Mathematics of Knowledge Refinement: Probabilistic Arithmetic, with no unknowns and no infinity. Part I. Generalized Probabilistic Arithmetic. Basic definitions and properties

TL;DR

This paper introduces a novel framework called Generalized Probabilistic Arithmetic that refines correlated random variables' values and accuracies without unknowns or infinities, with promising applications in computational and quantum physics.

Contribution

It defines and explores properties of Probabilistic Generalized Addition and Multiplication, including inverse operations and continuity at zero, advancing mathematical tools for probabilistic computation.

Findings

Operations have inverse counterparts, enabling subtraction and division.

Division by near-zero values is possible without resulting in infinity.

The approach's hyperbola is continuous at zero, unlike traditional hyperbolas.

Abstract

An approach to build Probabilistic Arithmetic in which initial values of all correlated random variables are known, but with varying degrees of accuracy. As a result of the proposed Probabilistic Arithmetic operations, variable values, degrees of their accuracy and correlations are refined. Probabilistic Generalized Addition (PGA) and Probabilistic Generalized Multiplication (PGM) operations on correlated random variables are defined and their basic properties identified and described: \bullet Proposed PGA and PGM operations possess inverse operations - subtraction and division. \bullet There is no difference between direct and inverse operations: addition and subtraction, multiplication and division (this is why these operations are called "Generalized"). \bullet Division by approximately zero is possible and the result never equals to \infty, making this approach promising in computational and computer mathematics. \bullet Unlike the usual hyperbola, which, when the argument is changing from + \infty to 0, has a gap at 0, the Generalized Probabilistic Hyperbola, under certain combination of initial accuracies, is continuous at 0. First, as the argument changes from +\infty to 0, it behaves like a typical hyperbola, monotonically increasing. However, after reaching certain maximum value, it starts to decrease to 0 monotonically. Last property gives a hope that the proposed approach might be promising in Quantum Physics, as it would allow a more adequate description of macro and micro physics, as well as of the transition from one to the other.