Arithmetic logical Irreversibility and the Turing's Halt Problem

TL;DR

This paper explores how arithmetic logical irreversibility and information loss contribute to the halting problem, linking computational uncertainty with entropy and the limits of algorithmic predictability.

Contribution

It introduces the concept of arithmetic logical entropy to quantify information loss and explains how it underpins the fundamental unpredictability in Turing machine halting behavior.

Findings

Information loss leads to computational uncertainty.

Arithmetic logical entropy quantifies irreversibility.

Limits of predictability in recursive functions.

Abstract

The Turing machine halting problem can be explained by several factors, including arithmetic logic irreversibility and memory erasure, which contribute to computational uncertainty due to information loss during computation. Essentially, this means that an algorithm can only preserve information about an input, rather than generate new information. This uncertainty arises from characteristics such as arithmetic logical irreversibility, Landauer's principle, and memory erasure, which ultimately lead to a loss of information and an increase in entropy. To measure this uncertainty and loss of information, the concept of arithmetic logical entropy can be used. The Turing machine and its equivalent, general recursive functions can be understood through the {\lambda} calculus and the Turing/Church thesis. However, there are certain recursive functions that cannot be fully understood or predicted by other algorithms due to the loss of information during logical-arithmetic operations. In other words, the behaviour of these functions cannot be completely determined at every stage of the computation due to a lack of information in their definition. While there are some cases where the behaviour of recursive functions is highly predictable, the lack of information in most cases makes it impossible for algorithms to determine if a program will halt or not. This inability to predict the outcome of the computation is the essence of the halting problem of the Turing machine. Even in cases where more information is available about the program, it is still difficult to determine with certainty if the program will halt or not. This also highlights the importance of the Turing oracle machine, which introduces information from outside the computation to compensate for the lack of information and ultimately decide the result of the computation.