dice throwing exhibits all possible behaviours, given thestatistical constraints of a Markovian system, it is a stochasticmodel (analogous to a determinate model) for simulating thebehaviour of an assembly. The constraints represent stockholding parameters, demand functions, and value fluctuations(o r any other statistically known quantity), pertinent to a

, , ; '. , hi

r

THE STATE DETERMINED BEHAVIOURbusiness or an industrial process. The simulation is cMonte Carlo procedure and is programmcd on a digital comEach illitial state of the stochastic model corresponds willitial displacement of the determinate model. Ench set ofsentative systems started from a given state, correspondssillgle behaviour of the determinate model. The pointemphasis perhaps, because each representative system inis, of course, a determinate system, which is however statemined by the dice awl by the stalistical constraints jointlyStatistical Equilibrium (see Appendix 4)By analogy with a state determined system any Markovianreaches statistical equilibrium. In equilibrium it is characby averages '!'j and, regarded as an information source, imeasurable variety. For n states, the maximum variety is Lthe variety of the reference frame, without any statisticastraints. Bu t by learning about the '!'j an observer canthe variety of the system, as he sees it, to a minimumwhich depends upon P. This variety is equivalent to Shannstatistical information measure on the system. I t is a mawhen the equilibrium distribution p . = ! " ~ " " ~ i n d e e d ,

II II ncase, it is Log n. Unequal probabi!itiesp"PJ' reduce the vConditional constraints P'j render the slate of the systempredictable and decrease the variety sli11 further, by an acalled the redundancy of the source. (seC' Appendix 5.)NOll-Stationary SystemsSuppose there is an honest to goodness statistical whirligidice throwers and bits and pieces of mechanism to determP'I' all enclosed in a black box. The whirHgig has n distates and each of these is accessible 10 an observer - whmodel is in a particular state a particular lamp is illumHowever, it could be rather a subtle device, a 'learning' main which the P,} changed from moment to moment, incase we write P ij (t) in place of P ;j and nO-lice that the ouour learning machine is non-stationary.

Taking an omniscient view, the rules which change thetical constraints are part of the specification (the rules withe form 'P;; (t) is some mathematical function of the pr

, ,,


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46 AN APPROACH TO CYBERNETICSstates'), and thc whole thing, rules and all, is an expandedMarkovian system. An obse rver who looks long enough canmake valid estimates '1'1 of the constraints. 'Memory' or theability to 'learn' is not a property of the system, but of therelation between the system and an observer. As Ashby pointsout, any system with many equilibria will exhibit ' m c ~ o r y 'if some of its states are indistinct. or two observers, lookmg atthe same assembly, one - who is able to distinguish few stateswill say his system has a 'memory', whereas the other - able todistinguish many states - will say his system has nonc (seeAppendix 6).Discarding omniscience let us look at a black box through theeyes of an observer who can only form estimates TjIJ of a limitedset of states. The behaviour of the system may be wholly intractable. On the other hand, the behaviour may be described by aMarkovian system, say PI> which reaches a temporary stable ormetastable equilibrium and remains there for an interval. Then,rather suddenly, the behaviour changes. Th e new behaviour isrepresented by a different Markovian system, say P t , which againreaches a metastable equ ilibrium, then, in turn, gives place top. and p.Animalleaming is a case in point. When primates arc learningto solve problems, their behaviour, though not strictly stationary,remains p p r o ~ i m a t e l y so; the learning curves can be extrapolatedwith confidence, and the behaviour is predictable. Then, rathersuddenly, the creature learns a new concept an d subsequentlydeals with problems in a different way which it sticks to for afurther appreciable interval. Once again, the learning curves canbe extrapolated and a different kind of behaviour becomespredicable. But in between the two behavioural modes there isa discontinuity and prediction of the subsequent mode, given theinitial mode, is impossible unless we make use of averages o\eran ensemble of animals. H. Harlow, for exampIeu, distinguishesbetween repetitive learning which is predictable an d the processof concept or 'set" learning which entails discont inuities that canbe interpreted as 'insightful' behaviour.Th e statistical system we have examined is tractable because,by analogy with a determinate system, it can be partitioned intostatistical subsystems. An equivalent black box would contain awhirligig having a set P of possible transition matrices P; and a

THE STATE DETERMINED BEHAVIOUR.selective operation F to choose different membersdifferent instants. On the other hand, if the system capartitioned (or if the selt.'Ctive operation acts too fa stan observer to sample each P;) the estimates q'J are wand the observer must rely upon ensemble avcragesJloj.The difficulty is to decide which systems aTe macrossimilar. Given a lot of identical molecules, we arc on safein saying that 'macroscopically similar collections aretained at the same temperature an d pressure. But,convincing to hear that 'macroscopically similar' learindividuals selected from the same breed of rat.The Se(fOrganizing SystemA non-stationary system becomes 'self-organizing' whenuncertainty about the criteria of macroscopic similarity.tions are offered by Beeru , Pringle"", Von F o e r s t and" . An observer is impelled to change his criteria of s(hence. also, his reference frame) in order to make sensself-organizing systems, behaviour and he changes it onof what he has already learned (by his interaction wsystem). Typically self-organizing systems are alive' thoshall examine some which have been embodicd in 'inamaterials. Let us take 'man', whom most of us would agself-organizing system. A man is any member of a well-sset of men. But this set can be well-specified (that is, spea way that meets common approval) in a vast number oaccording to an obscrvers objective. /I.-13n, for example,specified anatomically (two legs, head , and so on), or alteras a decision maker which influences and is influencedcircle of acquaintances. Each specification is equally vaentails criteria of similarity. The poiO[ is, there are objecwhich neither the first specification (and the criteria itnor the second (and Ihe criteria it entails) are sufficient.versation, when trying to control a man, to persuade hi

s o m e t ~ i n g , how do I define him? Manifestly I do not,I continually change my specification in such a wayappears 10 me as a self-organizing system.Hence, the phrase 'self-organizing system, entails abetween an observer and an assembly. It also entails the obobjective (an assembly may be a self-organizing system


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48 AN APPROACH TO CYBERNETICSobserver but not another, or for one objective but not another).Again it is possible that an assembly will appear as a selforganizing system initially and become stationary after interaction (the conversation partner does, on average, what I askhim). The dependence is also evident in measures of organization;for example, Von Foerster proposes to use Shannon's Redundancy' (Appendix 5) for this purpose. A system is 'self-organizing' ifthe rate of change of its redundancy is positive. From Appendix 5redundancy is a function of V and Vmax (two informationmeasures) of which V depends chiefly upon constra ot s developedwithin the speculed system but VmQX depends upon the specification and the observer's frame of reference.

I

4 Control SystemsA CONTROLLER is a natural or constructed asseminteracts with its environment to bring about astability calIed the 'goal' or 'objective'. Hence the pobservers are controllers (with 'goals' or 'objectives'Indeed, whenever there is a stable system, then, in prica n envisage a subsystem acting as the controller that this stability. More often, though, we come across cthat have been deliberately built (thermostats, protrollers) and the partitioning which separates these devthe environment is given by their construction.

VOLT""E :'LCONTROLLER

'7 -

Power Supplywith"fludualions:t V

" 51X ~ , 0 nbutpCOMPAA(WITH Yo

Fig. 130. A simple controllerThe voltage controller of Figure 130 is a case iPhysically it is a neat, meehanical1y distinct entity. Thits environment is represented by the value of one This is a first order linear servo dJ'fdl __ b(y-y .) with bconstant. Solving for y ..."c have y = , ( I ~ thus as t iapproaches Yo. We shall not discuss the mathematics of servisms because it is a subject in its own r ight". ReferenceM a ~ 1 1 and, for, applications to behavioural science, inSOCiology, to Tustm I

49


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50 AN APPROACH TO CYBERNETICSnamely the voltage y which is to be stabilized at a chosen valueyo> In the absence of the controller the supply voltage v fluctuatesabout Yo - To maintain y = Yo a 'difference signal' Y-Yo is appliedin negati\'c feedback to the controller. Now, from inspection,y = x + \' where x is the controller output derived from apotentiometer placed across an auxiliary power source. Th e'negative feedback' connection means that the motor which movesthis potentiometer is driven at a rate - (Y-Yo) hence that therate of change of x, is equal to - (Y-Yo). The controller is inequilibrium if and only if x is unchanging and this is the case onlywheny=yo,x= v -yo.

Fig. Db . Abstract image of simple controllerSuch a control is formally represented in Figure Db by asubsystem A (the controller) with states X, a subset beingequilibrial, a subsystem B, (the environment) with states Y, ofwhich a subset,; is the objective (i.e., includes the state we wantthe environment to assume) and coupling functions f and g,whereby A and B interact (Le. states of A displace states of Band vice versa). The coupling functions and the behavioural

equation of A are so chosen that Y is in .; if and only if X isequilibria!. The behavioural equation of A is often called thecontroller's 'decision rule' since it determines what correctivedisplacement attends each change of state in the environmentan d there is a sense in which A's tendency to equilibrium forces Yinto The formalism adequately describes any 'Automatic'controller like the voltage regulator, (any device which has afixed 'decision' rule) and any simple homeostasis. In order todesign such a thing we must, of course, know what the ruleshould be (which entails having a model to r epresent the environment and determine what is and what is no t a corrective response).

No t all controller s are so simple, An 'Adaptive' or ultrastab1e'ucontroller is shown, formally in Figure 13c. Its designer need no thave a comprehensive model of the environment - hence, in thepicture, we show a source of unpredictable disturbances per-

I

"U"pr r I C L o ~d slotbane I

CONTROL SYSTEMS

.1!nIU"ol SOurCe ()f d ; ~ l u r b o n < ..

, B f8

A

Fig. 13c. Abstract image of adaptive or ultraslablc contrturbing the states of B. No r is there a unique decisiInstead there is a set of possible rules - possible staformations. An internal source of disturbances perturbsof A, (as designers, we should say that this source into make trial actions). Now whatever rule (or transfois currently selected we know, from our previous argthat the behaviour of A on its own would be equilibriaequilibrial behaviour also forces Y into'; then Ihe syswhole is equilibrial an d the currently selected decisioleft unchanged. On the other hand, if the whole (A an dacting) does no t reach equilibrium, the rule (or a 'statformation) is changed an d the process is repeated until equis achieved.Important Restrictions(i). The controller in Figure 13a is Slable alld succesfora {imiled rangeo f luctuations. I f vgoes plusorminus tox does also, and the potentiometer arm comes of f the ewinding, which is an irreversible change. If v changes toomotor cannot keep pace and the controller fails to cofluctuation which may lead to cumulative instability.(ii). The rariety o f actions mllst be at least as greqrariefy o f he fluctuations to be corrected.

This principle, which Ashby calls 'requisite variety'strikingly il!ustrated if we suppose:1. The potentiometer replaced by a switch (this is no for a real potentiometer is like a switch and x does chdiscrete units).


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52 AN APPROACH TO CYBE RNETICSIT. That v also assumes discrete values. In this case, switchpositions (controller's actions) select columns in the table ofFigure 13d and values of y select rows. The states of h e e n v ! r o ~ -

menl, now more conveniently called outcomes, are the entnes Inthe table. For convenience, it is assumed that )' 0 = 0 when theoutcomes 0 become the set e. Since the potentiometer onlym o \ ~ s one right , one left, or stays where it is in each interval Atthe sdectivc variety per. tlt is Log.: 3.

" .t ... ; . - 1

OUTc.oME"""TRI)(

Fig. I3d. Outcome matrix used to determine a decision rule.If the same restrictions apply to disturbances occurring nomore often than once per J t the environment variety is also

Log.,) and, by inspection, whatever value v assumes the con-troller can maintain an outcome = 0 in e. On the other hand,if " changes more rapidly, say, two moves per Jt , this is nolonger the case, nor is it the case for magiludes greater than

= x___ or less than ,, = X .. 'Requisite variety' appliesequally for any well-defined set of actions and outcomeschanges in the environment. (Since in the general case, the entnesare unrestricted the 'table' is isomorphic with the 'outcomematri:-:.' which, in the theory of games, specifies the ou tcomeattending a pair of moves, one by each of two participants selecting columnsj and rows i respectivc:ly. In the theory of games anumber 61 is assigned, for each participant, to each ~ t r y and thematri:-:. of numbers is called the pay-off matrIX, for It says howmuch of some desirable commodity each participant receivesfor each possible combination of moves. The present participantsare A and B. It is feasible to assign number OJ' related in some way

CONTR OL SYSTEMSto :lchievement of to each outcome and thus determioff matri:-:.. We have, in fact, done this in our table. Buthe numbers in the table lead to a rather obvious decithe decision rule for the general casc is fa r from obviouII f. To extend the principle, 'The amount0 /control(meoJ"Oriet;y) depends upon the all/ounlo/in/ormation Ihe colltrufrom it s e n ~ i r n m e n l ' , In stating 'requisite variety' wethat A had complete information about B (regarded acipant, A could inspect B moves and the pay-off matriselecting an A move). Commonly, of course, the sysenclosed in a 'black box' (A receiving imperfect eviden8). Hence, we distinguish two kinds of controller - th'perfectly informed' type, and 'imperfectly informed' cwhich we shall discuss in a moment.IV. A voltage controller acts in a well-defined referen0/ voltage at c. I t cannot appreciate voltages other th:lis notoriously unable to deal with humidity changeexert a very adverse effect upon its behaviour and it reacbadly to kicks. This is not trlle of every controller. Bcontrollers, in particular, can change their reference frChapter 7).Alltomatie ConrrollersAutomatic controllers receive perfect information asystem they control and have fixed decision rules, that dtheir actions. They are the Sluff that automation used tosometimes still is, made from. Personally J am more iby pianolas and calliopes than any grim automalOn rproduction line. Do not despize the: machines even if yospare my childish wonderment. I have seen a kind ofmade in 1920, which includes a fourth order non-linesystem, and the most elaborate code transformation input music roll. These beautiful machines reached aingenuity years ago and, for all the ta lk, automationclassical sense, is a hoary old art. The best place [0 learthe music hall. beside Sutros, on the cliff at San Francisecond best place is D isneyland - I admit a prefernorthern California, In England we have Battersca ParTypically. an industrial controller senses a certain comof events, for example, that all of r different welding pr


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'4 AN APPROACH TO CYBERNETICShave been completed, upon the lIth piece of metalwork at boothi, via a logical network. As a re sul t of this information, the automatic controller takes an action determined by its decision rule,i.e. moves the n-th job to booth i + I. It then awaits the r + I thevent, a feedback signal say the metalwork has arrived, afterwhich it is free to accept the n + !th job at booth. i, and thewhole cycle is repeated. The automatic controller is inert. I f themetalwork runs ou t it does nothing, or at best rings a bell tosay it is idle. It cannot prod its environment, looking forwork, and, unfortunately, the same is true of my favouritecalliopes.The Distinction between Perfectly and Imperfectly InformedControllersIn tne simplest case a perfectly informed, automatic controllerreduces to Figure 14. (i) in which the switch A is turned by thecontroller to actions a, p, whilst switch B is turned by thebehaviour of the environment to stages a. b. at each. instant dr.For the moment we ca n neglect the small devil G, who altersthe structure of the environment, because he is quiescent.According to the circuit, the lamp is illuminated if and only ifA .., a when B _ a, and A = Pwhen B = b, an d this is indicatedin the pay-off matrix. We sha ll call the lamp a knowledge ofresults signal, since it tells the controller the result of its actionafter it has selected an action. In addition the controllerreceives complete information about the state of theenvironmcnt(B switch position) through channel F. Hence, assuming it canselect one action each At , and given the decision rule A = a. ifB = a, A = p, if B =b. it can keep the lamp illuminated bymatching its actions (A switch positions) to the state of theenvironment. Notice the 'decision rule' entails 'a model' of theenvironment which, in this case, is built illto the controller.

Strictly a servomechanism, like the voltage regulator, receivesonly k n o w 1 e d ~ of resullS since it mus t make $Orne trial displacementin order to ehdt a difference signal. Indeed, in the region of J' = y.the servomechanism does make 'hunting' actions. These can beobliterated by suitable design which relies on the fact that .:e is a'continuous' variable. Given continuity the distinction between knowlege of results and direct information is tenuous. But it becomesimportant when, as at x = .:eM .. there are discontinuities and in thepresent discussion discontinuity is the rule.

II

I

,-- . ._-- -,,,,,,,,, I G

CONTROL SYSTEMS-,- -- -- --. . _-_. "

,,,,,"F- -" -- .. . -- - -- - -- --.: --- - 01' "' /i -:, A '1 :, :e' ..,. . . __ . . . . . .P (' ) t ..

,; ( o . t ) . )i:(- , ( b . t l

,,,

,,

! B"~I .

" ,. . . . . . . .,

,'. ::.,

h d ~ l \ u fo< C( r .d.ncr for fJ Eyidl'ftC-rr n u (l 01In a ( g


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56 AN APPROACH TO CYB ERNETICSFigure 14 (ii). which provides evidence'" (a, f) . 4> (h, f) . aboutthe states of the environment, according to the convention thatthe more positive the average value of p (,) in the interval LItpreceeding this instant, the more likely is B = a (and the higher'" (a, I the morc negative p( t) the more likely is B = b (and thehigher (h , t. With two mutually exclusive and exhaust ivestates, one or the other must be the case so.p (a, I) + '" h, t) = I.Obviously if .p (a, t) = .p (h, I ) = 0-5, no information is conveyed.(ii). The knowledge of results data may be disconnected ormutilated.

Given either impairment i or ii. completely accurate matchingis impossible and we must consider statistical rather than determinate matching between A and B. Statistical matching canmaximize the pay-off on aw!ragl!, i.e. illuminate the lamp as oftenas possible.To illustrate the idea consider the biased dice thrower inFigure 14 (iii). It can th row a 'two-sided' dice each .dr, the out-come determining either A _ IX, or A = p. I f the bias is uniformthe dice th rower will produce a sequence in which IXS and ps areequiprobable. Now this sequence is matched to an environmentwherein p (a) = p (b) = O'S (using the letter p as in Chapter 3,for the actual value of an a priori probability, which depends uponsome physica l constraint in the environment). Obviously p (a) +p (b) = I. Suppose we happened to know that p (a) = 08 andp (b) = 02 (which is a rudimentary sta tistical model of theenvironment). T he activity of the dice thrower can be matchedby adjusting the bias so that the probability of IX = P (a) = 0'8and of P= p (b) = 02. [n other words, by building in ou r'statistical model'. There ar

Pask Gordon an Approach to Cybernetics

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