The computational inevitability of life: self-replication under resource-bounded nested algorithmic probability

TL;DR

The paper addresses why self-replication emerges so ubiquitously across diverse computational substrates by positing a computational inevitability under resource-bounded universal induction. It introduces resource-bounded algorithmic probability and its nested variant to show that self-constructing programs (quines) form stable fixed points and dominate program space as attractors under resampling. The key theoretical contribution is a formal framework that treats replication as an intrinsic consequence of constrained computation, supported by an empirical study on a 2-state, 2-symbol linear-bounded automaton where quines P0 and P4095 prevail across nesting levels. This work unifies observations from artificial life, self-modifying systems, and cellular automata, suggesting that life represents the simplest persistent structure in bounded computation and that memory for self-description is algorithmically unavoidable. The results have implications for biology, AI, and physics by linking replication, complexity, and persistence to fundamental algorithmic principles rather than external optimization or fitness functions.

Abstract

Recent computational experiments have demonstrated the spontaneous emergence of self-replicating programs across universal automata, artificial chemistries, and self-modifying code systems. Remarkably, these results arise without explicit fitness functions, reward shaping, or predefined objectives, indicating a gap in our formal understanding of the underlying computational process. In this work, we argue that self-replication is computationally inevitable under resource-bounded automata. Building on algorithmic information theory, we show that when universal inductive bias is applied under finite constraints of time, memory, and description length, programs that construct descriptions of themselves, i.e., quines, emerge as stable fixed points of nested algorithmic probability. We formalize this argument and demonstrate that self-replicating programs act as attractors in program space, independent of external optimization criteria. Thus, resource bounds transform universal induction into a competitive ecological process over programs, in which self-constructing programs dominate by stabilizing their own measure under resampling. We reinterpret recent results from computational life experiments and self-improving artificial agents as empirical realizations of this theoretical principle. More broadly, we propose that life is the simplest persistent structure available to constrained computation. A living system remembers itself because doing so is algorithmically and thermodynamically unavoidable.

Figures (5)

• Figure 1: The space of programs typically maps to a smaller set of output strings. The algorithm probability $P(x^0)$ of the output strings $x^0$ is based on a uniform probability $2^{-l}$ of the programs in the highest meta-level (left circle).
• Figure 2: Attractor (green double-circled nodes) and repeller strings (red nodes).
• Figure 3: Level 3 of nesting algorithmic probability for a 2-state 2-symbol LBA (the high-resolution SVG is available at QuBio_2020).
• Figure 4: The 8 largest attractor basins of level 3 of the nesting algorithmic probability.
• Figure 5: Level 2 of nesting algorithmic probability for a 2-state 2-symbol LBA.