Beyond Directed Acyclic Graphs: Causal Zeros and Causal Differential Equations

arXiv cs.LG Papers
causal-inference causal-models structural-causal-models differential-equations feedback extended-causal-model

Summary

This paper extends Pearl's structural causal model framework by introducing causal zeros and causal differential equations to handle symmetric constraints and feedback cycles, which are not allowed in directed acyclic graphs.

arXiv:2607.22910v1 Announce Type: new Abstract: Pearl's structural causal model (SCM) framework, built on directed acyclic graphs (DAGs) and the do-calculus, is the dominant formal language for causal reasoning. Yet it carries two structural restrictions: every relationship must be pre-specified as a directed causal edge, and feedback cycles are forbidden. This paper examines two classes of phenomena that strain these restrictions. First, symmetric physical and economic constraints, the ideal gas law being the canonical case, carry no intrinsic causal direction. Direction emerges only under intervention, and which variable is solved for must be specified as part of the intervention. We formalize such constraints as causal zeros within an Extended Causal Model by adding an activation operator, subject to local solvability and graph-admissibility conditions. Second, for the class of finite-propagation state-space systems considered here, we treat apparent instantaneous cycles as artifacts of suppressed time and ground both causal zeros and feedback in Causal Differential Equations (CDEs). In these, the transient regime is a time-unrolled acyclic causal process, and causal zeros arise as the defining functions of attracting equilibrium manifolds; periodic and chaotic attractors define further regimes of the same dynamics, treated through attractor-relative intervention. We give the extended do-calculus, identifiability conditions, counterfactual semantics, and open problems.
Original Article
View Cached Full Text

Cached at: 07/28/26, 06:23 AM 

# Beyond Directed Acyclic Graphs: Causal Zeros and Causal Differential Equations
Source: [https://arxiv.org/abs/2607.22910](https://arxiv.org/abs/2607.22910)
[View PDF](https://arxiv.org/pdf/2607.22910)

> Abstract:Pearl's structural causal model \(SCM\) framework, built on directed acyclic graphs \(DAGs\) and the do\-calculus, is the dominant formal language for causal reasoning\. Yet it carries two structural restrictions: every relationship must be pre\-specified as a directed causal edge, and feedback cycles are forbidden\. This paper examines two classes of phenomena that strain these restrictions\. First, symmetric physical and economic constraints, the ideal gas law being the canonical case, carry no intrinsic causal direction\. Direction emerges only under intervention, and which variable is solved for must be specified as part of the intervention\. We formalize such constraints as causal zeros within an Extended Causal Model by adding an activation operator, subject to local solvability and graph\-admissibility conditions\. Second, for the class of finite\-propagation state\-space systems considered here, we treat apparent instantaneous cycles as artifacts of suppressed time and ground both causal zeros and feedback in Causal Differential Equations \(CDEs\)\. In these, the transient regime is a time\-unrolled acyclic causal process, and causal zeros arise as the defining functions of attracting equilibrium manifolds; periodic and chaotic attractors define further regimes of the same dynamics, treated through attractor\-relative intervention\. We give the extended do\-calculus, identifiability conditions, counterfactual semantics, and open problems\.

## Submission history

From: Sergei Kalinin \[[view email](https://arxiv.org/show-email/95cd3e22/2607.22910)\] **\[v1\]**Fri, 24 Jul 2026 20:54:42 UTC \(588 KB\)

Similar Articles

Causal Reasoning with Bipartite Graphical Causal Models

arXiv cs.AI

The paper proposes bipartite graphical causal models (BGCMs) to resolve ambiguities in causal interventions for systems at equilibrium with cyclic dependencies, generalizing existing frameworks like causal Bayesian networks and structural causal models.

Compiling VGDL into Causal Models

arXiv cs.AI

The paper proposes a deterministic framework that compiles games specified in VGDL into Dynamic Structural Causal Models to guarantee causal fidelity, enabling transparent causal pathways for counterfactual reasoning and causal reinforcement learning.

Relational Structural Causal Models

arXiv cs.AI

This paper introduces relational structural causal models, extending structural causal models to settings with varying objects and relations. It provides theoretical results for identification and proposes relational neural causal models that outperform non-relational baselines on simulated traffic scenes.

Towards Continuous-time Causal Foundation Models

arXiv cs.LG

Proposes a continuity criterion for extending discrete-time causal prior-data fitted networks to continuous time using stochastic differential equations, introducing a taxonomy and fine-grid integration method that outperforms naive integration on irregular observation schedules.