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.
Highlights
Artificial life studies “life as it could be” [1]. One approach to this is to start at the level of physics and look at artificial systems—usually dynamical systems—as artificial universes and study properties of life within them
We look at a quantification of this notion for the set of elementary cellular automata (ECA)
Constructor theory [6] conceives of physical laws as statements that rule out the possibility of particular kinds of transformations
Summary
Artificial life studies “life as it could be” [1]. One approach to this is to start at the level of physics and look at artificial systems—usually dynamical systems—as artificial universes and study properties of life within them. Constructor theory [6] conceives of physical laws as statements that rule out the possibility of particular kinds of transformations. Our investigation here investigates physical laws that hold not for constructors in general but for the transformations that particular kinds of finite constructors can achieve in finite time on finite substrates. We compute all possible transformations that an environment can induce on a volume. Is means each such impossible transformation corresponds to a resource constrained physical law in the sense of constructor theory Any transformations that we do not find are impossible. is means each such impossible transformation corresponds to a resource constrained physical law in the sense of constructor theory
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