Exploring the Space of Substitution Systems

Substitution systems, where strings are rewritten according to local rules, have many applications. They are used to model the development of plants, as well as to generate music and architectural designs. Many substitution systems can generate highly complex patterns using only simple rules. This feature can make substitution systems difficult to analyze mathematically. A different approach, pioneered by Stephen Wolfram, is to use computer searches to reveal simple systems with interesting properties. This approach is used to explore a class of systems we call symmetric sequential substitution systems within which a string is repeatedly updated by applying rewrite rules in a non-overlapping way. In this paper several simple examples of these systems are exhibited that produce complex behavior. The dynamics of several of these systems are studied and a system is exhibited that is computationally universal. Applications of symmetric sequential substitution systems are discussed, such as compression and the evaluation of numerical functions.

[1]  A. Lindenmayer.  Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs. , 1968, Journal of theoretical biology.

[2]  Susan Stepney,et al.  Growing Music: Musical Interpretations of L-Systems , 2005, EvoWorkshops.

[3]  R. Landauer,et al.  The Fundamental Physical Limits of Computation. , 1985 .

[4]  Narendra S. Goel,et al.  Some non-biological applications of L-systems , 1991 .

[5]  Jarkko Kari,et al.  Reversible Cellular Automata , 2005, Developments in Language Theory.

[6]  Mark Nelson,et al.  The Data Compression Book , 2009 .

[7]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..

[8]  Przemyslaw Prusinkiewicz,et al.  The Algorithmic Beauty of Plants , 1990, The Virtual Laboratory.

[9]  Kenichi Morita,et al.  Reversible Cellular Automata , 2009, Encyclopedia of Complexity and Systems Science.

[10]  Stephen Wolfram,et al.  Universality and complexity in cellular automata , 1983 .

[11]  Ming Li,et al.  Clustering by compression , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[12]  Abraham Lempel,et al.  Compression of individual sequences via variable-rate coding , 1978, IEEE Trans. Inf. Theory.

[13]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[14]  G. A. Hedlund,et al.  Unending chess, symbolic dynamics and a problem in semigroups , 1944 .

[15]  Stephen Wolfram,et al.  A New Kind of Science , 2003, Artificial Life.

[16]  G. Chaitin.  Meta Math!: The Quest for Omega , 2004, math/0404335.

[17]  Hector Zenil,et al.  Compression-Based Investigation of the Dynamical Properties of Cellular Automata and Other Systems , 2009, Complex Syst..

[18]  Marcin Seredynski,et al.  Block cipher based on reversible cellular automata , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[19]  Jordan B. Pollack,et al.  Evolving L-systems to generate virtual creatures , 2001, Comput. Graph..

[20]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .