# Formal: Theory

This document is part of the eXtensible Binary Universal Protocol project documentation. Provides description of the basic thesis of the protocol's formalization.

## Basic Thesis

XBUP as a universal protocol should allow to represent any data as much unambiguously as possible way. The following document describes efforts to formalize this claim.

### Data Representation

The aim of Protocol is to represent data in the form of bit sequences, where this sequence may be potentially infinitely long. Bit]] is the basic unit of information and for the formalization bits will be represent as the empty set for bit 0 and set containing only an empty set as a bit 1, which means value range is the set B = { ∅, {∅} }, also known directly as 2.

Range field is then sequence of bits N x B. There is then defined order by components. For the set of natural numbers N there is standard order < defined as ∅ < {∅}

This order is the total order, as defined:

- irreflexive - not exist any x: x < x
- asymmetric - for each x and y: x < y ∧ y < x ⇒ x = y
- transitive - for every x, y, and z: x < y ∧ y < z ⇒ x < z
- Every nonempty subset has a smallest element - for each set P: P ⊆ S exits x such as for each y ∈ P holds: x = y ∨ x < y

Instances of data representation is then function f: N → B ∈ N x B. Each instance can keep only countable amount of data. Instance is called finite if and only if: ∃ x: ∀ y: y > x ⇒ f(y) = 0.

### Universal Protocol Theory

The author of this protocol, believes that as in the case of Church thesis, it is possible for the above-defined conditions to define a hierarchy and that it is possible to define exactly one form of protocol, which is in some sense minimal.

There is a universal protocol that can represent each object of the universum with the enumerable complexity in the basic form, which holds:

- If an object can be split into multiple objects, the object is not in the basic form.
- Representation must provide way how to store countable quantity of final instances of itself and at least one infinite instance

### Finite Set

For the representation of the final sets, it is possible to use the fact, that the total ordered set of the same sizes are mutually isomorphic.

For the representation of these sets it is possible to use binary representation of the element index with as n-tuple of bits with length at least …

### Countable Set

For the representation of the countable sets it is not possible to use previous method, or that can be used just once respectively.