Philosophy of Science Association

A General Black Box TheoryAuthor(s): Mario BungeSource: Philosophy of Science, Vol. 30, No. 4 (Oct., 1963), pp. 346-358Published by: The University of Chicago Press on behalf of the Philosophy of ScienceAssociationStable URL: http://www.jstor.org/stable/186066Accessed: 23/02/2009 13:41

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A GENERAL BLACK BOX THEORY*

MARIO BUNGE Physics Department, Temple University, Philadelphia, Pa.

A mathematical theory is proposed and exemplified, which covers an extended class of black boxes. Every kind of stimulus and response is pictured by a channel connecting the box with its environment. The input-output relation is given by a postulate schema according to which the response is, in general, a nonlinear functional of the input. Several examples are worked out: the perfectly transmitting box, the damping box, and the amplifying box. The theory is shown to be (a) an extension of the S-matrix theory and the accompanying channel picture as developed in microphysics; (b) abstract and applicable to any problem involving the transactions of a system (physical, biological, social, etc.) with its milieu; (c) superficial, because unconcerned with either the structure of the box or the nature of the stimuli and responses. The motive for building the theory was to show the capabilities and limitations of the phenomenological approach.

A black box is a fiction representing a set of concrete systems into which stimuli S impinge and out of which reactions R emerge. The constitution and structure of the box are altogether irrelevant to the approach under consideration, which is purely external or phenomenological. In other words, only the behavior of the system will be accounted for.

The various kinds of stimuli and responses will be pictured as so many channels C along which signals travel.' A channel, then, is assigned to each class of stimuli or responses, regardless of the intervening organs or mechanisms-the investigation of which is left to nonphenomenological, or representational theories. The number of gates of the same kind will be irrelevant to the present approach:

S

FIG. I * Received, February 1962. 1 The channel picture has been taken over from the kinetic theory of nuclear reactions.

See John M. Blatt and Victor F. Weisskopf, Theoretical Nuclear Physics (New York: Wiley, 1952), pp. 313 and 517 ff.

346

A GENERAL BLACK BOX THEORY 347

thus a transformer-a four-terminal box-will count as a single channel box(see Fig. 1), since both the stimuli and the responses at the terminals are electrical. In a two- channel box one of the channels may specialize, say, in pressure inputs and outputs, whereas the other channel may specialize in thermal stimuli and responses.

Since the aim is to relate stimuli to responses without regard to the parts or organs of the system, the adequate mathematical tool will be integral relations rather than differential equations. In other words, a CCnonlocal" theory seems to be called for if the box is regarded as a unit rather than as a system of interdependent parts.

The theory will first be developed for the single channel box and will then be extended to the multiple channel box.

1. Single Channel Box. The simplest black box reacts to stimuli of a single kind with responses of the same kind: in our picture it has a single channel C even though the corresponding concrete system itself may have several gates or terminals. We shall assume that the intensity R(t) of the output at time t is determined by the intensity of the input S at times u prior to and up to u - r, and that the form of the dependence is

[1] R(t) f 0 du M(t, u)F[S(u -)],

where u is the integration variable and -r the time delay (reaction time) of the box for the given channel. For many purposes (e.g., particle scattering), r 0. The functions M and F, which may be real or complex, sum up the global properties of the box. Notice that the response is not, in general, a function but a functional of the stimulus; moreover, R need not be a linear functional of S, so that [1] covers both linear and nonlinear systems. The instantaneous response of the box to a stimulus is a function of its entire history, unless M (the memory or heredity function) vanishes for certain past intervals. If either S or M is a random variable, R will be a random variable as well.

The postulate schema [1] is broad enough to account for any kind of box and stimulus. Three main sets of problems can be approached with the help of [1]:

(1) The problem of prediction: given the kind of box (i.e., given the functions M and F) and the stimulus S, find the response R.

(2) The inverse problem of prediction: given the kind of box (M and F) and the response R, find the stimulus S responsible for the known behavior. Notice that, contrary to the corresponding situation in systems described by means of differential equations, here we have, in general, no symmetry between prediction and retrodiction, since an infinity of past histories may end up in the final state R(t).

(3) The problem of explanation: given the behavior (R) under a known stimulation (S), find the kind of box (i.e., determine M and F) that will account for that behavior.

In general, problems (2) and (3) are not well-determined, i.e., they have no unique solutions. They become well-determined if it is further assumed that the hereditary function M is invariant under time displacements (i.e., is a function of the time differences u - t alone) and is extended to a limited past (i.e., vanishes before a given instant). This wide class of problems is dealt with in Appendix I.

We shall illustrate the postulate schema [1] with the cases of the perfect transmitter, the damping box, and the amplifying box. In each case we shall distinguish three kinds of typical stimuli:

348 MARIO BUNGE

(a) S is constant over the time interval [0, T] and zero elsewhere, i.e., [2] S(u) = const. if u E [0, T];

o ~~otherwise. (b) S acts only at time to, i.e.,

[3] F[S(u)] - F(S) - 8(u-to), to [0, T], where W' denotes Dirac's singular "function", the most important property of which is

T

[4] f duf(u) 8(u - to) = f(to), to c [0, T] o

for an arbitrary function f(u). (c) S is a periodic function, in particular [5] F[S(u)] = SF(S) cos wu, u > 0; [5] F[S(u)] 0 otherwise. Let us now specify the heredity or memory function M, without however making

any assumption regarding the mechanism responsible for M. 1.1. Perfectly Transmitting Box. This box neither distorts nor improves qualitatively the information it receives: it just converts the inputs into outputs of the same form, though in general of different intensities and duration. That is, M does not depend upon time. Hence our postulate schema [1] becomes

[6] R(t) M J duF[S(u -)]. 0

S ~~~~~~

I R

I - I _I 2

FIG. 2

A GENERAL BLACK BOX THEORY 349

(a) Constant stimulus Recalling [2], we see that the effective stimulus, i.e., the retarded input, is

(0, u < T [7] S(u-)z)= S =const., Ir u T+T

tO, u>T+-r In fact, (u - r) E (0, T) is equivalent to u E [r, X + T]. Hence, the response is [8] R(t)-= M JT+T )du TMTF(S), t E [r, - + T]; [8] R(t)== M F(S 0u otherwise. That is, the output is constant over the interval T regardless of the time elapsed since the application of the stimulus; and the response is delayed both in the beginning and in the end (see Fig. 2).

In particular, if [91 F(S) =- S2, we get [10] R _ kSP, with k = MT. Consequently, the psychophysical law fits -the model of the perfectly transmitting box subject to constant stimuli.

(b) Sudden Stimulus According to [3], the effective stimulus is now

[11] F[S(u - r)] F(S) 8(u - - to).

sIR

S

O to to + Z FIG. 3

350 MARIO BUNGE

Using [4] we obtain

[12] R(t) = ME(S)f duS8(u - - - to) - M() >r1t0>O Jo 0 otherwise. That is, the response to an instantaneous input is time-independent; moreover, it does not cease while the box lasts. (The problem of the energy supply for such a continued response is, of course, beyond the scope of our approach.)

(c) Periodic Stimulus Recalling [5], the retarded stimulus is

[13] F[S(u -T)] = F(S) cos c(u - r), t > r > 0; Hence,

[14] R(t) MF= S) M du cos co(u -T) - MF(S) [sinCoT +sin o(t-T)] if t > T.

S j [R~~~~~~~~ o z \ 7Jo Ct

FIG. 4

If the reaction time is zero, the bracket reduces to sin ct; the only difference between the stimulus and the response is then, apart from the intensity, a 90? shift. In any case, the output is periodic with the same period as the input.

1.2. Damping Box. The reaction capacity of this box deteriorates with time in an exponential form: [15] M(t, u) = M. exp [- k(t - u)], M = const., u < t, where k' is the relaxation time. The response is then

[16] R(t) = M { du e-(t-u)F[S(u -T)]. 0

A GENERAL BLACK BOX THEORY 351

(a) Constant Stimulus In view of [7], the response is

[17] R(t) = MF(S) f du e-k(t-u) ekr(ekT e-kt for t > T. T ~~~~k

S iR

__ _. _ ____ I\ S

0 ~~~~~~~~T IS

FIG. 5

After a long enough time (t > T), the response becomes negligible. But, if the stimulus is kept up to the observation time t, i.e., if T= t, we have the case of stimulus reinfor- cement and the response increases steadily in accordance with

[18] R(t) ==ME(S) ekTr( - ekt) k Incidentally, this is Hull's function, well known to psychologists. In words: constant stimulation can make up for defective memory. The asymptotic value of [18] is [19] R(oo) == MF(S) ek?/lk = const.

(b) Sudden Stimulus Taking [3] and [16] into consideration, the response is

R(t) M(S) fdu ek(t-u) (u- to) [20] MF ME(S) eW-r-to), t > T + to > 0;

0 otherwise. The response to a sudden stimulus decays exponentially and dies out completely after an infinite lapse.

4

352 MARIO BUNGE

(c) Periodic Stimulus In view of [5] and [16],

R(t) = MF(S) du e-k(t-u) cos w0(u-) [211 ?M(S)

k2] & + , [k cos c(t-r) + c sin c(t-r) - (k cos ci-- c sin w) e kt]. The response, too, is a periodic function with the same period as the stimulus, only with an exponentially decreasing amplitude. Notice that, for k = 0, [21] reduces to [13]. 1.3. Amplifying Box. Suppose our black box amplifies stimuli in accordance with the law [22] M(t, u) = M[1 - e-k(t1U)], M = const., u < t. The response will then be

[23] R(t) = M f du[1 - e-k(t-u)] F[S(u -T)] 0

(a) Constant Stimulus Inserting [2] in [23], we get

-rT R(t) MF(S) du[1 e-k(t-u)]

[24] MF(S) - k [kT -(eT -1)ek(t?)], t > T.

FIG. 6

A GENERAL BLACK BOX THEORY 353

For either very small excitation periods or very rapidly amplifying boxes, ekT- 1 - kT, and

[25] R(t) M (S) [I - e-k(tT) T [ If the stimulus is reinforced up to the observation time (t = T),

[26] R(t) -= MF(S) [kt - ek(l - e-kt)]. k For large observation periods the bracket approaches the straight line kt - ekr. That is, the behavior of the box tends to become linear.

(b) Sudden Stimulus R(t) = MF(S) j du[1 - ek(t-u)] S3(u - -to)

[273 - ME(S) [10- ek(T+to) e-kt], t > to + T. The amplifying box subjected to an impulsive stimulus behaves, then, like the damping box subjected to constant reinforcement (cf. [18]).

(c) Periodic Stimulus g

R(t) = MF(S) f du[l - e-kt-t')] cos w(u - T)

[28] = MF(S) | [sin wi + sin (t -r)] k2 + -, [k cos w(t- T) + sin co(t - T) -(k cos COT - J sin wT) e-kt

The period of the output is, again, the same as that of the input.

2. Multiple Channel Box. A box with two channels has the following four (mutually compatible) possibilities of behavior:

SI Rl, SI R2 S2 R1, S2 R2.

In general, a box with N channels will be able to make N2 possible transitions, which may conveniently be ordered in a square matrix [Sm --* Rn]. This matrix has in general no simple structure. Thus, it need not be symmetric, since reversibility is an exception, and it need not be unitary either, since there need not be conservation of the incoming stimulus: the box will in general either absorb or dissipate some information and/or energy through some channels. In a general black box theory the scattering matrix (as it is called in physics) is then much more complicated than in the case, say, of "celementary" particle collision, where reciprocity (reversibility) ensures the matrix symmetry, and flux conservation corresponds to unitarity.

Still, it may be advantageous to retain part of the vocabulary coined in the field of the scattering matrix theory, or S-matrix theory for short.2 Thus the diagonal

2 See, e.g., N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (New York: Interscience Publishers, Inc., 1959), pp. 197 ff.

354 MARIO BUNGE

element S. -* Rn of our generalized scattering matrix may be called the reflection coefficient of the n-th channel. And the off-diagonal term Sm -* R, of the scattering matrix may be called the transmission coefficient from channel Cm into channel Cn.

St C S

S2 Ca

RI

FIG. 7

Furthermore, if the black box is statistical, the elements of the scattering matrix will be related to the transition probabilities or, rather, to the probabilities for the conver- sion of inputs of given kinds into outputs of other given kinds. In short, our theory is a generalization of the S-matrix theory; this will emerge more clearly in Appendix I.

We will generalize the postulate schema [1] by assuming that the m-th response Rm(t) at time t depends on a linear combination of the various inputs, in the form

[29] Rm(t) = 0 du EMmn(t, u) F.[S.(u -T)]

where CMmn' designates the memory function associated with the m -* n transition; in particular, Mmm is the memory of the box for the m-th class of stimuli. In general, the matrix [Mmn] will not be symmetric. The contributions Fn[Sn] of the various kinds of stimuli are, in general, different from one another, and every channel has its own delay constant.

We shall specify this postulate schema for the cases treated above, correspondingly generalized:

(a) each Sn acts constantly during a time interval T,; (b) each Sn acts suddenly at a particular time tn; (c) each Sn, acts periodically with a frequency co, and a phase 0n of its own. That is, (a) Constant Stimuli

[30] S.(u) 0:S, = const.ifueT, n( l otherwise. (b) Sudden Stimuli

[31] FP[S,(u)] Fn(Sn) 3(u - tn), tn E [0, t].

A GENERAL BLACK BOX THEORY 355

(c) Periodic Stimuli [32] F.[S.(u)] = F.[S.] cos(wnu + u > 0. 2.1. Perfectly Transmitting Box. The memory matrix is time-independent:

Mmn (t, u) Mmn const. (a) Constant Stimuli

[33] Rm(t) MmnFn(Sn) T, t e [I) n + T.

By introducing the column matrices

R, ~ ~ [F1T1 R2 F2T2

[34] R= F| S

_RN __FN TN

the N relations [33] are abreviated to [35] R = MS, where 'M' designates the memory matrix, or scattering matrix. The time-indepen- dence of the over-all process is reflected in the time-independence of M.

(b) Sudden Stimuli N

[36] Rm(t) = InMmnFn(Sn), t > Tn + tn 1

In matrix form we write again R = MS, where now and in the following

-Fl(Sl) F2(S2)

[37] S=

L FN(SN) (c) Periodic Stimuli

[38] Rm(t) = M F( {sin (cwn-rT + i) + sin [cn(t - -T,) + n]},

for t > Max (Tn).

2.2. Damping Box. Let the memory matrix elements be

[39] Mmn(t, u) M mne-kmn(t-u), Mmn = const., u < t, where k-J1 is the relaxation time for the m -* n "reflection".

356 MARIO BUNGE

(a) Constant Stimuli

[40] Rm(t) mk ( ) nTn (e mnT 1 e-kmnt 1 mn

For t > Max(TJ), the response is negligible. But if all stimuli are kept up to the observation time (i.e., if t Tn), the scattering matrix elements become proportional to 1 - e-mnt and the response tends asymptotically to a constant column matrix.

(b) Sudden Stimuli N

[41] Rm(t) = I MmnFn(Sn) e-kmn(tTt_), t > tn + Tn.

(c) Periodic Stimuli N M Fn(S'1

[42] Rm(t) = jkk2+ .;I {kmn[cos (wnt + w,nTn - ?n) - cos ( -cn n)] 1 mn n

+ cajsin (cont - conTn + qn) + sin (nTn -n)]} 2.3. Amplifying Box. Let the memory matrix elements be

[43] Mmn(t, u) = Mmn[l - ekmn(tu)]. (a) Constant Stimuli

[44] Rm(t) N ! Fn(Sn) [kmnTn - (ekmnTn -1) ekmn(t )] kmn

If the stimuli are all reinforced up to the observation time t,

mn [45] Rm~(t) Xn kmn [kmt - ekmnt,Tn (1 + e-kmn,t)] F (S ) With increasing time the individual contributions of the various stimuli tend to become linear:

N [46] Rm(t) s 1 MmnFn(Sn) t

Incidentally, the growing automaton fits this schema.

(b) Sudden Stimuli N

[47] Rm(t) n Mmn[ + e mn(rn+tn) ekmnt] Fn(sn).

(c) Periodic Stimuli N

[48] Rm(t) = In Mmn [sin (wnTn + k.) + sin (wnt - WnTn + jn)] 1

-- k I 2- [kmn cos (cwnt - WjnTn + qj) + ovn sin (wnt - &nTn + Ojn)]

-[kmn cos (wnTh + 0n) - wn sin (wnTn + 0,n) e-kmnt] Fn(Sn).

A GENERAL BLACK BOX THEORY 357

3. Discussion. A general theory of systems of any kind (physical, biological, social, etc.) has been presented which has the following main features:

(1) The theory assumes no law of nature save the principle of antecedence ("The effect cannot occur before the cause"). Moreover, no specific variable, except time, is involved in the theory; and even this interpretation is dispensable: if found necessary some other meaning could be attributed to the variables 't', 'u', 'T', and 'i'. In short, the theory is almost abstract.

(2) A consequence of the abstract character of the theory, together with the broadness of its postulate schema, is a high degree of generality. In fact, it may be regarded as a generalization of the physical S-matrix theory (see Appendix II).

(3) The theory can be extended to systems of interconnected boxes. Thus, if two single-channel boxes are connected in series, the input S(2) on the second box will be equal to the output RM1) of the first box, so that the output R(2) of the second box will be obtained by replacing S(2) by RM. In a sense, though, a black box theory of a system of black boxes is self-defeating, since a beginning of internal structure is being recognized.

(4) Since the theory is unconcerned with either the basic structure of the box or the specific nature of the stimuli and responses, it is, literally speaking, an extremely superficial theory. This superficiality is the price that must be paid for its large coverage.

(5) The theory is nonlocal, in the sense that it treats systems as units and introduces no space coordinates to localize its parts. The intensity of the output R does not depend on the strength of the input in the immediate vicinity and at an immediately preceding time, but it depends on the over-all conditions prevailing throughout the box and along all or part of its history.

(6) The theory is extremely simple both as regards its presuppositions and its form: in fact, it just presupposes mathematical analysis and the principle of ante- cedence; and once the various functions are specified, everything boils down to a relation of the type of R = MS, where M (the scattering matrix) summarizes the behavioral properties of the box, whereas R and S contain all the empirical data concerning the controllable (non-intervening) variables. M may be regarded as an operator converting the set of stimuli data S into the set of responses data R.

(7) The theory is untestable because it is almost abstract (uninterpreted). The only relevant test would be that of usefulness: i.e., whether the proposed schema can cover a large number of kinds of black boxes. Moreover, even if all the variables of the theory were interpreted, there would be no way of independently testing for M and F[S]. In fact, one and the same (S, R) situation can be covered by an unlimited number of combinations of functions M and F[S]. Thus, a shock-absorbing black box can be represented either by a rapidly decaying M or by an F[S] insensitive to small variations of S, such as the time average of S.

Given the above characteristics, it is conceivable that the proposed schema could be of some use in general behavior theory in the preliminary stage where mechanisms are not sought, either because of an anachronistic methodological conviction or because the grapes are still sour.3

3The structure, scope, and function of phenomenological theories are studied in detail in the author's "Phenomenological Theories", in M. Bunge (Ed.), The Critical Approach: Essays in Honor of Karl Popper (Glencoe, Ill.: The Free Press, 1964).

358 MARIO BUNGE

APPENDIX I. AN IMPORTANT CLASS OF INVERSE PROBLEMS

In the text only the prediction problem (cf. section 1) was treated in some elementary cases. The inverse prediction problem and the explanation problem become well- defined if the "memory" of the box is limited and if it depends only on the interval u - t between the stimulus action and the observation time. If

O ,u O then for the single-channel box we have

t [50] R(t) = f du M(t - u)F[S(u -T)]

0

and similarly for the multiple-channel box. Multiplying by e-8t and integrating between 0 and oo, we obtain

oo co

rt [51] f dt e-stR(t) = f dt e-st f du M(u - t)F[S(u-r)].

0 0 0

Now, if the following integrals exist,

r(s) f dt e-stR(t), (s) f dt e-stM(t), f(s) f dt e-stF[S(t - 0 00

then by the Faltung theorem [51] becomes [52] r(s) = m(s)f(s). The inverse problem of prediction is then solved by computing the inverse Laplace transform of

f(s) = r(s)/m(s). And the problem of explanation is solved by finding the inverse Laplace transform of

m(s) = r(s)/f(s). In most cases the tables of Laplace transforms will do a large part of the job.

APPENDIX II. SUBSUMPTION OF THE S-MATRIX THEORY

The scattering operator (equivalent to our M-matrix) connects the initial state in the remote past, 0(- oo), with the final state in the remote future, b(oo), in the form [53] p(oo) = S(- oo). This formula is obtained from our postulate schema [1] by putting

R =S= S k,F(S) = S =0 0 = O, M(t, u)- M' a(u + t), M = const.

With these substitutions, we get

+(t)-M(t, - 00(-t) In particular,

O(oo) = M(oo, - oo)0(- oo), where M(oo, - 00) is the scattering operator, which in physics is usually denoted 'S'.

Article Contentsp. 346p. 347p. 348p. 349p. 350p. 351p. 352p. 353p. 354p. 355p. 356p. 357p. 358

Issue Table of ContentsPhilosophy of Science, Vol. 30, No. 4 (Oct., 1963), pp. 305-416Volume Information [pp. 411-415]Front MatterRemarks on the Language of Physics [pp. 305-306]DiscussionRemarks on Myhill's Remarks on Coordinate Languages [pp. 307-308]

Meaning and Action [pp. 309-324]Discrete State Systems, Markov Chains, and Problems in the Theory of Scientific Explanation and Prediction [pp. 325-345]A General Black Box Theory [pp. 346-358]Reflexive Predictions [pp. 359-369]DiscussionComments on Professor Roger Buck's Paper "Reflexive Predictions." [pp. 370-372]Rejoinder to Grnbaum [pp. 373-374]

Diversity in the Behavioral Sciences [pp. 375-395]Book ReviewsReview: untitled [p. 396]Review: untitled [pp. 396-398]Review: untitled [pp. 398-399]Review: untitled [pp. 399-401]Review: untitled [pp. 401-402]Review: untitled [pp. 402-403]Review: untitled [pp. 403-404]Review: untitled [pp. 404-405]Review: untitled [pp. 405-406]Review: untitled [pp. 406-407]Review: untitled [pp. 407-408]Review: untitled [pp. 408-409]

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