p-Adic physics as physics of cognition and imagination and counterparts of recursive functions

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+ kpⁿ) 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_npⁿ→ ∑_n x_np^-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+p²+..) are both mapped to real unit and one can have f_p(1)≠ f_p((p-1)(p+p²+..)). 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 (x₀, x₁,...x_n, 0, 0,...) and (x₀,x₁,...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.

Recursion plays a key role in the theory of computation and it would be nice if it would generalize in a non -trivial manner to the realm of p-adic integers (or general p-adic numbers).

1. From Wikipedia one finds a nice article about primitive recursive functions. Primitive recursive functions are very simple. Constant function, successor function, projection function. From these more complex recursive functions are obtained by composition and primitive recursion. These functions are trivially recursive also in p-adic context and satisfy the conditions of p-adic continuity and smoothness. Composition respects these properties tool. I would guess that same holds also for primitive recursion.
   

   It would seem that there is nothing new to be expected in the realm of natural numbers if one identifies p-adic integers as real integers as such. Situation changes if one uses canonical identification mapping p-adic integers to real numbers (for instance, 1+2+2²→> 1+1/2+1/4= 7/4 for 2-adic numbers). One could think of doing computations using p-adic integers and mapping the results to real numbers so that one could do computations with real numbers using p-adic integers and perhaps p-adic differential calculus so that computation using analytic computations would become possible instead of pure numerics. This could be very powerful tool.
   
   

2. One can consider also real valued recursive functions and functions having values in (not only) algebraic extensions of rationals. Exponent function is an interesting primitive recursive function in real context: in p-adic context exp(x) exists p-adically if x has p-adic norm smaller than 1). exp(x+1) does not exist as p-adic number unless one introduces extension of p-adic numbers containing e: this is necessary in physically interesting p-adic group theory. exp(x+kp) however exists as p-adic number. The composition of exp restricted to p-adic numbers with norm smaller than 1 with successor function does not exist. Extension of rationals containing e is needed if one wants successor axiom and exponential function.
   
   
3. The fact that most p-adic integers are infinite as real numbers might pose problems since one cannot perform infinite sums numerically. p-Adic continuity would of course allow approximations using finite number of pinary digits. The real counterparts of functions involved using canonical identification would be p-adic fractals: this is something highly non-trivial physically.
   

   One could also code the calculations at higher level of abstraction by performing operations for functions rather than numbers. The finite arithmetics would be for the labels of functions using tables expression the rules for various operations for functions (such as multiplication). Build a function bases and form tables for various operations between them like multiplication table of algebra, computerize the operations using these tables and perform pinary cutoff at end. The rounding error would emerge only at this last step.
   
   

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.

Articles and other material related to TGD.