A Decomposition Method for Global Evaluation of Shannon Entropy and Local Estimations of Algorithmic Complexity

TL;DR

This paper introduces the Block Decomposition Method (BDM), a novel approach for estimating algorithmic complexity that extends the Coding Theorem Method, providing more accurate complexity estimates for various objects including strings, arrays, and graphs.

Contribution

The paper presents BDM, a new decomposition-based method that improves local estimations of algorithmic complexity and connects it more closely to algorithmic probability theory.

Findings

BDM provides efficient complexity estimates for diverse objects.

It performs comparably to Shannon entropy when accuracy diminishes.

The method is adaptable to multi-dimensional objects like arrays and tensors.

Abstract

We investigate the properties of a Block Decomposition Method (BDM), which extends the power of a Coding Theorem Method (CTM) that approximates local estimations of algorithmic complexity based upon Solomonoff-Levin's theory of algorithmic probability providing a closer connection to algorithmic complexity than previous attempts based on statistical regularities e.g. as spotted by some popular lossless compression schemes. The strategy behind BDM is to find small computer programs that produce the components of a larger, decomposed object. The set of short computer programs can then be artfully arranged in sequence so as to produce the original object and to estimate an upper bound on the length of the shortest computer program that produces said original object. We show that the method provides efficient estimations of algorithmic complexity but that it performs like Shannon entropy when it loses accuracy. We estimate errors and study the behaviour of BDM for different boundary conditions, all of which are compared and assessed in detail. The measure may be adapted for use with more multi-dimensional objects than strings, objects such as arrays and tensors. To test the measure we demonstrate the power of CTM on low algorithmic-randomness objects that are assigned maximal entropy (e.g. $\pi$) but whose numerical approximations are closer to the theoretical low algorithmic-randomness expectation. We also test the measure on larger objects including dual, isomorphic and cospectral graphs for which we know that algorithmic randomness is low. We also release implementations of the methods in most major programming languages---Wolfram Language (Mathematica), Matlab, R, Perl, Python, Pascal, C++, and Haskell---and a free online algorithmic complexity calculator.