Temporal Structural Equation Models

Context

Temporal structural equation models reinterpret causal equations as stepwise mechanisms over trajectories. Gladyshev2025 - Temporal Causal Reasoning with Non-Recursive SEMs develops the CPLTL reasoning layer; Gladyshev2026 - Temporal Causal Models as a Model of Computation studies TSEMs as computational devices.

Formal Statement

For deterministic temporal SEMs with exogenous temporal context u, a computation starts from default endogenous assignment v and evolves by repeatedly applying the structural functions to the previous step:

$$C(0)=v,C(i{)}_X=F_X(u(i-1),C(i-1)).$$

For nondeterministic TSEMs over signature (V,R,D), each structural equation has type

$$F_Y:\prod _{X\in D(Y)}R(X)\to 2^{R(Y)}.$$

A computation is a tree of configurations, with v_1 -> v_2 iff every next variable value belongs to its structural-equation output.

Time-indexed interventions fix values at specific steps, for example Y(n)<-y or do(Y^n<-y). CPLTL formulas combine such interventions with past/future temporal logic over the generated computation.

The computation paper proves:

• an LBA accepts a string iff its constructed TSEM causal calculator accepts it;
• a Turing machine accepts a string iff its corresponding TSEM causal calculator accepts it;
• nondeterministic TSEMs can encode nondeterministic Turing machines step by step.

Derivation / Construction

• A temporal context is an infinite sequence of exogenous assignments.
• Non-recursive dependencies are interpreted as feedback across time, so a self-edge means dependence on the previous value rather than simultaneous self-definition.
• Temporal equivalence compares observable variables across all interventions; rescalable equivalence allows one model’s time step to correspond to multiple steps of another model.
• The LBA causal calculator represents current control state, head movement, written symbol, and tape symbols as TSEM variables.
• Interventions support counterfactual computation questions: flipping an input bit or intermediate value can test causal necessity for an output.

Implications

• Temporal SEMs unify interventionist counterfactuals with temporal-logic reasoning.
• TSEM expressivity reaches classical computation models, so causal reasoning can be applied to computational traces.
• The framework is promising for explaining failures, soft errors, and causal influence in programs or verified systems.
• Undecidability can reappear when outcome specifications use rich temporal properties such as eventual halting.

Links

• Sources: Gladyshev2025 - Temporal Causal Reasoning with Non-Recursive SEMs, Gladyshev2026 - Temporal Causal Models as a Model of Computation
• Concepts: Temporal Causal Models, Structural Equation Models, Actual Causality, Linear Temporal Logic