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The Genuine Symmetry on Natural Numbers:

Truncated Universal Distribution is Enough

Artem S. Shafraniuk

Master of Scientific and Technical Computing; Loyola University Chicago, Illinois, United States of America

Bachelor of Computer Science, Master of Information Control Systems and Technologies; European University, Kiev, Ukraine

(Dated: September 5, 2015)

author's e-mail: artem.shafraniuk@gmail.com

The author analyzed the structure of universal computation, and came to the result that the structure

of its limit on infinity consists of constraints on its results. Then he has found out that the non-halting

programs can be taken out of consideration, by using point-free topology & quantale representation.

At last, for effective purposes, the universal distribution can also be pruned. The main result is that

there is a symmetry on natural numbers, which is their infinite random permutation. This one is

unique: any other sequence can be obtained from the author's one by swapping any two elements of

the sequence one or more times. But this always, and only, decreases the randomness of the sequence

in question. So this sequence is also the maximally random sequence of natural numbers.

hereby the author claims an exclusive discovery of an idea made for the first time and copyright to its formulation

I.

INTRODUCTION TO FORMAL

COMPUTATION

First, let's take a look at formal computation

models from Computer Science. It is known that

some are universal, and the rest are not. A

computation model is universal if it can execute

any algorithm possible. (Examples are Turing

Machine, Random Access Machine, Universal

Recursive Function, Universal mu-Function,

Queue Machine, etc.) Also, it, a machine, has an

input, and an output, and what it does is, it

transforms the input (an algorithm or a program)

into the output, if possible. The reason for the

exceptions is, not all algorithms halt, and this is

the only case in which it is impossible for the

computation model, also called a machine, to

halt, and give an output, in finite time.

Evidently, if the machine doesn't halt in finite

time, it doesn't halt at all, so it cannot provide

any output. It is also known that any

computation universal machine can simulate any

other computation universal machine, using a

program (adaptor) that simulates that other one.

This is clear from the fact that any computation

universal machine can run any algorithm, and

also that any computation universal machine can

be represented by an algorithm. G. Chaitin has

discovered a very curious real number, which he

called Omega. In binary expansion it starts with

zero, and then at each position i (index),

progressing towards the right of the number's

expansion, the bit x_i is '1' if the program

number i halts, and '0' otherwise (if the program

number i doesn't halt.) Now, these programs

represent algorithms, so this is a matter of

coding. Also, since a program is a finite discrete

object, it can be directly encoded by a natural

number. Then, it's clear to see that all Omega's

(for all possible different computation universal

machines) are bit-wise isomorphic, as are

algorithms indexed by the i's] in a Omega's. So,

any computation universal machine takes an

input, runs, and then either halts at a finite step,

and provides an output; or doesn't halt at all. It

runs in steps (sometimes called tacts). It is easy

to see that, for any classical computation

universal machine, any of its states, including

the start state, which is the input itself, can be

represented by a natural number, and the

machine itself then can be presented as an

iterated function o=f(i), where i is input, o is

output. This function then iterates on itself,

starting from the input, until it reaches a limit

cycle, which is a state s=f(s). This s then is taken

to be the output. Such a machine, then, can be

otherwise interpreted being "tabulated" into an

infinite sequence of natural numbers f(0), f(1),

...f(i), ... . Executing a program then means

picking start index s, and looking up what's at

that index, then changing state, s:=f(s). The

author of this paper has published such a

sequence before, based on a pairwise encoding

of the calculus of combinators S & K. Any

combinator possible is then encoded as a natural

number. This sequence was published online, in

the Encyclopedia of Integer Sequences.

Actually, any computation universal machine,

represented by a function, is a homomorphism,

and, represented as a sequence, is a

pseudorandom one. In the limit it (computation /

iteration) converges to a random sequence,

except for the infinities at the positions

corresponding to the non-halting input

programs. Now, if we knew Omega, we could

construct an oracle for the halting problem, and

compute the limit, which latter, again, will come

out to be random.

II.

COMPLEXITY OF UNIVERSAL

COMPUTATION

Second, there was a famous mathematician A.

Kolmogorov, who discovered two brilliant

things that we need to know. The first is the

Axiomatic Probability Theory, and the second is

called Kolmogorov Complexity. From the first

we only need to look at the sample space, which

is the set of all sets of n possible elementary

items, so the number of sets is then 2^n. The

second, Kolmogorov Complexity is written

down KC(o) = min{l(p)} : o=M(p), where KC Kolmogorov Complexity, M - computation

universal machine, p - any program, min{l(p)}:

o=M(p) - the length of the minimal program,

such that if M takes it as input and runs, it's

output is o. Next, let's take a Universal Turing

Machine (UTM) as our model for computation.

One should note, however, that it, a UTM, is a

program on the standard Turing Machine (TM),

being an adaptor, which is also computation

universal. Each of its states, either from input to

the output, or from input to infinity (if it doesn't

halt on that input), is a finite string of bits. Let's

call it a bitstring. Now, observe that if we add a

'0.' on the left to a finite bitstring, we always get

a rational number. Also, all the possible inputs

to our UTM are, then, all the rational numbers,

and so its, UTM's, one-step iteration is a

homomorphism on the rational numbers. Let's

call the UTM with bitstring states the Bitstring

Universal Turing Machine, or BUTM.

III.

QUANTAL TOPOLOGY &

COMPUTATION

Third, let's introduce a concept, from Point

Free Topology, called a quantale, discovered by

Mulvey. A quantale is, by definition, an upperjoin semi-lattice, and its original primary role is

quantization of space, or something spatial.

Next, the sample space from Kolmogorov's

Probability Theory can, in general, be

represented by a quantale, as there is an

isomorphism from the first to the second. The

sample space contains items (usually events,

constructed from elementary events, as their

power set. So the sample space then contains all

the possible combinations of the elementary

events.) Let's call the spatial object that results

from the quantization of space by such a

quantale, a quantal figure or spatial figure, such

that they coincide. It should be constructed,

using point-free topology, from the sample

space, represented by a labeled quantale,

isomorphic to the sample space; with labels

being the elementary items / "events". Now,

bitstring-programs -to- bitstring-results, using

BUTM, *is* a homomorphism, except for the

non-halting programs. But the latter, nonhalting, cannot be excluded by all the means we

all have believed since Turing, Goedel, Berry,

and Chaitin. But the next idea doesn't agree!

IV.

QUANTAL STRUCTURE OF

UNIVERSAL COMPUTATION

Fourth, using a sample space from probability

with input programs as bitstrings, ordered as

binary numbers, as elementary items of this

sample space, we get the source quantal figure;

it's a piece of space, resulting from the

quantization of space by the sample-spacequantale. What does it look like? The list of all

the binary numbers (bitstrings) can be

represented by a binary tree, with all branches

finite, but a tree with infinite breadth. So, the

definition of the quantal figure we're searching

for is this: fractal spatial figure. Next, let's try to

imagine what the limit of the computation looks

like on infinity. Well, there's clearly one option:

random quantal figure <==> random spatial

figure. Let's not yet bother with how it looks

like, but believe me: it's unique. It must be

unique because all c.u. (computation universal)

machines are equivalent. We also get a mapping

from the fractal spatial figure to the random

spatial figure.

V. UNIVERSAL & INTRICATE SYMMETRY

ON THE NATURAL NUMBERS

Fifth, let's remove labels (bitstrings) from the

random spatial figure. It's something new, a

random spatial object, this figure. To understand

it, we need to find the genuine symmetry on it.

The symmetry is again a quantale, precisely

because such a spatial figure results from the

quantization of space by a quantale. Then, what

we get is a random quantale. But such a

quantale, too, can be represented as a sequence

of natural numbers. It's easy to see that, first, this

is the maximally random sequence of natural

numbers, and, second, this is the infinite random

permutation of them, the natural numbers. If we

swap two elements once or more, it will become

less random. This means, it is the maximally

random of all the possible infinite sequences of

natural numbers, which simply means that it's

unique. There is an analogy of this sequence

uniqueness proof to the proof that the infinite

random directed graph is unique. This must be

for a reason. The sequence in question encodes

the infinite random isomorphism, unique again.

The transitive closure of this isomorphism is a

directed graph. This graph is, evidently, random,

except that there's precisely one directed edge

going out of a node. The nodes are labeled by

the natural numbers, and there's a unique

number at each node, and a unique node labeled

by a natural number.

V.

TRUNCATED UNIVESAL

DISTRIBUTION IS SUFFICIENT

Last, consider the so-called Universal

Distribution. Let's take the random spatial figure

(there's just one such unique object, as shown

above in the paper.) It's easy to see that it puts a

discrete geometrical framework on the universal

distribution, and this framework consists of

"boxes" (parallepipeds). These latter are

constraints, and what the framework represents

is a truncated universal distribution, where all,

and only, halting computation destinations (on

termination)/ are present.

PS.

Please spread the word,

feel free to send, and forward,

this article to the people you

know, & don't know, if allowed!

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