The Walrus & the Carpenter

“The time has come,” the Walrus said, “to talk of many things…”

It’s a bumper year for mathematical related anniversaries.  Alexandre DeCastro has an upcoming post relating to a very significant anniversary we passed this May. As it’s also coming up on the 150th. year since publication of Alice’s Adventures in Wonderland and, as Dodgson was so much enamored of inverting logic, I will use him to segue my way into this review…

…I am not the walrus.

In review; the initial presentation on this blog was of a one-way trapdoor function (& PRG), the cryptographical ‘hardness’ of which is yet to be formally determined. (Tentatively termed ‘sticky predicates’, not hard nor soft, yielding yet impenetrable?)
From my remarks upon the similarities between trapdoor functions and the information preserving properties of ‘adiabatic’ reservoirs as outlined in his paper. ‘One-way-ness in the input-saving (Turing) machine‘, Alexandre DeCastro was motivated to explore this line of inquiry from a mathematical viewpoint. His distillation of the finite-field, one-way paradox was presented in; ‘A plain proof that Levin’s combinatorial function is one-way(draft)‘.
Consequently over the past year or so, we’ve presented herein, a total of three instantiations of the principle; that finite quanta or number fields yield  information paradoxes via the agency of involutionary mathematical operations. To wit; the Bennett’s paper by Alex appearing in Physica A and the mathematical exploration of Levin’s function over GF₂, along with the trapdoor-permutation by the author. You may ask, in what way is my trapdoor permutation a finite (or finitary?) construction? This goes back to the direct analogy with Alexandre’s reversible Bennett’s engine. This real numbered compression hashing represents a ‘bit-bucket’, a forward-feed reservoir of entropy arising from quantized precision overflow. As such, it iterates a temporalization of finite mathematical precision. It was Alex DeCastro’s genius to see that this quasi-involution could be distilled down to the smallest Gallois field while still retaining an anomalous dual identity.
Whilst it is long acknowledged that ZFC set theory admits paradoxes through unspecificity of power set hierarchies, the work presented here provides the first ever numerical-logic explanation for such behaviors.
Those self-same anomalies which lie at heart of the ultimate barrier to our complete understanding of the physical world, as Alex will elucidate upon in the next post.

Goo, goo, g’joob!