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.remain the same absolute system the values of the probabilitiesThis is the basic theory, in terms of communication, of the simplemust be unchanging.feedback regulator.In S.2/10 it was shown that a single-valued transformationcould be specified by a matrix of transitions, with 0 s or 1 s in the12/7.The reader may feel that excessive attention has just been cells (there given for simplicity as 0 s or + s).In S.9/4 a Markovgiven to the error-controlled regulator, in that we have stated with chain was specified by a similar matrix containing fractions.Thuscare what is already well known.The accuracy of statement is, a determinate absolute system is a special case of a Markovianhowever, probably advisable, as we are going now to extend the machine; it is the extreme form of a Markovian machine in whichsubject of the error- controlled regulator over a range much wider all the probabilities have become either O or 1.(Compare S.9/3.)than usual.A machine with input was a set of absolute systems, distin-This type of regulator is already well known when embodied in guished by a parameter.A Markovian machine with input musta determinate machine.Then it gives the servo-mechanism, the similarly be a set of Markovian machines, specified by a set ofthermostat, the homeostatic mechanism in physiology, and so on.matrices, with a parameter and its values to indicate which matrixIt can, however, be embodied in a non-determinate machine, and is to be used at any particular step.it then gives rise to a class of phenomena not yet commonly occur- The idea of a Markovian machine is a natural extension of the224 225 AN INTRODUCTION TO CYBERNETICS THE ERROR-CONTROLLED REGULATOR*Ex.4: (Continued.) What general rule, using matrix multiplication, allows theidea of the ordinary, determinate machine the type consideredanswer to be written down algebraically? (Hint: Ex.9/6/8.)throughout Part I.If the probabilities are all O or I then the two are*Ex.5: Couple the Markovian machine (with states a, 6, c and input-states ±,identical.If the probabilities are all very near to O or I, we get a²)machine that is almost determinate in its behaviour but that occa-a b c a b csionally does the unusual thing.As the probabilities deviate fur-ther and further from O and 1, so does the behaviour at each stepa 0.2 0.3 0.3 a 0.3 0.9 0.5± : ² :become less and less determinate, and more and more like that of b.7 0.2 b 0.6 0.1 0.5c 0.8.5 c 0.1.one of the insects considered in S.9/4.It should be noticed that the definition, while allowing someto the Markovian machine (with states e, f and input-states ´, µ, ¸ )indeterminacy, is still absolutely strict in certain respects.If themachine, when at state x, goes on 90% of occasions to y and one f e f e f10% of occasions to z, then those percentages must be constante 0.7 0.5 e 0.2 0.7 e 0.5 0.4´ : µ : ¸ :(in the sense that the relative frequencies must tend to those per-f 0.3 0.5 f 0.8 0.3 f 0.5 0.6centages as the sequence is made longer; and the limits must beunchanging as sequence follows sequence).What this means inby the transformationspractice is that the conditions that determine the percentages musta b c e fremain constant.µ ´ ¸ ² ±The exercises that follow will enable the reader to gain somefamiliarity with the idea.What is the Markovian machine (without input) that results ? (Hint: Trychanging the probabilities to O and 1, so as to make the systems determi-Ex.1: A metronome-pendulum oscillates steadily between its two extremenate, and follow S.4/8; then make the probabilities fractional and followstates, R and L, but when at the right (R) it has a 1% chance of sticking therethe same basic method.)at that step.What is its matrix of transition probabilities ?*Ex.6: (Continued.) Must the new matrix still be Markovian?Ex.2: A determinate machine ± has the transformation*Ex.7: If M is a Markovian machine which dominates a determinate machine N,show that N s output becomes a Markov chain only after M has arrived atA B C Dstatistical equilibrium (in the sense of S.9/6).B D D DA Markovian machine ² has the matrix of transition probabilities12/9.Whether a given real machine appears Markovian or deter-A B C Dminate will sometimes depend on how much of the machine isA 0 0 0 0observable (S.3/11); and sometimes a real machine may be suchB 0.9 0 0 0that an apparently small change of the range of observation mayC 0 0 0.2 0be sufficient to change the appearances from that of one class toD 0.1 1.0 0.8 1.0the other.How do their behaviours differ? (Hint: Draw ± s graph and draw ² s graphThus, suppose a digital computing machine has attached to it aafter letting the probabilities go to 1 or 0.)long tape carrying random numbers, which are used in some proc-Ex.3: A Markovian machine with input has a parameter that can take three val-ues p, q, r and has two states, a and b, with matrices ess it is working through.To an observer who cannot inspect thetape, the machine s output is indeterminate, but to an observer(p) (q) (r)who has a copy of the tape it is determinate.Thus the question Isa b a b a bthis machine really determinate? is meaningless and inappropri-a 1/2 1 a 1/4 3/4 a 1/3 3/4ate unless the observer s range of observation is given exactly.Inb 1/2 0 b 3/4 1/4 b 2/3 1/4other words, sometimes the distinction between Markovian anddeterminate can be made only after the system has been definedIt is started at state b, and goes one step with the input at q, then one step withaccurately.(We thus have yet another example of how inadequateit at r, then one step with it at p.What are the probabilities that it will now beat a or b?is the defining of the system by identifying it with a real object.226 227 AN INTRODUCTION TO CYBERNETICS THE ERROR-CONTROLLED REGULATORReal objects may provide a variety of equally plausible sys- single-valued, more than one arrow can go from each state.Thustems , which may differ from one another grossly in those prop- the Markovian machineerties we are interested in here, and the answer to a particulara b cquestion may depend grossly on which system it happens to beapplied to.) (Compare S.6/22.)a 0.2 0.3 0.1b 0.8 0.7 0.5c.412/10.The close relation between the Markovian machine and thedeterminate can also be shown by the existence of mixed forms.has the graph of Fig.12/11/1, in which each arrow has a fractionThus, suppose a rat has partly learned the maze, of nine cells, shownindicating the probability that that arrow will be traversed by thein Fig.12/11/1,representative point.Fig.12/10/1Fig.12/11/1in which G is the goal.For reasons that need not be detailed here,the rat can get no sensory clues in cells 1, 2, 3 and 6 (lightly shaded),In this particular example it can be seen that systems at c will allso when in one of these cells it moves at random to such other cellssooner or later leave it, never to return.as the maze permits.Thus, if we put it repeatedly in cell 3 it goesA Markovian machine has various forms of stability, whichwith equal probability to 2 or to 6.(I assume equal probabilitycorrespond to those mentioned in Chapter 5.The stable region ismerely for convenience.) In cells 4, 5, 7, 8 and G, however, cluesa set of states such that once the representative point has enteredare available, and it moves directly from cell to cell towards G
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