Symmetry and Algorithmic Complexity of Polyominoes and Polyhedral Graphs

We introduce a definition of algorithmic symmetry able to capture essential aspects of geometric symmetry. 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 enumeration 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 properties of polyominoes, polytopes, regular and quasi-regular polyhedra as well as polyhedral networks, thereby demonstrating its profiling capabilities.

[1]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences , 1966, JACM.

[2]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[3]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences: statistical considerations , 1969, JACM.

[4]  Christopher G. Langton,et al.  Studying artificial life with cellular automata , 1986 .

[5]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[6]  Cristian S. Calude.  Information and Randomness: An Algorithmic Perspective , 1994 .

[7]  Shu-Kun Lin,et al.  Correlation of Entropy with Similarity and Symmetry , 1996, J. Chem. Inf. Comput. Sci..

[8]  Danail Bonchev.  Overall Connectivity and Topological Complexity: A New Tool for QSPR/QSAR , 2000 .

[9]  I. Gutman.  The Energy of a Graph: Old and New Results , 2001 .

[10]  A. ADoefaa,et al.  ? ? ? ? f ? ? ? ? ? , 2003 .

[11]  Leonid Peshkin,et al.  Structure induction by lossless graph compression , 2007, 2007 Data Compression Conference (DCC'07).

[12]  P. Fowler.  Energies of Graphs and Molecules , 2008 .

[13]  Jean-Paul Delahaye,et al.  Numerical evaluation of algorithmic complexity for short strings: A glance into the innermost structure of randomness , 2011, Appl. Math. Comput..

[14]  Hector Zenil,et al.  Algorithmic complexity of motifs clusters superfamilies of networks , 2013, 2013 IEEE International Conference on Bioinformatics and Biomedicine.

[15]  Hector Zenil,et al.  Correlation of automorphism group size and topological properties with program−size complexity evaluations of graphs and complex networks , 2013, 1306.0322.

[16]  Jean-Paul Delahaye,et al.  Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines , 2012, PloS one.

[17]  Hector Zenil,et al.  Methods of information theory and algorithmic complexity for network biology. , 2014, Seminars in cell & developmental biology.

[18]  Thomas Böttcher,et al.  An Additive Definition of Molecular Complexity , 2016, J. Chem. Inf. Model..

[19]  Hector Zenil,et al.  A Computable Measure of Algorithmic Probability by Finite Approximations , 2015, Complex..

[20]  Hector Zenil,et al.  Low Algorithmic Complexity Entropy-deceiving Graphs , 2016, Physical review. E.

[21]  Hector Zenil,et al.  A Decomposition Method for Global Evaluation of Shannon Entropy and Local Estimations of Algorithmic Complexity , 2016, Entropy.

[22]  Hector Zenil,et al.  Predictive Systems Toxicology. , 2018, Methods in molecular biology.