Part 12

cos 2β cos 2Θ₁ = -1 + 2 ——————— 1-cos 2γ

cos Θ = cos Θ₁ cos Θ₂.

We have obtained a complete description of the NPO which involves 74 formulas. These treat the noise in the various outputs, invariances of the NPO and other interesting features. A presentation of these would be outside of the scope of this paper and would tend to obscure the main features of the NPO. Thus, we show here only a typical sample of the computer simulation, Figure 8 and Figure 9. Conditions for these runs are shown in Table I. Run No. 6 duplicates run No. 5 except for the fact that i₁ and i₂ were disabled in run No. 6.

Observe that all our descriptions of the NPO and the space it is to decompose have been time invariant while the signals shown in the simulation are presented as functions of time. The conversion may be effected as follows: Given a measurable (single-valued) function

x = x(t)t ∊ T where μ(T) > 0 we define the space X = {x = x(t) ∍ t ∊ T}

and a probability distribution

μ(x⁻¹(X′)) P(X′) = —————————— X′ open ⊂ X μ(T) on that space.

TABLE I Legend for Traces of Figures 8 and 9 ---------+-------+-------+--------+-----+----------+-------+-------- Trace | | | | | | | Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 ---------+-------+-------+--------+-----+----------+-------+-------- Symbol | X₂ | X₁ | γ | β | i | dξ₂/dτ | dξ₁/dτ ---------+-------+-------+--------+-----+----------+--------+------- run No. 5| | | | | | | | | | | | | | signal |7½ Vrms|7½ Vrms| π ptop | |35.6 m cps| | | | | | | | | noise |16 Vrms|15 Vrms| π/9 | | | | | | | ptop| |sine wave | | | | | | | | | DC | 0 | 0 | | | | | | | | | | | | power s/n| 1/4 | 1/4 | 81/1 | | | 0 | 1/2 | | | | | | | terminal | | | | | | | value | | | π/4 | π/4 | | | ---------+-------+-------+--------+-----+----------+--------+------- run No. 6| | | | | | | | | | | | | | signal |7½ Vrms|7½ Vrms| π ptop | |35.6 m cps| | | | | | | | | noise | 0 | 0 | 0 | |sine wave | | | | | | | | | DC | -30V | 0 | | | | | | | | | | | | power s/n| ∞ | ∞ | ∞ | | | 0 | ∞ | | | | | | | terminal | | | | | | | value | | | π/4 | π/4 | | | ---------+-------+-------+--------+-----+----------+--------+-------

Observed from Oscillogram

Computed

Observed from Oscillogram

Then (X,p(X)) is a stochastic space in our usual sense and x(T) is a stochastic variable. Two immediate consequences are:

P(X) is stationary (P(X) is not a function of t ∊ T), and no question of ergodicity arises.

NETWORKS OF NPO’S

A network of NPO’s may constitute anything from a SOM to a preprogrammed detector, depending upon the relative amount of preprogramming included. Two methods of preprogramming are: (1) Feeding a signal out of a permanent storage into some of the inputs of the network of NPO’s. This a priori copy need not be perfect, because the SOM will measure the angles Θᵢ anyhow. (2) Feedback, which, after all, is just a way of taking advantage of the storage inherent in any delay line. (We implicitly assume that any reasonable physical realization of an NPO will include a delay T between the x input and the ξ output which is not less than perhaps 10⁻¹ times the time constant of the internal feedback loop in the γ computation.)

Simulation of channels that possess a discrete component requires feedback path(s) to generate the required free products of the finitely generated groups. Then, such a SOM converges to a maximal subgroup of the group describing the symmetry of the signal that is a free product available to this SOM.

Because a single NPO with 1 ≤ n₀ ≤ K₀ is isomorphic (provides the same input to output mapping) to a suitable network of NPO’s with n₀ = 1, it suffices to study only networks of NPO’s with n₀ = 1.

Figure 10 is largely self-explanatory. Item a is our schematic symbol for a single NPO with n₀ = 1. Items b, d (including larger feedback loops), and f are typical of artificial intelligence networks. Item c is employed to effect the level changing required in order to apply the three channels in cascade algorithm to the solution of one-dimensional coding problems. Observe that items c and e are the only configurations requiring the γ output. Item d may be used as a limiter by making T⁻¹ high compared to the highest frequency present in the signal. Observe that item e is the only application of NPO’s that requires either the ξ₂ or β outputs. Item f serves the purpose of handling higher power levels into and out of what effectively is a single (larger) NPO.

CONCLUSION

The definition of self-organizing behavior suitably represented has permitted the use of Information Theoretic techniques to synthesize a (mathematical) mechanism for a self-organizing machine. Physical mechanization in the form of an NPO has been accomplished and has introduced the experimental phase of the program. From among the many items deserving of further study we may mention: more economical physical mechanization through introduction of modern technology; identification of networks of NPO’s with their group theoretic descriptions; analysis of the dimensionality of tasks which a SOM might be called on to simulate, and prototype SOM applications to related tasks. It is hoped that progress along these lines can be reported in the future.

REFERENCES

1. Ścibor-Marchocki, Romuald I., “A Topological Foundation for Self-Organization,” Anaheim, California:Northrop Nortronics, NSS Report 2828, November 14, 1963

2. It is true that our definition is very similar to that proposed by Hawkins (reference 5). Compare for example his definition of learning machines (page 31 of reference 5). But the subsequent developments reviewed therein are different from the one we have followed.

3. Ashby, W. R., “The Set Theory of Mechanism and Homeostasis,” Technical Report 7, University of Illinois, September 1962

4. Ashby, W. R., “Systems and Information,” Transactions PTGME =MIL-7=:94-97 (April-July, 1963)

5. Hawkins, J. K., “Self-Organizing Systems—A Review and Commentary,” Proc. IRE. =49=:31-48 (January 1961)

6. Mesarovic, M. D., “On Self Organizational Systems,” Spartan Books, pp. 9-36, 1962

7. Braverman, D., “Learning Filters for Optimum Pattern Recognition,” PGIT =IT-8=:280-285 (July 1962)