[ Pobierz całość w formacie PDF ]
.are the various messages and C1, C2, & arelea, and then becomes a pattern of nerve-impulses moving up thetheir coded forms.A coding, then, is specified by a transforma-auditory nerve.Here we can leave it, merely noticing that this very140 141 AN INTRODUCTION TO CYBERNETICS TRANSMISSION OF VARIETYEx.10: The concentrations high or low of sex-hormone in the blood of a cer-tion.tain animal determines whether it will, or will not, go through a ritual ofOften the method uses a key-word or some other factor thatcourtship.If the sex-hormone is very complicated chemically and the ritualis capable of changing the code from one form to another.Such avery complicated ethnologically, and if the variable behaviour is regardedfactor corresponds, of course, to a parameter, giving as many par-as a coded form of the variable concentration , how much variety is thereticular codings (or transformations) U1, U2,.as there are values in the set of messages ?to the factor. Decoding means applying such a transformation to the trans-form Ci as will restore the original message Mi :8/5.Coding by machine.Next we can consider what happens whena message becomes coded by being passed through a machine.C1 C2 C3 &That such questions are of importance in the study of the brainV:M1 M2 M3 &needs no elaboration.Among their other applications are thosepertaining to instrumentation the science of getting informa-Such a transformation V is said to be the inverse of U; it may thention from some more or less inaccessible variable or place, suchbe written as U-1.In general, only one-one transformations haveas the interior of a furnace or of a working heart, to the observer.single-valued inverses.The transmission of such information almost always involvesIf the original message Mi is to be recoverable from the codedsome intermediate stage of coding, and this must be selected suit-form Ci, whatever value i may have, then both U and U-1 must beably.Until recently, each such instrument was designed simply onone-one; for if both Mi and Mj were to be transformed to one formthe principles peculiar to the particular branch of science; today,Ck, then the receiver of Ck could not tell which of the M s hadhowever, it is known, after the pioneer work of Shannon andbeen sent originally, and Ck cannot be decoded with certainty.Wiener, that certain general laws hold over all such instruments.Next suppose that a set of messages, having variety v, is sentWhat they are will be described below.coded by a one-one transformation U.The variety in the set ofA machine was defined in S.3/4 as any set of states whosecoded forms will also be v.Variety is not altered after coding bychanges in time corresponded to a closed single-valued transfor-a one-one transformation.mation.This definition applies to the machine that is totally iso-It follows that if messages of variety v are to pass through sev-lated i.e.in constant conditions; it is identical with the absoluteeral codes in succession, and are to be uniquely restorable to theirsystem defined in Design.In S.4/ I the machine with input wasoriginal forms, then the process must be one that preserves thedefined as a system that has a closed single-valued transformationvariety in the set at every stage.for each one of the possible states of a set of parameters.This isEx.1: Is the transformation x' = log10 x, applied to positive numbers, a one-oneidentical with the transducer of Shannon, which is defined as acoding? What is decoding it usually called?system whose next state is determined by its present state and theEx.2: Is the transformation x' = sin x, applied to the positive numbers, a one-onepresent values of its parameters.(He also assumes that it can havecoding?a finite internal memory, but we shall ignore this for the moment,Ex.3: What transformation results from the application of, first, a one-one trans-returning to it in S.918.)formation and then its inverse ?Assume then that we have before us a transducer M that can beEx.4: What transformation is the inverse of n' = n + 7?in some one of the states S1, S2,., Sn, which will be assumedEx.5: What transformation is the inverse of x' = 2x + y, y' = x + y?here to be finite in number.It has one or more parameters that canEx.6: If the coded form consists of three English letters, e.g.JNB, what is thevariety of the possible coded forms (measured logarithmically) ? take, at each moment, some one of a set of values P1, P2,., Pk.Ex.7: (Continued.) How many distinct messages can be sent through such a Each of these values will define a transformation of the S s.Wecode, used once?now find that such a system can accept a message, can code it, andEx.8.Eight horses are running in a race, and a telegram will tell Mr.A.whichcan emit the coded form.By message I shall mean simply somecame first and which second.What variety is there in the set of possible mes-succession of states that is, by the coupling between two systems,sages ?at once the output of one system and the input of the other.OftenEx.9: (Continued.) Could the set be coded into a single letter, printed either asthe state will be a vector.I shall omit consideration of any mean-capital or142 143 AN INTRODUCTION TO CYBERNETICS TRANSMISSION OF VARIETYEx.6: Pass the message 314159. (the digits of ) through the transducer n'ing to be attached to the message and shall consider simply what= n + a 5, starting the transducer at n = 10.will happen in these determinate systems.Ex.7: If a and b are parameters, so that the vector (a,b) defines a parameter state,For simplicity in the example, suppose that M can take any oneand if the transducer has states defined by the vector (x,y) and transformationof four states: A, B, C, and D; that the parameters provide threestates Q, R, and S.These suppositions can be shown in tabular x' ax + by=ó y' = x + (a b)y,form, which shows the essentials of the transducer (as in S.4/l):complete the trajectory in the table:A B C Da 1 2 0 1 2 5 2Q C C A Bb 1 1 1 0 1 2 0RA C B Bx 2 1 2 ? ? ? ?y 1 4 11 ? ? ? ?S B D C D*Ex.8: A transducer, with parameter u, has the transformation dx/dt = (u + 4)x;Given its initial state and the sequence of values given to theit is given, from initial state x = 1, the input u = cos t; find the values of x asparameter, its output can be found without difficulty, as in S.4/1.output.Thus, suppose it starts at B and that the input is at R; it will change*Ex.9: If a is input to the transducerto C.If the input goes next to Q, it will go from C to A.The resultsdx/dt = yso far can be shown in tabular form: dy/dt = x 2y + a,with diagram of immediate effectsInput-state: R Q!a ! y x,!Transducer-state: B C Awhat is the output from x if it is started at (0,0) with input a = sin t? (Hint:It can now easily be verified that if the initial state is B and theUse the Laplace transform.)input follows the sequence R Q R S S Q R R Q S R, the output*Ex.10: If a is input and the transducer iswill follow the sequence B C A A B D B C B C C B.dx/dt = k(a x)There is thus no difficulty, given the transducer, its initial state,and the input sequence, in deducing its trajectory
[ Pobierz całość w formacie PDF ]