Logical depth for reversible Turing machines with an application to the rate of decrease in logical depth for general Turing machines

TL;DR

This paper explores the concept of logical depth in reversible Turing machines, equating it with the shortest runtime of minimal programs, and applies this to validate previous results on the rate of decrease in logical depth despite earlier errors.

Contribution

It establishes that the logical depth of reversible Turing machines equals the shortest runtime of their minimal programs and applies this to confirm prior findings on logical depth reduction.

Findings

Logical depth of reversible Turing machines equals shortest program runtime.

Validation of previous results on the rate of decrease in logical depth.

Application of reversible computation concepts to logical depth analysis.

Abstract

The logical depth of a {\em reversible} Turing machine equals the shortest running time of a shortest program for it. This is applied to show that the result in L.F. Antunes, A. Souto, and P.M.B. Vit\'anyi, On the Rate of Decrease in Logical Depth, Theor. Comput. Sci., 702(2017), 60--64 is valid notwithstanding the error noted in Corrigendum P.M.B. Vit\'anyi, Corrigendum to "On the rate of decrease in logical depth" by L.F. Antunes, A. Souto, and P.M.B. Vit\'anyi [Theoret. Comput. Sci. 702 (2017) 60--64], {\em Theoret. Comput. Sci.}, https://doi.org/10.1016/j.tcs.2018.07.009 . /