Investigating Transformational Complexity: Counting Functions a Region Induces on Another in Elementary Cellular Automata

Abstract

Over the years, the field of artificial life has attempted to capture significant properties of life in artificial systems. By measuring quantities within such complex systems, the hope is to capture the reasons for the explosion of complexity in living systems. A major effort has been in discrete dynamical systems such as cellular automata, where very few rules lead to high levels of complexity. In this paper, for every elementary cellular automaton, we count the number of ways a finite region can transform an enclosed finite region. We discuss the relation of this count to existing notions of controllability, physical universality, and constructor theory. Numerically, we find that particular sizes of surrounding regions have preferred sizes of enclosed regions on which they can induce more transformations. We also find three particularly powerful rules (90, 105, 150) from this perspective.

Canonical Computations in Cellular Automata and Their Application for Reservoir Computing

“…A more closely related line of work studies the computation capacity in given family of automata, rather than constructing new CAs. This has been done, e.g., by assessing how many automata a particular CA can simulate (Israeli and Goldenfeld, 2006;Hudcová and Mikolov, 2021), by exploring the transformational effect a local area of a CA has on another finite area (Biehl and Witkowski, 2021), or by measuring the capacity of ECAs to implement simple mappings with one input bit and assesing the ECA's semantic capacity (Dittrich, 2018). Intrinsic computations in CA has also been analysed in (Feldman et al, 2008) by assessing the statistical properties of such systems via complexity-entropy diagrams.…”