突破因果图局限,用零约束和微分方程建模对称关系与反馈循环。
Beyond Directed Acyclic Graphs: Causal Zeros and Causal Differential Equations
- 引入因果零概念,通过激活算子处理无方向的物理约束。
- 将瞬时循环视为时间压缩结果,用因果微分方程描述动态平衡。
- 适用于需要处理对称约束或反馈系统的科学建模场景。
Pearl 的结构因果模型(SCM)基于有向无环图(DAG)和 do-演算,是主流因果推理框架。但其存在双重限制:所有关系必须预设为有向因果边,且禁止反馈环。本文探讨两类突破此限制的现象。第一类为对称的物理与经济约束(如理想气体定律),无内在因果方向,方向仅在干预中显现,需指定求解变量。我们通过扩展因果模型引入因果零,并加入激活算子,满足局部可解性与图适配性条件。第二类为有限传播状态空间系统,将看似瞬时的环视为被压缩时间的产物,以因果微分方程(CDE)为基础,瞬态过程为时间展开的无环因果过程,因果零定义吸引子流形的特征函数;周期与混沌吸引子构成同一动态的不同阶段,通过吸引子相对干预处理。本文给出扩展 do-演算、可识别性条件、反事实语义及开放问题。
原文摘要 · Abstract (English)
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.
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