Our understanding of the “foundations of math” taught to Turing by Newman have advanced to the point where we understand how one builds up from sets to metric spaces, to normed spaces, to complete normed spaces, to such spaces that are potentially separable, etc. All in all, we can think in the concepts that Turing thought. In particular we see how one builds further structure and one gets to measures spaces – upon which foundation one gets to Lesbesgue integration.

IN the world of Colossus, and the General Report on Tunny in particular, we recall our struggle mapping their terminology onto modern theory (or even 1930s presentations of the same theorems). Applying the above foundations, however, we seem to be able now to make such a mapping.

IN measure theory, for Tunny-era cryptanalysis, one recalls how a tape of 5 bit symbols provides events – elements in the set to be measured. There are 32 such events, and there are multiple experiments to be performed – one for each (of the 32) character to be read from the tape. In any one experiment, one may now consider a custom algebra – whose terms use such characters as atoms in the expression. We know that folks designed a custom algebra (of “proportional bulges”) to suite their problem domain. This algebra is much like the typical boolean algebra whose expressions work over subsets of the (32 element) event space, fashioning a Banach Space. Being a measure space, furthermore, any such expression, in the syntax of the algebra and subject to the proof theorems of that algebra too, can be measured – in order that the distance between any two can be asserted. In probability spaces, and in Colossus work in particular, the functional-measure applied to any such expression is the PB(exp) of the expression (exp) – where PB is proportional bulge. one set of expressions are the 32 constant expressions – of the 32 values enumerated in the 5 bit field. Thus one can compute PB(‘J’) say, where J is just some bit pattern. If this is hard, then make ‘J’ into a more obvious boolean expression PB(Ej) – where Ej is the expression (‘J’ xor ‘J’). The expression would always have value 0, and that would be assigned a PB value. Obviously, in this form, 32 constant functions all have the same PB!

Now, we note how Banach space theory of the day also extended to applying the “spectral” class of measures, in addition to the more conventional distance measures, as above. And, we saw this idea presented in the General Report on Tunny, too. Folks indicated that they thought, at least theoretically, in terms of PB*(exp), where the * means the fourier transform of PB. In other words, use a spectral measure.

As the Tunny documents indicate, in the world of binary boolean algebras is quite easy to calculate spectral measures, in the fourier domain, since its largely a matter of counting the transitions between 1 and 0, in a bit sequence. Or, as they say, the “sign changes”, assuming that the 2 binary states are known as +1 and –1.

Now the “design concept” of Colossus makes a lot of sense, given Newman’s background. He knows that a general machine needs to be able to work over all boolean functions that can be expressed in the algebra (and so he builds such a “expressive” capability into the machine, represented by lots of switching knobs) and furthermore knows his expression algebra has to work with 2 parameters, allowing computation over 2 characters in sequence, delta’ed.

IN the general report, we then see various theorems expressed about the algebra, which show that its terms are NOT 5 bit characters but terms from a more abstract symbol set – the various Phi and Chi and DeChi terms of an algebra that models the particular measures found in the statistical bulges found, over the expectation number of the counts of sign changes in the values read from overlapping tapes.

Once folk I the Colossus 2 re-design get around to thinking about wheel breaking, the machine is extended so that counts can be performed assuming a particular basis centered around letting each wheel bit position, set to 1 when all other bits are 0, be a hamming-weight basis vector. Thus the Colossus can do early signal processing – computing convolutions necessary for wheel breaking.

If we continue this thinking and apply it to Turings “On Permutation’s” manuscript, Turing’s sigma algebra (whose expressions are subejct to probability measures) consists of expressions involving R and U (step the rotor, and swap wired points per the wheel wiring) and S – states or spaces. Expressions are cayley graphs, with R and U being generators and with the logic being that of a group.

To calculate with the above, the expressions can be “compiled” by a homomorphism rule into a group involving cosets group elements – the factors of H/H1, where H is the set from which S were take above. H1 is a constrained and normal subgroup of H, defined in terms of constrained initial values of the Rs and U – whose unitary transforms retain the invariants in the computed-group imposed by a particular (hamming cube centric) 7 point geometry).

Since we are now computing a group whose elements are cosets, with a coset-centric operator, we get to automorphisms – which are the functional/lambda output of the unitary expression in the original group (J). Since automorphisms are just terms in another (Paut) group, we see that one calculated-lambda can be applied to the lamba from s second coset-expression to produce a third lambda (yet another relabelling automorphism), allowing a recursive functional-expressions – since any two automorphisms can further relabel what was just relabeled…

In order that the computation engine, which is a very abstract compilation system, calculates lambda expressions, Turing not only works over the particular 7-point geometry which gives him a complete normed L2 spaced but he works with the particular form of the adjacency matric that allows his value space to be a differentiable manifold, giving him custom limiting functions for condensation points that produces the desired limiting distributions.