Properties of optimal prefix-free machines as instantaneous codes

TL;DR

This paper explores the properties of optimal prefix-free machines, viewed as instantaneous codes, focusing on the structure and distribution of codewords associated with each symbol using algorithmic information theory.

Contribution

It provides a detailed analysis of the set of codewords for each symbol in optimal prefix-free machines, revealing their structural properties and distribution patterns.

Findings

Analyzes the number of codewords per symbol

Examines the distribution of codewords for each symbol

Uses algorithmic information theory tools for analysis

Abstract

The optimal prefix-free machine U is a universal decoding algorithm used to define the notion of program-size complexity H(s) for a finite binary string s. Since the set of all halting inputs for U is chosen to form a prefix-free set, the optimal prefix-free machine U can be regarded as an instantaneous code for noiseless source coding scheme. In this paper, we investigate the properties of optimal prefix-free machines as instantaneous codes. In particular, we investigate the properties of the set U^{-1}(s) of codewords associated with a symbol s. Namely, we investigate the number of codewords in U^{-1}(s) and the distribution of codewords in U^{-1}(s) for each symbol s, using the toolkit of algorithmic information theory.