Temporal Causal Past-and-Future LTL

• Temporal Causal Past-and-Future LTL (CPLTL) is a logic framework that extends classical temporal logic by incorporating precise, time-indexed causal interventions and counterfactual reasoning.
• It formalizes the relationship between causal interventions, temporal evolution, and logical formulas, effectively handling systems with feedback and cyclic dependencies.
• Its efficient, polynomial-time model-checking procedure supports scalable analysis and verification of discrete-time dynamical processes in various application domains.

Temporal Causal Past-and-Future Linear Temporal Logic (CPLTL) is a formal framework for representing and reasoning about temporal causality in systems modeled by Structural Equation Models (SEMs). CPLTL extends classical temporal logic by integrating explicit, time-indexed causal interventions and counterfactual reasoning, accommodating systems with feedback and cyclic dependencies. Its semantics provide a precise correspondence between causal interventions, temporal evolution, and logical formulas, admitting an efficient (polynomial time) model-checking procedure. The approach addresses long-standing limitations in the expressiveness of standard SEMs and temporal logics, enabling compositional and scalable analysis of discrete-time dynamical processes with rich causal structure (Gladyshev et al., 17 Jan 2025).

1. Syntax and Formal Structure

CPLTL is defined over a fixed causal signature $$\mathcal{S} = (\mathcal{U}, \mathcal{V}, \mathcal{R})$$, where $\mathcal{U}$ is a finite set of exogenous variables, $\mathcal{V}$ a finite set of endogenous variables, and $\mathcal{R}$ assigns every variable its (finite) range of values. The atomic propositions are equalities of the form $(X=x)$, with $X \in \mathcal{U} \cup \mathcal{V}$ and $x \in \mathcal{R}(X)$.

CPLTL formulas have two major layers:

• PLTL (Past-and-Future LTL): Classical temporal operators $\bigcirc$ (“next”), $\blacktriangleleft$ (“previous”), $\,\mathcal{U}\,$ (“until”), and $\,\mathcal{S}\,$ (“since”) apply over equalities of endogenous variables.
• Causal interventions: Denoted $[\vec{Y}(\vec{n}) \leftarrow \vec{y}]\varphi$, where interventions set variables $\vec{Y}$ to values $\vec{y}$ at times $\vec{n}$. Temporal indices allow precise control over when causal manipulations are enacted.

The grammar is: $$\Phi ::= [\vec{Y}(\vec{n})\leftarrow\vec{y}]\varphi \mid \neg\Phi \mid \Phi \wedge \Phi$$ where $\varphi$ is any PLTL formula.

2. Causal and Temporal Semantics

A causal scenario $(M, \uvec, \vvec)$ consists of a SEM $M = (\mathcal{S}, F)$, an infinite context $\uvec : \mathbb{N} \to \prod_{U \in \mathcal{U}} \mathcal{R}(U)$ specifying exogenous variable assignments over time, and an initial endogenous assignment $\vvec$.

• Standard Computation: The system’s evolution $C$ is generated by recursively applying $F$:

$\begin{aligned} C(0) &= \vvec \ C(i) &= \left(F_X(\uvec(i-1), C(i-1))\right)_{X \in \mathcal{V}} \quad (i \geq 1) \end{aligned}$

• Intervention Computation: For intervention $\intv = \vec{Y}(\vec{n})\leftarrow\vec{y}$:

$C^{\intv}(i)\bigr|_X = \begin{cases} y_j & \text{if } X(n_j)\leftarrow y_j \in \intv \text{ with } n_j = i \ F_X(\uvec(i-1), C^{\intv}(i-1)) & \text{otherwise} \end{cases}$

• PLTL Semantics: Evaluation ($\models_{PLTL}$) is standard, using the traced path of assignments over time.
• CPLTL Satisfaction: $(M, \uvec, \vvec), t \models [\vec{Y}(\vec{n})\leftarrow\vec{y}]\varphi$ iff $$\left(C^{\vec{Y}(\vec{n})\leftarrow\vec{y}}, t\right) \models_{PLTL} \varphi$$.

3. Interventions, Counterfactuals, and Examples

CPLTL encodes counterfactual queries by representing interventions as time-indexed assignments, permitting precise “what if” analyses.

Examples from (Gladyshev et al., 17 Jan 2025):

• Rocks Example: To assess whether the bottle shatters at time 1 if Billy throws at time 0,

$$(M, u_0, u_1, 000), 0 \models [BT(0)\leftarrow 1] \; \bigcirc(BS = 1)$$

• Treatment Example: To evaluate if a treatment at step 0 causes eventual recovery,

$(M, \uvec, \vvec), 0 \models [T(0)\leftarrow 1] \; F(R = 1)$

These illustrate CPLTL’s ability to formalize temporally granular and causally specific scenarios.

4. Non-Recursive SEMs, Feedback, and Temporal Unfolding

Standard SEMs are often recursive (acyclic), but CPLTL supports non-recursive SEMs—allowing for cyclic dependencies and feedback. The temporal interpretation treats each equation’s right-hand side as referencing the previous time step for all arguments; thus, even cyclic systems yield well-defined, unique infinite computations.

Illustrative Example (Memory Loop):

Step ($i$)	$T(i)$	$R(i)$
0	$0$	$0$
1	$0$	$0$
2	$1$	$\tfrac{1}{2}$
3	$0$	$0$
4	$1$	$\tfrac{1}{2}$
...	...	...

A self-loop in $R$ (i.e., $R(t) = f(T(t-1), R(t-1))$) creates periodic, memory-dependent behaviors.

This approach obviates algebraic fixed-point ambiguities associated with cyclic (simultaneous) dependencies in static SEMs.

5. Model-Checking and Computational Properties

The CPLTL model-checking problem asks whether a given formula $\Phi$ holds at time $t$ in a scenario $(M, \uvec, \vvec)$. Key algorithmic observations:

• Exogenous sequences ($\uvec$) modeled as ultimately periodic words $(n,m)$.
• The execution under any intervention is also ultimately periodic, with its period computable by period-finding algorithms.
• Standard PLTL path-model-checking (including Karp-Miller style procedures) applies to periodic paths, with polynomial-time complexity in both period and formula length.

Theorem: CPLTL model-checking is decidable in polynomial time in the size of $M, \uvec, \vvec, t$, and $\Phi$ (Gladyshev et al., 17 Jan 2025).

Outline of the model-checking procedure:

1. Parse $\Phi$ to identify intervention subformulas.
2. For each intervention, compute the periodic trace of the system under that intervention.
3. Apply a PLTL checker to the resulting path.
4. Recursively combine truth values as per the Boolean structure of $\Phi$.

A plausible implication is that this efficiency, combined with CPLTL’s expressiveness, makes CPLTL suitable for scalable temporal-causal analysis in automated verification and simulation settings.

6. Worked Example: Feedback and Counterfactual Query

A detailed case involving repeated treatment and memory loop demonstrates the full power of CPLTL (Gladyshev et al., 17 Jan 2025):

• Variables: $\mathcal{U} = \{U\}$ (binary), $\mathcal{V} = \{T, R\}$ ($T$ binary, $R = \{0, \tfrac{1}{2}, 1\}$).
• Feedback structure: $R(t)$ depends on $T(t-1)$ and $R(t-1)$.
• Periodic exogenous sequence: $\uvec = (0,1,0,1, 0,1, \dots)$.
• Default state: $T=0, R=0$.

Query: "If $T$ is forcibly set to $1$ at $t=0$, will $R$ eventually reach and stay at $1$?"

• The computation, under the intervention $T(0)\leftarrow 1$, cycles between $(1,0)$ and $(1,\tfrac{1}{2})$, so $R$ never stabilizes at $1$.
• Consequently,

$(M, \uvec, \vvec), 0 \not\models [T(0)\leftarrow 1] \; F(R=1) \wedge G(R=1)$

This concrete procedure exemplifies both how interventions alter temporal causal dynamics and how fine-grained queries can be definitively answered.

7. Significance and Extensions

CPLTL unifies the semantics of temporal logic and structural causal models, generalizing prior counterfactual frameworks to handle arbitrary (including cyclic) temporal-causal structures. The absence of acyclicity requirements enables direct modeling of feedback-rich systems, which are prevalent in control, biology, and economics. The ability to perform model-checking in polynomial time supports integration into formal verification, diagnosis, and simulation workflows.

A plausible implication is that CPLTL provides a foundation for future developments targeting expressivity (e.g., richer intervention languages, probabilistic causality) and scalability in causal temporal reasoning. The correspondence between causal interventions and temporal logic opens directions for compositional specification and reasoning in cyber-physical and automated decision systems.

For foundational exposition and worked proofs: Gladyshev et al., “Temporal Causal Reasoning with (Non-Recursive) Structural Equation Models” (Gladyshev et al., 17 Jan 2025).