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Tuesday, October 4, 2016 1:48

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In TGD Universe p-adic physics is physics of cognition and imagination and real physics also carries signatures about the presence of p-adic physics as p-adic fractality: this would explain the unexpected success of p-adic mass calculations. The outcome would be a fusion of real and various p-adic number fields to form adeles. Each extension of rationals giving rise to a finite-dimensional extension of p-adic numbers defines an adele, and there is hierarchy of adeles defining an evolutionary hierarchy. The better the simulation p-adic space-time sheet is for real space-time sheet, the larger the number of common algebraic points is. This intuitive idea leads to the notion of monadic geometry in which the discretization of the imbedding space causal diamond is central for the definition of monadic space-time surfaces. They are smooth both in real and p-adic sense but involve discretization by algebraic points common to real and p-adic space-time surfaces for some algebraic extension of rationals inducing corresponding extension of p-adics.

How this could relate to computation? In the classical theory of computation recursive functions play a key role. Recursive functions are defined for integers. Can one define them for p-adic integers? At the first glance the only generalization of reals seems to be the allowance of p-adic integers containing infinite number of powers of p so that they are infinite as real integers. All functions defined for real integers having finite number of pinary digits make sense p-adically.

What is something compeletely new that p-adic integers form a continuum in a well-defined sense and one can speak of differential calculus. This would make possible to pose additional conditions coming from the p-adic continuity and smoothness of recursive functions for given prime p. This would pose strong constraints also in the real sector for integers large in the real sense since the values f(x) and f(x+ kp^{n}) would be near to each other p-adically by p-adic continuity and p-adic smoothness would pose additional strong conditions.

How could one map p-adic recursive function to its real counterpart? Does one just identify p-adic arguments and values as real integers or should one perform something more complex? The problem is that this correspondence is not continuous. Canonical identification for which the simplest form is I: x_{p}=∑_{n} x_{n}p^{n}→ ∑_{n} x_{n}p^{-n}=x_{R} would however relate p-adic to real arguments continuously. Canonical identification has several variants typically mapping small enough real integers to p-adic integers as such and large enough integers in the same manner as I. In the following let us restrict the consideration to I.

Basically, one would have p-adic valued recursive function f_{p}(x_{p}) with a p-adic valued argument x_{p}. One can assign to f_{p} a real valued function of real argument – call it f_{R} – by mapping the p-adic argument x_{p} to its real counterpart x_{R} and its value y_{p}=f_{p}(x) to its real counterpart y_{R}: f_{R}(x_{R}) = I(f(x_{p})=y_{R}. I have called the functions in this manner p-adic fractals: fractality reflects directly to p-adic continuity.

f_{R} could be 2-valued. The reason is that p-adic numbers x_{p}=1 and x_{p} =(p-1)(p+p^{2}+..) are both mapped to real unit and one can have f_{p}(1)≠ f_{p}((p-1)(p+p^{2}+..)). This is a direct analog for 1=.999… for decimal expansion. This generalizes to all p-adic integers finite as real integers: p-adic arguments (x_{0}, x_{1},…x_{n}, 0, 0,…) and (x_{0},x_{1},…x_{n}-1,(p-1),(p-1),…) are mapped to the same real argument x_{R}. Using finite pinary cutoff for x_{p} this ceases to be a problem.

The unexpected success of deep learning is conjectured to reflect the simplicity of the physical world: only a small subset of recursive functions is needed in computer simulation. The real reason could be p-adic physics posing for each value of p very strong additional constraints on recursive functions coming from p-adic continuity and differentiability. p-Adic differential calculus would be possible for the p-adic completions of integers and could profoundly simplify the classical theory of computation.

For background see the article TGD Inspired Comments about Integrated Information Theory of Consciousness.

For a summary of earlier postings see Latest progress in TGD.