TY - JOUR

T1 - A Decomposition Method for Global Evaluation of Shannon Entropy and Local Estimations of Algorithmic Complexity

AU - Zenil, Hector

AU - Hernández-Orozco, Santiago

AU - Kiani, Narsis

AU - Soler-Toscano, Fernando

AU - Rueda-Toicen, Antonio

AU - Tegner, Jesper

N1 - KAUST Repository Item: Exported on 2021-02-23
Acknowledgements: This research was funded by Swedish Research Council (Vetenskapsrådet) grant number [2015-05299].

PY - 2018/8/15

Y1 - 2018/8/15

N2 - 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 on Solomonoff-Levin's theory of algorithmic probability providing a closer connection to algorithmic complexity than previous attempts based on statistical regularities such as 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. 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., π) 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 an online algorithmic complexity calculator.

AB - 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 on Solomonoff-Levin's theory of algorithmic probability providing a closer connection to algorithmic complexity than previous attempts based on statistical regularities such as 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. 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., π) 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 an online algorithmic complexity calculator.

UR - http://hdl.handle.net/10754/628775

UR - https://www.mdpi.com/1099-4300/20/8/605

UR - http://www.scopus.com/inward/record.url?scp=85052194796&partnerID=8YFLogxK

U2 - 10.3390/e20080605

DO - 10.3390/e20080605

M3 - Article

AN - SCOPUS:85052194796

VL - 20

SP - 605

JO - Entropy

JF - Entropy

SN - 1099-4300

IS - 8

ER -