Next Article in Journal
A Simple Chaotic Map-Based Image Encryption System Using Both Plaintext Related Permutation and Diffusion
Next Article in Special Issue
Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box
Previous Article in Journal
From Identity to Uniqueness: The Emergence of Increasingly Higher Levels of Hierarchy in the Process of the Matter Evolution
Previous Article in Special Issue
Entropy of Iterated Function Systems and Their Relations with Black Holes and Bohr-Like Black Holes Entropies
Article Menu
Issue 7 (July) cover image

Export Article

Open AccessArticle
Entropy 2018, 20(7), 534; https://doi.org/10.3390/e20070534

Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations

1,2,3,4,* , 1,2,3,4
and
2,3,5,6
1
Algorithmic Dynamics Lab, Centre for Molecular Medicine, Karolinska Institute, Stockholm 171 77, Sweden
2
Unit of Computational Medicine, Department of Medicine, Karolinska Institute, Stockholm 171 77, Sweden
3
Science for Life Laboratory (SciLifeLab), Stockholm 171 77, Sweden
4
Algorithmic Nature Group, Laboratoire de Recherche Scientifique (LABORES) for the Natural and Digital Sciences, Paris 75005, France
5
Biological and Environmental Sciences and Engineering Division (BESE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia
6
Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia
An online implementation to estimations of graph complexity is available online at http://www.complexitycalculator.com (accessed on 17 July 2018).
*
Author to whom correspondence should be addressed.
Received: 25 June 2018 / Revised: 12 July 2018 / Accepted: 14 July 2018 / Published: 18 July 2018
Full-Text   |   PDF [871 KB, uploaded 18 July 2018]   |  

Abstract

We introduce a definition of algorithmic symmetry in the context of geometric and spatial complexity able to capture mathematical aspects of different objects using as a case study polyominoes and polyhedral graphs. We review, study and apply a method for approximating the algorithmic complexity (also known as Kolmogorov–Chaitin complexity) of graphs and networks based on the concept of Algorithmic Probability (AP). AP is a concept (and method) capable of recursively enumerate all properties of computable (causal) nature beyond statistical regularities. We explore the connections of algorithmic complexity—both theoretical and numerical—with geometric properties mainly symmetry and topology from an (algorithmic) information-theoretic perspective. We show that approximations to algorithmic complexity by lossless compression and an Algorithmic Probability-based method can characterize spatial, geometric, symmetric and topological properties of mathematical objects and graphs. View Full-Text
Keywords: Kolmogorov–Chaitin complexity; algorithmic probability; algorithmic coding theorem; Turing machines; polyominoes; polyhedral networks; molecular complexity; polytopes; information content; Shannon entropy; symmetry breaking; recursive transformation Kolmogorov–Chaitin complexity; algorithmic probability; algorithmic coding theorem; Turing machines; polyominoes; polyhedral networks; molecular complexity; polytopes; information content; Shannon entropy; symmetry breaking; recursive transformation
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Zenil, H.; Kiani, N.A.; Tegnér, J. Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations. Entropy 2018, 20, 534.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top