#
Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations^{ †}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Polyhedra, Polytopes and Polyhedral Networks

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 1.2. Basics of Graph Theory

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

#### 1.3. Polyominoes

## 2. Methodology

#### 2.1. Computability/Recursivity

**Definition**

**10.**

**Definition**

**11.**

#### 2.2. Group-Theoretic Symmetry

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

#### 2.3. Information Theory

**Definition**

**16.**

#### 2.4. Graph Entropy

**Definition**

**17.**

**Definition**

**18.**

#### 2.5. Algorithmic Complexity

**Definition**

**19.**

#### 2.6. Algorithmic Probability

**Definition**

**20.**

**Definition**

**21.**

#### The Coding Theorem and Block Decomposition Methods

#### 2.7. The Algorithmic Complexity of a Graph

**Definition**

**22.**

**Definition**

**23.**

## 3. Results

#### 3.1. Algorithmic Characterization of Geometric Symmetry

**Definition**

**24.**

**Definition**

**25.**

#### 3.2. Correspondence of Algorithmic Ranking

#### 3.3. Classification of Polyominoes

#### 3.4. Polytope Profiling

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Another illustration of the distribution of polyominoes according to different order parameters. Entropy is sometimes too sensitive or insensitive, lossless compression (Compress) is always too insensitive and the AP-based measure BDM appears to be robust, assigning similar complexity and respecting relative order as shown in Figure 3.

## References

- Lin, S.-K. Correlation of Entropy with Similarity and Symmetry. J. Chem. Inf. Comput. Sci.
**1996**, 36, 367–376. [Google Scholar] [CrossRef] - Zenil, H.; Soler-Toscano, F.; Kiani, N.A.; Hernández-Orozco, S.; Rueda-Toicen, A. A Decomposition Method for Global Evaluation of Shannon Entropy and Local Estimations of Algorithmic Complexity. arXiv, 2016; arXiv:1609.00110. [Google Scholar]
- Kiani, N.A.; Shang, M.; Zenil, H.; Tegnér, J. Predictive Systems Toxicology. In Computational Toxicology—Methods and Protocols, Methods in Molecular Biology; Nicolotti, O., Ed.; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Zenil, H.; Kiani, N.A.; Shang, M.-M.; Tegnér, J. Algorithmic Complexity and Reprogrammability of Chemical Structure Networks. Parallel Process. Lett.
**2018**, 28, 1850005. [Google Scholar] [CrossRef] - Zenil, H.; Kiani, N.A.; Tegnér, J. Low Algorithmic Complexity Entropy-deceiving Graphs. Phys. Rev. E
**2017**, 96, 012308. [Google Scholar] [CrossRef] [PubMed] - Levin, L. Laws of information conservation (non-growth) and aspects of the foundation of probability theory. Probl. Form. Trans.
**1974**, 10, 206–210. [Google Scholar] - Solomonoff, R.J. A formal theory of inductive inference. Parts 1. Inf. Control
**1964**, 7, 1–22. [Google Scholar] [CrossRef] - Solomonoff, R.J. A formal theory of inductive inference. Parts 2. Inf. Control
**1964**, 7, 224–254. [Google Scholar] [CrossRef] - Calude, C.S. Information and Randomness: An Algorithmic Perspective, EATCS Series, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; Wiley-Blackwell: Oxford, UK, 2009. [Google Scholar]
- Soler-Toscano, F.; Zenil, H. A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences. Complexity
**2017**, 2017, 7208216. [Google Scholar] [CrossRef] - Delahaye, J.-P.; Zenil, H. Numerical Evaluation of the Complexity of Short Strings: A Glance Into the Innermost Structure of Algorithmic Randomness. Appl. Math. Comput.
**2012**, 219, 63–77. [Google Scholar] [CrossRef] - Zenil, H.; Soler-Toscano, F.; Dingle, K.; Louis, A. Correlation of automorphism group size and topological properties with program-size complexity evaluations of graphs and complex networks. Phys. A Stat. Mech. Appl.
**2014**, 404, 341–358. [Google Scholar] [CrossRef][Green Version] - Soler-Toscano, F.; Zenil, H.; Delahaye, J.-P.; Gauvrit, N. Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines. PLoS ONE
**2014**, 9, e96223. [Google Scholar] [CrossRef] [PubMed] - Langton, C.G. Studying artificial life with cellular automata. Phys. D Nonlinear Phenom.
**1986**, 22, 120–149. [Google Scholar] [CrossRef] - Peshkin, L. Structure induction by lossless graph compression. In Proceedings of the 2007 Data Compression Conference, Snowbird, UT, USA, 27–29 March 2007; pp. 53–62. [Google Scholar]
- Chaitin, G.J. On the length of programs for computing finite binary sequences. J. ACM
**1966**, 13, 547–569. [Google Scholar] [CrossRef] - Kolmogorov, A.N. Three approaches to the quantitative definition of information. Probl. Inf. Trans.
**1965**, 1, 1–7. [Google Scholar] [CrossRef] - Zenil, H.; Soler-Toscano, F.; Delahaye, J.-P.; Gauvrit, N. Two-Dimensional Kolmogorov Complexity and Validation of the Coding Theorem Method by Compressibility. PeerJ Comput. Sci.
**2013**, 1, e23. [Google Scholar] [CrossRef] - Zenil, H.; Kiani, N.A.; Tegnér, J. Algorithmic complexity of motifs clusters superfamilies of networks. In Proceedings of the IEEE International Conference on Bioinformatics and Biomedicine, Shanghai, China, 18–21 December 2013. [Google Scholar]
- Zenil, H.; Kiani, N.A.; Tegnér, J. Methods of Information Theory and Algorithmic Complexity for Network Biology. Semin. Cell Dev. Biol.
**2016**, 51, 32–43. [Google Scholar] [CrossRef] [PubMed] - Weisstein, E.W. “Polyomino.” From MathWorld—A Wolfram Web Resource. Available online: http://mathworld.wolfram.com/Polyomino.html (accessed on 17 July 2018).
- Bonchev, D. Overall Connectivity and Topological Complexity: A New Tool for QSPR/QSAR. In Topological Indices and Related Descriptors in QSAR and QSPR; Devillers, J., Balaban, A.T., Eds.; Gordon & Breach: Langhorne, PA, USA, 1999; pp. 361–401. [Google Scholar]

**Figure 1.**

**Top**: platonic solid networks with order parameter values (Block Decomposition Method (BDM), entropy and lossless compression by Compress);

**Bottom left**: different polyhedra sorted by algorithmic symmetry (BDM);

**Bottom right**: platonic graphs and their duals classified by order parameters, duals are in same colour. Numerical approximations of graphs and duals have similar values, as theoretically expected, given that there is an algorithm of fixed size that sends a graph to its unique dual and back.

**Figure 2.**Symmetry breaking by edge removal.

**Top**: starting from a growing complete graph (perfect symmetry), removing a node produces another perfectly symmetrical object (another complete graph) hence preserving the symmetry. However, as soon as a random edge is deleted from the complete graph, the symmetry is broken and information is generated as quantified by the difference between the original algorithmic content of the complete graph and the mutated one without a single edge;

**Bottom**: in a random graph, symmetry breaking does not produce algorithmic information. Entropy and lossless compression both fail to characterize these instances of graph-theoretic symmetry breaking.

**Figure 3.**(

**a**) Spearman rank correlation values between the same polyominoes represented as graphs from their adjacency matrices were: $\rho =0.99,p=3.216\times {10}^{-14}$ for Algorithmic Probability (AP)-based (BDM) ranking; $\rho =0.44,p=0.0779$ for Compress; and $\rho =0.227,p=0.38$ and thus only statistically significant for the AP-based ranking. This means that it is the AP-based measure that also assigns the same complexity to qualitatively similar polyominoes, something that neither compression nor entropy was able to do consistently due to under- or over-sensitivity; (

**b**,

**c**) display their AP-estimated values for some examples. Adjacency matrices of net-form representations of simply connected free polyominoes (without holes). Adjacency matrices representing networks and solid objects may look very different yet the AP-based measure produces a robust classification more independent of the object description than other measures (entropy and compression). Some functions used to generate the objects from a library of polyominoes and net polyominoes written by Eric Weinstein’s and published in Wolfram’s MathWorld were adapted for our purposes [22].

**Figure 4.**

**Left**: Numerical approximation of algorithmic complexity by BDM over a polytope (hypercube) of growing dimensions, showing how algorithmic complexity monotonically decreases;

**Right**: Graphs belonging to different symmetric groups have different growing numerical algorithmic complexity values as theoretically expected.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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.
https://doi.org/10.3390/e20070534

**AMA Style**

Zenil H, Kiani NA, Tegnér J. Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations. *Entropy*. 2018; 20(7):534.
https://doi.org/10.3390/e20070534

**Chicago/Turabian Style**

Zenil, Hector, Narsis A. Kiani, and Jesper Tegnér. 2018. "Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations" *Entropy* 20, no. 7: 534.
https://doi.org/10.3390/e20070534