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Such a constraint is not possible in the process construction. Instead we calculate the probability that the construction remains at the first three levels. For this purpose we assume a generalised ergodic hypothesis — that when there are several possibilities in the process, each will occur with equal probability. At the first level, a dcs may be one of the three at that level or none of them (because it is at a higher level) so each has probability 1/4. Similarly at the next levels there are probabilities 1/8 and 1/128, so long as Parker-Rhodes’choice of a normal set of matrices is made.

This technique appeared in most of Pask’s actual learning and teaching machines. Input to the 16level was randomised to compensate for having to cut off the construction somewhere. Dimensionality 29 It was intended to use the machine by imposing a constraint on its working corresponding to the physical requirement that one of the 4 units differed from the others — distinguishing space from time. One would then see if this imposition produced numerical effects that might be interpreted as the coupling constants, and perhaps exhibit these in a larger class of numbers that were more like continuous variables.

Define yzz = p, zyz = q, zzy = r. Then qr = zyy, rp = yzy, pq = yyz, and pqr = yyy, which we write Y . The automorphisms p, q, r and their products form the centre of the algebra, together with their squares, the identity Z. This centre is the group C2 × C2 × C2 . The generators of L3 are 132∗ , 321∗ , and 123. To see what is happening it is useful to construct the table for L3 : 132∗ Z 213 z11 321∗ 132∗ 321∗ 123 2∗ 1∗ 3∗ 321∗ 2∗ 1∗ 3∗ Z 2∗ z2∗ 1∗ 3∗ 2 123 ∗ ∗ ∗ z 1 1 2z2 Z 33∗ y 2∗ 1∗ 3 3∗ 2∗ 1 132 3∗ 3y Z Taking a different basis as in the corresponding table in the skeleton, 1, 2, 4 for 132∗ , 321∗ , 123 and using the signals, this can be written: 1 2 3 4 1 Z Y3 2 qr5 2 3 Z Y 6 3 Y2 r1 Z pq7 4 5 rp6 7 Z Evidently instead of elements coming in dual pairs they come in octets.