Quantify spontaneous complexity in open-ended computational systems

Determine how much complexity can spontaneously arise in open-ended computational systems composed of interacting, self-modifying programs without explicit fitness functions, such as the Brainfuck-family substrates, Forth variants, and real-world instruction set soups analyzed in the paper, using well-defined complexity measures (e.g., high-order entropy) to characterize the attainable levels and dynamics over time.

Background

The paper demonstrates the emergence of self-replicators and subsequent dynamics across several computational substrates without explicit fitness functions. It introduces a practical complexity measure (high-order entropy) and reports state transitions where self-replicators take over. The authors observe increasing and then decreasing complexity patterns, with transitions becoming more likely over epochs.

However, the extent to which complexity can grow and sustain itself in such open-ended systems remains unresolved. This question concerns the limits and typical trajectories of complexity that arise purely from random interactions and self-modification, rather than externally imposed selection or objectives.