Self-replication and Computational Universality

TL;DR

The paper tackles the core question of whether Turing universality implies self-replication in physical-like dynamics. It introduces two notions of CA universality—local (bounded-region simulation) and global (intrinsic simulation via packing)—and proves a strict hierarchy, with GloballyUniversal ⊊ LocallyUniversal, while showing local universality does not entail universal self-replication. It formalizes a minimal local self-replication condition and demonstrates a 1D locally universal CA that cannot sustain non-trivial self-replication, alongside a 1D universal self-replicator constructed via bounded-height encoding. The analysis of Rule 110 provides a concrete, rigorous treatment of encoder/decoder complexity in CA computation, showing local universality can arise with polynomial-time encoders and linear-space decoders. Overall, the work lays mathematical foundations clarifying when physical dynamics can host living-like replication, separating the challenges of computation from replication, and outlining a rich agenda of open problems and future directions for self-reproduction theory.

Abstract

Self-replication is central to all life, and yet how it dynamically emerges in physical, non-equilibrium systems remains poorly understood. Von Neumann's pioneering work in the 1940s and subsequent developments suggest a natural hypothesis: that any physical system capable of Turing-universal computation can support self-replicating objects. In this work, we challenge this hypothesis by clarifying what computational universality means for physical systems and constructing a cellular automaton that is Turing-universal but cannot sustain non-trivial self-replication. By analogy with biology, such dynamics manifest transcription and translation but cannot instantiate replication. More broadly, our work emphasizes that the computational complexity of translating between physical dynamics and symbolic computation is inseparable from any claim of universality (exemplified by our analysis of Rule 110) and builds mathematical foundations for identifying self-replicating behavior. Our approach enables the formulation of necessary dynamical and computational conditions for a physical system to constitute a living organism.

Figures (32)

• Figure 1: (a) Physical or chemical systems, abstracted as cellular automata (CAs), may in appropriate circumstances evolve from random initial conditions into self-replicating organisms. We develop a theory constraining which dynamics and configurations can give rise to self-replicating organisms. (b) We rigorously define what it means for a CA to be Turing-universal and establish its relationship to self-replication. We identify and study three classes of CAs with increasing capabilities: (i) locally universal CAs can host a universal Turing machine within their dynamics, (ii) universal self-replicating CAs support self-replicating structures that also perform Turing-universal computation, and (iii) globally universal CAs can simulate any other CA of the same dimension via local, translation-invariant encodings. We prove these form a strict hierarchy: Locally Universal $\supset$ Universal Self-Replicating $\supset$ Globally Universal. Consequently, some CAs (such as the one shown in the 'locally universal' block) are Turing-universal and yet provably cannot support self-replicating organisms, analogous to biological systems capable of transcription but not replication.
• Figure 2: (a) Space-time diagram of the elementary CA, Rule 110. Each cell assumes a binary state (black or white) and evolves deterministically based on its current state and those of its two nearest neighbors. Time progresses downward from a random initial configuration (top row), revealing the emergence of complex spatiotemporal structures. (b) Schematic of a computational process, as implemented by a Turing machine, where an input bit string undergoes computation to produce an output bit string. To simulate this computation using Rule 110, three requirements must be satisfied: (i) the input bit string must be encoded into the CA's initial configuration, (ii) the CA's evolution must faithfully track the Turing machine's computational steps, and (iii) a decoding scheme must extract the final output from the CA dynamics such that it matches the Turing machine's output.
• Figure 3: (a) Local universality in a 2D CA. A spatially finite configuration (shown at a fixed time) can perform Turing-universal computation: any Turing machine can be encoded into such a configuration, with the CA's evolution tracking the machine's computation and its states recoverable through decoding. The CA is termed locally universal because Turing machines with finite input can be simulated within finite (though potentially growing) spatial regions, rather than requiring infinite configurations of the CA at the outset. (b) Global universality in a 1D CA. The CA $\mathcal{B}$ (shown evolving top to bottom) can simulate infinite configurations of any other 1D CA through local, translation-invariant encoding and decoding maps. By contrast, local universality only requires a CA to be able to simulate a Turing machine within a finite region. Here we exemplify how $\mathcal{B}$ globally simulates dynamics in three different CAs $\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3$ simultaneously.
• Figure 4: (a) Rule 110 can simulate any Turing machine via a sequence of intermediate encodings and decodings. In particular, Rule 110 (where certain "gliders" are colored orange for emphasis) simulates a cyclic tag system which itself simulates a clockwise Turing machine, which itself simulates an ordinary Turing machine. We carefully keep track of the computational complexities of the encodings and decodings as well as their compositionality properties, and establish that the composed encoding and decoding between Rule 110 and a Turing machine requires only polynomial time complexity and linear space complexity. (b) The encoding from a Turing machine to Rule 110 and the subsequent decoding are very intricate. Shown here are encoding components of Turing machine states mapping to modular regions of the Rule 110 CA that fit together like puzzle pieces.
• Figure 5: Space time diagram of a 1D CA with states 0 (white) and 1 (black), radius 1, and local rule $f(x, y, z) = x + z \bmod 2$ operating on a cyclic configuration of size 250. Time is progressing downwards.

Theorems & Definitions (111)

• Definition 2.1: Local universality of a CA, informal
• Definition 2.2: Global universality of a CA, informal
• Theorem 2.3: Computational universality hierarchy for CAs
• Theorem 2.4: Local universality of Rule 110
• Theorem 2.5
• Definition 2.6: Local self-replicating condition, informal
• Theorem 2.8: Non-talking heads CA does not have self-replication
• Theorem 2.9: Self-replication lies strictly between global and local universality
• Conjecture 3.1
• Definition B.1: Cellular automaton