[Paper Review] Causal sets from simple models of computation

Tommaso Bolognesi|arXiv (Cornell University)|Apr 19, 2010

Computability, Logic, AI Algorithms3 citations

TL;DR

This paper proposes causal sets as the fundamental physical structure in a computational universe framework, deriving them from simple, deterministic models of computation such as cellular automata and string-rewriting systems. By identifying causality among computation events in linear and planar models, it demonstrates the emergence of spacetime-like features—dimensionality, curvature, pseudo-randomness—without probabilistic growth, offering a deterministic foundation for quantum gravity.

ABSTRACT

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.

Motivation & Objective

• To identify causal sets as the only physically significant structure in the computational universe paradigm.
• To resolve the ambiguity in choosing among computationally universal models by abstracting to causal sets.
• To propose a deterministic alternative to probabilistic causal set growth in quantum gravity.
• To explore how causal sets arise from simple computational systems and exhibit spacetime-like properties.

Proposed method

• Analyzing computation models operating on linear supports (e.g., tapes, strings) and planar supports (e.g., grids, graphs).
• Defining causality among computation events based on dependency and order of operations.
• Implementing causal set generation through deterministic rewriting rules in cellular automata and string-rewriting systems.
• Conducting exhaustive exploration of the resulting causal set spaces to analyze structural features.
• Measuring dimensionality, curvature, planarity, and emergence of pseudo-randomness in the causal sets.
• Identifying substructures resembling particles and causal set symmetries.

Experimental results

Research questions

• RQ1 Can causal sets emerge naturally from simple, deterministic computational models without probabilistic assumptions?
• RQ2 What structural features—such as dimensionality and curvature—emerge in causal sets derived from computation?
• RQ3 How do causal sets from linear versus planar computation models differ in their geometric and topological properties?
• RQ4 Can pseudo-randomness and particle-like substructures arise in deterministic causal set models?
• RQ5 To what extent do causal sets derived from computation models resemble classical spacetime?

Key findings

• Causal sets derived from deterministic computation models exhibit measurable dimensionality consistent with 4D spacetime in certain regimes.
• Curvature-like features emerge in the causal sets, suggesting a geometric structure analogous to general relativity.
• Pseudo-randomness arises naturally in the causal set structure, even from deterministic rules, due to complex causal dependencies.
• Planar computation models produce causal sets with higher planarity and richer substructure compared to linear models.
• Particle-like substructures are observed as localized causal set configurations, suggesting potential for modeling elementary particles.
• The causal set space generated from deterministic models shows robustness and reproducibility, supporting their physical relevance.