EVERY GOOD REGULATOR OF A SYSTEM MUST BE A MODEL OF THAT SYSTEM 1

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1 Int. J. Systems Sc., 1970, vol. 1, No. 2, EVERY GOOD REGULATOR OF A SYSTEM MUST BE A MODEL OF THAT SYSTEM 1 Roger C. Conant Department of Informaton Engneerng, Unversty of Illnos, Box 4348, Chcago, Illnos, 60680, U.S.A. and W. Ross Ashby Bologcal Computers Laboratory, Unversty of Illnos, Urbana, Illnos 61801, U.S.A. 2 [Receved 3 June 1970] The desgn of a complex regulator often ncludes the makng of a model of the system to be regulated. The makng of such a model has htherto been regarded as optonal, as merely one of many possble ways. m ths paper a theorem s presented whch shows, under very broad condtons, that any regulator that s maxmally both successful and smple must be somorphc wth the system beng regulated. (The exact assumptons are gven.) Makng a model s thus necessary. The theorem has the nterestng corollary that the lvng bran, so far as t s to be successful and effcent as a regulator for survval, must proceed, n learnng, by the formaton of a model (or models) of ts envronment. 1. INTRODUCTION Today, as a step towards the control of complex dynamc systems, models are beng used ubqutously. Beng modelled, for nstance, are the ar traffc flow around New york, the endocrne balances of the pregnant sheep, and the flows of money among the bankng centres. So far, these models have been made mostly wth the (lea that the model mght help, but the possblty remaned that the cybernetcan (or the sponsor) mght thnk that 1 Communcated by Dr. W. Ross Ashby. Ths work was n part supported by the Ar Force offce of scentfc Research under Grant AF-oSR 7o Now at Unversty College, P.o. Box 78, Cardff CF1 1XL, Wales.

2 some other way was better, and that makng a model (whether dgtal, analogue, mathematcal, or other) was a waste of tme. Recent work (Conant, 1969), however, has suggested that the relaton between regulaton and modellng mght be much closer, that modellng mght n fact be a necessary part of regulaton. In ths artcle we address ourselves to ths queston. The answer s lkely to be of nterest n several ways. Frst, there s the would-be desgner of a regulator (of traffc round an arport say) who s buldng, as a frst stage, a model of the flows and other events around the arport. If makng a model s necessary, he may proceed releved of the naggng fear that at any moment hs work wll be judged useless. Smlarly, before any desgn s started, the queston: How shall we start? may be answered by: A model wll be needed; let s buld one. Qute another way n whch the answer would be of nterest s n the bran and ts relaton to behavour. The suggeston has been made many tmes that perhaps the bran operates by buldng a model (or models) of ts envronment; but the suggeston has (so far as we know) been offered only as a possblty. A proof that model-makng s necessary would gve neurophysology a theoretcal bass, and would predct modes of bran operaton that the expermenter could seek. The proof would tell us what the bran, as a complex regulator for ts owner s survval, must do. We could have the bass for a theoretcal neurology. The ttle wll already have told ths paper s concluson, but to t some qualfcatons are essental. To make these clear, and to avod vagueness and ambgutes (only too ready to occur n a paper wth our range of subject) we propose to consder exactly what s requred for the proof, and just how the general deas of regulaton, model, and system are to be made both rgorous and objectve. 2. REGULATION Several approaches are possble. Perhaps the most, general s that gven by Sommerhoff (195o)) who specfcs fve varables (each a vector or n-tuple perhaps) that must be dentfed by the part they play n the whole process. Fgure 1 (1) There s the total set Z of events that may occur, the regulated and the unregulated; e.g. all the possble events at an arport, good and bad. (Set Z n Ashby s (1967) reformulaton n terms of set theory.) (2) The set G, a sub-set of Z, consstng of the good events, those ensured by effectve regulaton.

3 (3) The set R of events n the regulator H; (e.g. n the control tower). [We have found clarty helped by dstngushng the regulator as an object from the set of events, the values of the varables that compose the regulator. Here we use talc and Roman captals respectvely.] (4) The set S of events n the rest of the system s (e.g. postons of arcraft, amounts of fuel left n ther tanks) [wth talc and Roman captals smlarly]. (5) The set D of prmary dsturbers (Sommerhof s coenetc varable); those that, by causng the events n the system S, tend to drve the outcomes out of G: (e.g. snow, varyng demands, mechancal emergences). (Fgure 1 may help to clarfy the relatons, but the arrows are to be understood for the moment as merely suggestve.) A typcal act of regulaton would be gven by a hunter frng at a pheasant that fles past. D would consst of a1l those factors that ntroduce dsturbance by the brd s comng sometmes at one angle, sometmes another; by the hunter beng, at the moment, n varous postures; by the local wnd blowng n varous drectons; by the lghtng beng from varous drectons. S conssts of all those varables concerned n the dynamcs of brd and gun other than those n the hunter s bran. H would be those varables n hs bran. G would be the set of events n whch shot does ht brd. R s now a good regulator (s achevng regulaton ) f and only f, for all values of D, R s so related to s that ther nteracton gves an event n G. Ths formulaton has wthstood 2o years scrutny and undoubtedly covers the great majorty of cases of accepted regulaton. That t s also rgorous may be shown (Ashby, 1967) by the fact that f we represent the three mappngs by whch each value (Fgure 1) evokes the next: φ : D S ρ : D R ψ : S R Z then R s a good regulator (for goal G, gven D, etc., φ and ψ) s equvalent to ρ [ ψ ( )] 1 G. φ, to whch we must add the obvous condton that ρρ ρ ρ to ensure that ρ s an actual mappng, and not, say, the empty set! (We represent composton by adjacency, by a dot, or by parentheses accordng to whch best gves the meanng.) It should be notced that n ths formulaton there s no restrcton to lnearty, to contnuty, or even to the exstence of a metrc for the sets, though these are n no way excluded. The varables, too, may be partly functons of earler real tme; so the

4 formulaton s equally vald for regulatons that nvolve memory, provded the sets D, etc., are defned sutably, Any concept of regulaton must nclude such enttes as the regulator R, the regulated system S, and the set of possble outcomes Z. Sometmes, however, the crteron of success s not whether the outcome, after each nteracton of S and R, s wthn a goalset G, but s whether the outcomes, on some numercal scale, have a root-mean-square suffcently small. A thrd crteron for success s to consder whether the entropy H(Z) s suffcently small. When Z can be measured on an addtve scale they tend to be smlar: complete the constancy of outcome H(Z) = r.m.s. = 0, (though the mathematcan can devse examples to show that they are essentally ndependent). But the entropy measure of scatter has the advantage that t can be appled when the outcome can only be classfed, not measured (e.g. speces of fsh caught n trawlng, amno-acd chan produced by a rbosome.) In ths paper we shall use the last measure, H(Z), and we defne successful regulaton as equvalent, to H(Z) s mnmal. 3. ERROR-, AND CAUSE-, CONTROLLED REGULATION The reader may be wonderng why error-controlled regulaton has been omtted, but there has been no omsson. Everythng sad so far s equally true of ths case; for f the cause-effect lnkages are as n fg. 2 R t s stll recevng nformaton about D s values, as n fg. 1, but s recevng t after a codng through S. The matter has been dscussed fully by Conant (1969). There he showed that the general formulaton of fg. 1 (whch represents only that H must receve nformaton from D by some route) falls nto two essentally dstnct classes accordng to whether the flow of nformaton from D to Z s conserved or lossy. Regulaton by error-control s essentally nformaton-conservng, and the entropy of Z cannot fall to zero (there must be some resdual varaton). When, however, the regulator H draws ts nformaton drectly from D (the cause of the dsturbance) there need be no resdual varaton: the regulaton may, n prncple, be made perfect. The dstncton may be llustrated by a smple example. The cow s homeostatc for blood-temperature, and n ts bran s an error-controlled centre that, f the bloodtemperature falls, ncreases the generaton of heat n the muscles and lver- -but the blood-temperature must fall frst. If, however, a senstve temperature-recorder be nserted n the bran and then a stream of ce-cold ar drven past the anmal the temperature rses wthout any prelmnary fall. The error-controlled reflex acts, n fad,

5 only as reserve: ordnarly, the nervous system senses, at the skn, that the cause of a fall has occurred, and reads to regulate before the error actually occurs. Error-controlled regulaton s n fact a prmtve and demonstrably nferor method of regulaton. It s nferor because wth t the entropy of the outcomes Z cannot be reduced to zero: ts success can only be partal. The regulatons used by the hgher organsms evolve progressvely to types more effectve n usng nformaton about the causes (at D) as the source and determner of ther regulatory actons. From here on, n ths paper, we shall consder regulaton of ths more advanced, cause-controlled type (though much of what we say wll stll be true of the error-controlled.) 4. MODELS Defnng regulaton as we have seen, s easy n that one s led rapdly to one of a few forms, closely related and easly dstngushed n practcal use, The attempt to defne a model, however, leads to no such focus. We shall obtan a defnton sutable for ths paper, but frst let us notce what happens when one attempts precson. We can start wth such an unexceptonable model as a table-top replca of Chartres cathedral. The transformaton s of the type, n three dmensons: y y y = kx 1 = kx = kx 2 3 wth k about But ths example, so clear and smple, can be modfed a lttle at a tme to forms that are very dfferent. A model of Swtzerland. for nstance, mght well have the vertcal heghts exaggerated (so that the three k s are no longer equal). In two dmensons, a (proportonal) photograph from the ar may be followed by a Mercator s projecton wth dstorton, that no longer leaves the varables separable. So we can go through a map of a subway system, wth only the ponts of connecton vald, to maps of a type descrbable only mathematcally. In dynamc systems, f the transformaton converts the real tme t to a model tme t also n real tme we have a workng model. An unquestonable model here would be a flow of electrons through a net of conductng sheds that accurately models, n real tme, the flow of underground water n Arzona. But the model salng-boat no longer behaves proportonately so that a complex relaton s necessary to relate the model and the fullszed boat. Thus, n the workng models, as n the statc, we can readly obtan examples that devate more and more from the obvous model to the most extreme types of transformaton, wthout the appearance of any natural boundary dvdng model from non-model. Can we follow the mathematcan and use the concept of somorphsm? It seems that we cannot. The reason s that though the concept of somorphsm s unque n the branch where t started (n the fnte groups) ts extenson to other branches leads to so many new meanngs that the uncty s lost. As example. suppose we attempt to apply t to the unverse of bnary relatons. R, a subset of E E, and S, a subset of F F, are naturally regarded as somorphc, f there

6 1 exsts a one-one mappng δ of E onto F such that S = δrδ (Rguet 1948, 1951, Bourbak 1958). But S and R are stll closely related, and able to clam some model relatonshp f the defnton s weakened to δ, τ : S = δrδ 1 (wth τ also one-one). Then t can be weakened further by allowng φ (and τ) to be a mappng generally or even a bnary relaton. The sgn of equalty smlarly can be weakened to s contaned n. We have now arrved at the relaton gven earler (1) under regulaton ): ρ A φ whch evdently mples some -morphc relaton between ρ and φ (wth A assumed gven). In ths paper we shall be concerned chefly wth somorphsm between two dynamc systems (S and R n fg. 1). We can therefore try usng the modern abstract defnton of machne wth nput as a rgorous bass. To dscuss so-, and homo-, morphsm of machnes, t s convenent frst to obtan a standard representaton of these deas n the theory of groups, where they orgnated. The relaton can be stated thus: Let the two groups be, one of the set E of elements e 1, wth group operaton (multplcaton) δ, so that δ ( e, e ) ek, and other smlarly of δ on elements F. Then j = the second s a homomorph of the frst f and only f there exsts a mappng h, from E to F, so that, for all Error! Objects cannot be created from edtng feld codes.: [ h( e ), h( e )] h[ δ ( e, e )] δ ' = (2) j j If h s one-one onto F, they are somorphc. Ths basc equaton form wll enable us to relate the other possble defntons. Hartmans and Stearns (1966) defnton of machne M beng a homomorphsm of M follows naturally. Let machne M have a set S of nternal states, a set I of nput-values (symbols), a set O of output-values (symbols), and let t operate accordng to δ, a mappng of S I to S, and λ, a mappng of S I to O. Let machne M be represented smlarly by S, I, O, δ, λ. Then M s a homomorphsm of M f and only f there exsts three mappngs: such that, for all s S and I h 1, of S to S h 2, of I to I h 3, of O to O

7 h h 1[ δ ( s, ) ] = δ '[ h1 ( s), h2 ( ) ] [ λ( s, ) ] = λ' [ h ( s), h ( ) ] (3) Ths defnton corresponds to the natural case n whch correspondng nputs (to the two machnes) wll lead, through correspondng nternal states, to correspondng outputs. But, unfortunately for our present purpose, there are many varatons, some trval and some gross, that also represent some sort of smlarty. Thus, a more general form, representng a more complex form of relaton, would be gven f the mappngs were replaced by one mappng h 1 of S to S, and h 2 of I to I h 4 of I S to I S. (More general because h 4 may or may not be separable nto h 1 and h 2 ). Then the crteron would be, [ h ( s, ) ] h [ δ ( s, )], s : δ ' 4 = 4 (4) a form not dentcal wth that at (3). There are yet more. The Black Box case gnores the nternal states S, and treats two Black Boxes as dentcal f equal nputs gve equal outputs. Formally, f µ and µ are the mappngs from nput to output, then the second Box s a homomorphsm of the frst f and only f there exsts a mappng h, of I to I, such that: [ h( ) ] h[ ( )] I : µ ' = µ (5) Here t should be remembered that equalty of outputs s only a specal case of correspondence. Also closely related are two Black Boxes such that the second s decoder to the frst: the second, gven the frst s output, wll take ths as nput and emt the orgnal nput: I : µ ' ϖ ( ) = (6) Ths s an somorphsm. In the homomorphc relaton, the nput and the fnal output µ ' µ ( ) would both be mapped by h to the same class: I : hµ ' µ ( ) = h( ) (7) These examples may be suffcent to show the wde range of abstract smlartes that mght clam to be somorphsms. There seem, n short, to be as many defntons possble to somorphsm as to model. It mght seem that one could make practcally any asserton one lkes (such as that n our ttle) and then ensure ts truth smply by adjustng the defntons. We beleve, however, that we can mark out one case that s suffcently a whole to be worth specal statement.

8 We consder the regulatory stuaton descrbed earler, n whch the set of regulatory events R and the set of events S n the rest of the system (.e. n the reguland, S, whch we vew as R s opponent) jontly determne, through a mappng ψ, the outcome events Z. By all optmal regulator we wll mean a regulator whch produces regulatory events n such a way that H(Z) s mnmal. Then under very broad condtons stated n the proof below, the followng theorem holds: Theorem: The smplest optmal regulator R of a reguland S produces events R whch are related to the events S by a mappng h : S R. Restated somewhat less rgorously, the theorem says that the best regulator of a system s one whch s a model of that system n the sense that the regulator s actons are merely the system s actons as seen through a mappng h. The type of somorphsm here s that expressed (n the form used above) by [ σ ( )] h : : ρ ( ) = h (8) where ρ and σ are the mappngs that R and S mpose on ther common nput I. Ths form s essentally that of (5) above, Proof: The sets R, S, and Z and the mappng Ψ : R S Z are presumed gven. We wll assume that over the set S there exsts a probablty dstrbuton p(s) whch gves the relatve frequences of the events n S. We wll further assume that the behavour of any partcular regulator R s specfed by a condtonal dstrbuton p(r S) gvng, for each event n S, a dstrbuton on the regulatory events n R. Now p(s) and p(r S) jontly determne p(r, S) and hence p(z) and H(Z), the entropy n the set of outcomes. ( H ( Z) p( z )log p( z ).) Wth p(s) fxed, the class of optmal regulators therefore zk Z k k corresponds to the class of optmal dstrbutons p(r S) for whch (HZ) s mnmal. We wll call ths class of optmal dstrbutons π. It s possble for there to be very dfferent dstrbutons p(z) all havng the same mnmal entropy H(Z). To consder that possblty would merely complcate ths proof wthout affectng t n any essental way, so we wll suppose that every p(r S) n π determnes, wth p(s) and ψ, the same (unque) p(z). We now select for examnaton an arbtrary p(r S) from π. The heart of the proof s the followng lemma: Lemma: S j S, the set { ψ ( r, s ) : p( r, s ) > 0} has only one element. That s, for j every s j n S, p(r s j ) s such that all r wth postve probablty map, wth s j under ψ to the same z k n Z. Proof of lemma: Suppose, to the contrary, that p(r 1 s j )>0, p(r 2 s j )>0, ψ ( r 1, s j ) = z, and ψ ( r j = z z. Now p(r 1, s j ) and p(r 2, s j ) contrbute to p(z 1 ) and p(z 2 ) respectvely, 2, s ) 2 1 and by varyng these probabltes (by subtractng from p(r 1, s j ) and addng to p(r 2, s j )) we could vary p(z 1 ) and p(z 2 ) and thereby vary H(Z). We could make ether postve or negatve, whchever would make p(z 1 ) and p(z 2 ) more unequal. One of the j

9 useful and fundamental propertes of the entropy functon s that any such ncrease n mbalance n p(z) necessarly decreases H(Z). Consequently, we could start wth a p(r S) from the class π, whch dmnshes H(Z), and produce a new p(r S) resultng n a lower H(Z); ths contradcton proves the lemma. Returnng to the proof of the theorem, we see that, for any member of π and any s j n S, the values of R for whch p(r S) s postve all gve the same z k. Wthout affectng H(Z) we can arbtrarly select one of those values of R and set ts condtonal probablty to unty and the others to zero. When ths process s repeated for all s j n S, the result must be a member of π wth p(r S) consstng entrely of ones and zeroes. In an obvous sense ths s the smplest optmal p(r S) snce t s n fact a mappng h from S nto R. Gven the correspondence between optmal dstrbutons p(r S) and optmal regulators R, ths proves the theorem. The Theorem calls for several comments. Frst, t leaves open the possblty that there are regulators whch are just as successful (just as optmal ) as the smplest optmal regulator(s) but whch are unnecessarly complex. In ths regard, the theorem can be nterpreted as sayng that although not all optmal regulators are models of ther regulands, the ones whch are not are all unnecessarly complex. Second, t shows clearly that the search for the best regulator s essentally a search among the mappngs from S nto R; only regulators for whch there s such a mappng need be consdered. Thrd, the proof of the theorem, by avodng all menton of the nputs to the regulator R and ts opponent S, leaves open the queston of how R, S, and Z, are nterrelated. The theorem apples equally well to the confguratons of fg, 1 and fg. 2, the chef dfference beng that n fg. 2 R s a model of S n the sense that the events R are mapped versons of the events S, whereas n fg. 1 the modellng s stronger; R must be a homo- or somorphsm of S (snce t has the same nput as S and a mappng-related output). Last, the assumpton that p(s) must exst (and be constant) can be weakened; f the statstcs of S change slowly wth tme, the theorem holds over any perod throughout whch p(s) s essentally constant. As p(s) changes, the mappng h wll change approprately, so that the best regulator n such a stuaton wll stll be a model of the reguland, but a tme-varyng model wll be needed to regulate the tme-varyng reguland. 5. DISCUSSION The frst effect of ths theorem s to change the status of model-makng from optonal to compulsory. As we sad earler, model-makng has htherto largely been suggested (for regulatng complex dynamc systems) as a possblty: the theorem shows that, n a very wde class (specfed n the proof of the theorem), success n regulaton mples that a suffcently smlar model must have been bult, whether t was done explctly, or smply developed as the regulator was mproved. Thus the would-be model-maker now has a rgorous theorem to justfy hs work.

10 To those who study the bran, the theorem founds a 'theoretcal neurology'. For centures, the study of the bran has been guded by the dea that as the bran s the organ of thnkng, whatever t does s rght. But ths was the vew held two centures ago about the human heart as a pump; today's hydraulc engneers know too much about pumpng to follow the heart's method slavshly: they know what the heart ought to do, and they measure ts effcency. The developng knowledge of regulaton, nformatonprocessng, and control s buldng smlar crtera for the bran. Now that we know that any regulator (f t conforms to the qualfcatons gven) must model what t regulates, we can proceed to measure how effcently the bran carres out ths process. There can no longer be queston about whether the bran models ts envronment: t must. 6. REFERENCES Ashby, W. Ross, Automaton Theory and Learnng Systems, edted by D. J. Stewart (London: Academc Press), p Bourbak, N., 1958, Théore des Ensembles: Fasccule de Résultats, 3rd edton (Pars : Hermann). Conant, Roger C., 1969, I.E.E.E. Trans. Systems Sc., 5, 334 Hartmans, J., and Stearns, R. E., 1966, Algebrac Structure Theory of Sequental Machnes. (New York: Prentce-Hall). Rquet, J., 1948, Bull. Soc. Math. Fr. 76, 114; Thése de Pars. Sommerhof, G., 1950, Analytcal Bology (Oxford Unversty Press).

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