Causal & Computational Closure

• Causal and computational closure are distinct forms of system self-sufficiency, where causal closure contains all relevant causes and computational closure ensures that formal rules fully describe intended behaviors.
• They underpin the analysis and prediction of complex systems by integrating multi-level interactions, deterministic evolution, and rigorous computational techniques.
• Achieving these closures entails overcoming significant computational challenges and complexity constraints, necessitating precise abstraction and constraint-based methodologies.

Causal and computational closure denote two interrelated yet distinct forms of domain closure relevant to the modeling, analysis, and interpretation of complex systems, particularly where physical, computational, or algorithmic structure is intended to account for all relevant phenomena or observations. Causal closure pertains to the self-sufficiency of a system concerning causes—that the system’s own dynamics, possibly including influences between levels or subsystems, suffice to explain its state evolution. Computational closure refers to the soundness and completeness of a system’s computational mechanisms or formal representations—ensuring that the operations, inferences, or computations performed within a given model or formalism exhaustively encapsulate the class of behaviors or consequences intended. Across physics, computer science, logic, neuroscience, and philosophy of computation, these forms of closure are foundational to both explanatory adequacy and the possibility (or impossibility) of inference, robust prediction, and implementation.

1. Foundations of Causal and Computational Closure

Causal closure is defined as the condition whereby all causes relevant to the behavior of a subsystem are contained within that system or, more technically, that the equations or mappings describing the system account for all relevant interventions, state transitions, and inter-level influences present in the phenomenon under investigation. In physics and biology, this includes both upward (emergent) and downward (constraining) causation, with Ellis (Ellis, 2020) emphasizing that effective causal closure emerges only when all mutually relevant levels—from microphysical to organismic or societal—are included, as no single level is, in practice, causally complete in isolation.

Computational closure, in contrast, captures the formal, algorithmic, or representational self-sufficiency of a system. It designates the property that the system’s formal rules or computational processes are sufficient to derive (and only derive) the set of outcomes or inferences that are possible by the system. This includes both logical closure (every inference that should be derivable is derivable) and algorithmic closure (the represented computation really “closes” over the intended domain). In the context of formal logic and model theory, computational closure often also refers to the completeness and soundness of inference systems, as seen in causal graph models and in the algebraic analysis of linear (Fock-space) dynamical systems (Hanckowiak, 2010, Geiger et al., 2013).

2. Causal Sets, Computational Models, and Determinism

In the “computational universe” perspective, causal closure is rigorously realized via the construction of causal sets (causets) from deterministic computational models (Bolognesi, 2010). A causet is a locally finite partially ordered set (C, ≺), encoding the causal relations x ≺ y between computational events. The “computational closure” is achieved by extracting only the causal ordering of events, abstracting away the unessential structural or implementation details of the underlying model (e.g., whether it is a Turing machine, a tag system, or a 2D automaton). Deterministic construction of causets—linking new events on the basis of explicit data dependencies such as reads/writes—provides a radical alternative to probabilistic “sprinkling” used in quantum gravity and embodies both closure principles: the entire structure is generated by rule-closed, deterministic evolution, and the only physically meaningful content—the causal network—is finitely and completely specified by the system’s internal computational history.

The significance is evident in the emergence of complex phenomena such as highways (recurring regular patterns), particle-like substructures, and regions of mixed pseudo-randomness, which show that even completely deterministic, locally closed computational processes can yield highly nontrivial causal architectures. For example, the node-shell growth and curvature analysis performed in (Bolognesi, 2010) reveals that deterministically closed systems still support rich geometric and statistical behaviors.

3. Formalisms and Closure Problems in Physical Theories

Universal linear formalisms, such as those represented by equations of the form

$A|V\rangle = |+\rangle$

in full Fock space (Hanckowiak, 2010), yield an explicit setting for analyzing closure problems. Here, $|V\rangle$ encodes all $n$-point correlation functions, with operator $A$ typically composed of both non-interacting (diagonal), interacting (upper-triangular), and externally driven (lower-triangular) components. The closure problem arises because the infinite hierarchy of equations for $n$-point correlations is empirically and computationally intractable—only a finite subset is accessible. To attain closure, additional conditions such as boundary/initial data, symmetry projectors, or rational nonlinearity must be imposed, producing a closed system in which all relevant correlations are uniquely determined by the equations and constraints.

In this setting, causal closure is ensured by the appropriate incorporation of “source” terms (accounting for environmental or external influences), while computational closure is realized by reducing the system to a finite or tractable hierarchy via imposed constraints. Thus, both forms of closure are essential for theoretical completeness and empirical computability.

4. Causal Closure in Hierarchically Structured Systems

Real-world complex systems—especially in engineering and biology—embody closure properties only when the appropriate inter-level perspectives are adopted (Ellis, 2020). Effective causal closure is achieved by integrating upward emergence (where higher-level behaviors arise from collective low-level dynamics) with downward causation (where macro-level structures or algorithms constrain micro-level dynamics). This is formalized by identifying suitable effective theories at each level

$ET_L: U_L \to O_L ,$

where $U_L$ and $O_L$ are inputs and outputs/resources at level $L$. However, no single $ET_L$ is causally complete outside of the context of other levels: the functioning of a digital computer, for example, cannot be deduced from microphysics alone unless the influence of software and circuit design (higher-level constraints) is included. The concept of “inextricably intertwined levels” encapsulates the irreducibility of causal closure to any single layer. Only by structurally coupling all relevant levels (often extending to the social or environmental context) can causal and computational closure, in the sense of full explanatory adequacy, be said to obtain.

5. Logical, Graphical, and Algorithmic Closure in Causal Inference

Directed acyclic graphs (DAGs) formalize causal closure by encoding explicit causal assumptions (edges) and conditional independence relations (Geiger et al., 2013). The key property is that the “closure” of the input list of independencies can be algorithmically completed using d-separation and the semi-graphoid axioms. This polynomial-time inference procedure constitutes computational closure: every consequence (conditional independence) that follows from the specified causal structure is discoverable by the graph-based mechanism. Furthermore, the Armstrong property guarantees consistency: for every DAG $D$, there exists some probability measure $P$ perfectly embodying all and only the DAG’s independencies, ensuring that the closure under the graphical model matches closure of real statistical relationships.

More generally, logical formulations (e.g., effective linear equations (Hanckowiak, 2010), model-theoretic structures (Halpern, 2011), and formal languages supporting probabilistic and interventional logic (Zander et al., 2023)) recast closure as the condition that the system’s set of rules, formulas, or proof procedures suffices to generate all valid consequences and no more. For languages expressive enough to match scientific notation (incorporating do-calculus, summation, and polynomial constraints), closure is computationally expensive: the satisfiability problem is complete for the class succ$\exists$R (Zander et al., 2023), so practical complete closure carries extreme complexity costs.

6. Constraints and Limits to Closure: Complexity and Approximation

The computational complexity of achieving causal or computational closure is often prohibitive. For instance:

• Invariant causal prediction, when framed as the search for non-trivial invariant predictors across environments, is NP-hard even for linear models (Gu et al., 29 Jan 2025). This fundamental hardness means that, absent additional assumptions, no polynomial-time algorithm can always identify the closed (uniquely invariant) predictor, even given population-level data. In estimation tasks, this implies a trade-off: computationally efficient algorithms can only estimate the causal parameter at arbitrarily slow rates unless structure is imposed.
• In reasoning about probabilistic and causal languages that encode standard inference tasks (including do-calculus), the satisfiability problem is succ$\exists$R-complete (Zander et al., 2023). Thus, forming logically and computationally closed causal models—even if conceptually possible—is algorithmically beyond the reach of standard inference and optimization techniques except in highly restricted or structured cases.
• In hidden-variable theories intended to capture quantum phenomena, computational closure interacts with causal closure via complexity-theoretic no-go theorems (Brogioli, 18 Sep 2024). If the sampling problem associated with a sequential or postselection-based hidden-variable model falls into a class lower in complexity than SampBQP (quantum sampling), then it cannot achieve closure with respect to quantum-mechanical predictions.

This spectrum of results demonstrates that closure is not merely a conceptual or theoretical property, but a resource constrained by computational hardness, algorithmic limits, and the structure of admissible models.

7. Synthesis: Closure in Causal, Computational, and Explanatory Practice

Across physical theories, computation, and the empirical sciences, causal and computational closure serve as criteria for explanatory adequacy—marking the boundary between models that are adequate for all relevant inferences and those that are fundamentally open or incomplete. Their joint analysis reveals the necessity of:

• Rigorous abstraction procedures that extract causal sets or DAGs embodying all and only meaningful dependencies (Bolognesi, 2010, Geiger et al., 2013);
• Model architectures that guarantee the closure of inference—whether via topological, algebraic, or algorithmic completeness (Hanckowiak, 2010, Carelli et al., 15 May 2025);
• Recognition of inter-level dependence and the necessity of integrating emergence and constraint across scales (Ellis, 2020);
• Realistic assessment of computational and algorithmic intractability, with appropriate relaxation or regularization strategies necessary for scalable or statistically efficient inference (Ren et al., 23 Dec 2024, Gu et al., 29 Jan 2025).

The investigation of closure—as property, problem, and limit—thus occupies a central place in the mathematical, computational, and philosophical foundations of modeling across the sciences. The articulation of when, how, and at what cost closure can be achieved remains a central research theme with profound implications for the design, validation, and deployment of explanatory and predictive systems in both theoretical and applied contexts.