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

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## 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.

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**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/).

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**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