Causal sets from simple models of computation

Causality among events is widely recognized as a most fundamental structure of spacetime, and causal sets have been proposed as discrete models of the latter in the context of quantum gravity theories, notably in the Causal Set Programme. In the rather different context of what might be called the ‘Computational Universe Programme’ – one which associates the complexity of physical phenomena to the emergent features of models such as cellular automata – a choice problem arises with respect to the variety of formal systems that, in virtue of their computational universality (Turing-completeness), qualify as equally good candidates for a computational, unified theory of physics. This paper proposes Causal Sets as the only objects of physical significance and relevance to be considered under the ‘computational universe’ perspective, and as the appropriate abstraction for shielding the unessential details of the many different computationally universal candidate models. At the same time, we propose a fully deterministic, radical alternative to the probabilistic techniques currently considered in the Causal Set Programme for growing discrete spacetimes. We investigate a number of computation models by grouping them into two broad classes, based on the support on which they operate; in one case this is linear, like a tape or a string of symbols; in the other, it is a two-dimensional grid or a planar graph. For each model we identify the causality relation among computation events, implement it, and conduct a possibly exhaustive exploration of the associated causal set space, while examining quantitative and qualitative features such as dimensionality, curvature, planarity, emergence of pseudo-randomness, causal set substructures and particles.

💡 Research Summary

The paper investigates the relationship between discrete computational models and causal sets (causets), which are regarded in quantum‑gravity research as the fundamental structure of spacetime. Starting from a general procedure, the authors map any computation into a directed acyclic graph: each elementary computational step (event) becomes a node, and a causal link is drawn from event e_i to a later event e_j whenever e_i last modified a component of the global state that e_j subsequently reads. This definition is applied in detail to Turing machines, where the global state consists of tape symbols, the head position, and the internal control state. Consequently each Turing‑machine event has at most two incoming and two outgoing links, one induced by the head’s state/position and one by the most recent write to the accessed tape cell. The authors prove that the causet of any Turing‑machine computation is planar; when the computation is visualised as a stack of successive tape configurations, the causal edges never cross. Moreover, because the head‑state links connect every event to its immediate successor, the transitive reduction of the causet collapses to a simple linear chain, making the partial order effectively total.

The study then classifies computational models into two broad families based on the substrate they operate on: linear substrates (tape or string) and two‑dimensional substrates (square grids or planar trivalent graphs). For all linear‑substrate models—including Turing machines, string automata, and string rewrite systems—the resulting causets are planar and exhibit a one‑dimensional character after reduction. By contrast, models that run on a two‑dimensional support generate richer causal structures. Grid‑based Turing machines produce causets whose dimensionality, measured via scaling of average path length with node count, lies near two and sometimes exceeds it, indicating an emergent “virtual dimension”. Mobile automata on trivalent graphs display even higher dimensionality, non‑planar features, and the spontaneous appearance of localized substructures that the authors liken to particles moving through the causal network.

Quantitative analyses of dimension and curvature are performed by fitting the relationship between the number of nodes N and the typical graph distance ℓ(N)≈N^{1/D}, where D is an effective dimension. Linear models yield D≈1, while two‑dimensional models give D≈2–2.5. Curvature estimates, derived from deviations from the Euclidean scaling law, reveal hyperbolic‑like behaviour in some grid‑based causets. The paper also examines pseudo‑randomness: computations that generate apparently random patterns (e.g., Wolfram’s rule 30) produce causets with high clustering coefficients and degree distributions reminiscent of random graphs, suggesting a possible analogue of quantum fluctuations in the causal‑set picture.

A major contribution is the proposal of a fully deterministic growth algorithm for causets, intended as an alternative to the stochastic “sprinkling” method traditionally used in causal‑set quantum gravity. In the deterministic scheme, each new event is inserted according to the same read‑write causal rule, without any probabilistic sampling. Simulations show that this approach can still generate complex, non‑planar, and high‑dimensional causets, thereby demonstrating that randomness is not a prerequisite for rich causal structure.

Overall, the work argues that causal sets provide a unifying abstraction that strips away the irrelevant details of specific universal computational models, allowing a direct comparison of their spacetime‑like properties. The observed diversity of causet geometries—planarity versus non‑planarity, low versus high effective dimension, presence or absence of particle‑like substructures—suggests that not all Turing‑complete systems are equally suitable as candidates for a “computational universe”. The authors conclude by outlining future directions: systematic comparison with Lorentzian manifolds, exploration of embedding properties, and the development of particle dynamics within the causet framework.