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A Counterfactual Cause in Situation Calculus

This paper proposes a counterfactual-based notion of achievement cause in the situation calculus that generalizes and refines the existing framework by Batusov and Soutchanski, while also clarifying its relationship to Halpern and Pearl's theory of actual causality, particularly regarding disjunctive goals.

Original authors: Daxin Liu, Vaishak Belle

Published 2026-05-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daxin Liu, Vaishak Belle

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a movie of a robot moving blocks around. At the end of the movie, a specific block is broken. You want to know: "Who or what actually caused the block to break?"

This paper is about building a better "detective tool" to answer that question, specifically for robots and computer programs that plan actions.

Here is the story of the paper, broken down into simple concepts:

1. The Problem: The "But-For" Test is Tricky

In philosophy and law, we often use a simple test to find a cause: "But-for" causality.

The Test: "But for this specific action, would the result have happened?"
The Logic: If the answer is "No, it wouldn't have happened," then that action is the cause.

The authors say this works well for simple stories. But in the complex world of robots (where they can do many things at once, or where different paths lead to the same result), the old tests get confused. They either miss the cause or blame the wrong thing.

2. The New Idea: The "Counterfactual" Detective

The authors propose a new definition of cause based on counterfactuals. This is a fancy word for "what if?"

Instead of just looking at the history of what happened, their tool asks: "If we erased this specific part of the robot's history, would the goal still be reached?"

They call this a "Counterfactual Achievement Cause."

The Goal: The robot wanted to break a block.
The History: The robot picked up the block, dropped it, then picked up another block, and dropped that one too.
The Test: If we delete the first "pick up and drop" from the history, does the block still break?
- If Yes: That first action wasn't the cause (maybe the second one did it).
- If No: That first action was the cause.

3. The "Filter" Mechanism: Cleaning Up the Timeline

Here is the clever part. Sometimes, if you remove an early action, the later actions become impossible to do.

Analogy: Imagine a recipe. If you remove "mix the eggs," you can't proceed to "bake the cake." The "bake" step becomes impossible.
The Paper's Solution: Their tool has a "Filter." When they remove a suspected cause, they automatically delete any future steps that can no longer happen because the cause is gone. They only look at the remaining legal actions to see if the goal is still met.

4. The "Forest Fire" Analogy (Disjunctive Goals)

The paper tackles a tricky scenario called a disjunctive goal.

The Scenario: A forest fire happens if either a match is dropped OR lightning strikes.
The Situation: Both the match is dropped AND lightning strikes. The forest burns.
The Confusion: Is the match the cause? Is the lightning the cause?
The Paper's Verdict: In their view, both are part of the cause. If you remove the match, the lightning still burns the forest. If you remove the lightning, the match still burns it. Because they are competing to achieve the same goal, the "cause" is the combination of both events.

This aligns with a famous theory by Halpern and Pearl (HP), which says that when two things compete to do the same job, they are both "part of the cause."

5. How It Compares to Other Detectives

The authors compare their tool to two other famous "detectives":

Batusov and Soutchanski: They had a previous tool that looked at the history and found the "minimal" sequence of steps needed. The new tool agrees with them in many cases but uses the "what if" (counterfactual) method to be more precise about why those steps matter.
Halpern and Pearl (HP): Their tool is very popular but relies on a rigid mathematical model (like a flowchart). The authors argue their tool is better for robots because it handles the flow of time and actions more naturally, without needing to force the story into a rigid flowchart first.

6. The Limitation: When Things Get Messy

The authors admit their tool isn't perfect.

The Issue: If two different "stories" (sequences of actions) happen at the same time and get mixed up (interleaved), the tool might get confused.
The Result: It might say, "The cause is the whole messy mix," rather than pinpointing exactly which specific step was the culprit. They acknowledge this is a hard problem that philosophers and computer scientists are still arguing about.

Summary

This paper introduces a new way for computers to understand causality. Instead of just looking at a list of events, it simulates "rewinding the tape" and removing specific actions to see if the result still happens. It handles complex situations where multiple actions compete to achieve a goal, offering a simpler, more natural way for robots to understand "who did what" in their own history.

Technical Summary: A Counterfactual Cause in Situation Calculus

Problem Statement
Recent work by Batusov and Soutchanski established a notion of "actual achievement cause" within the Situation Calculus, capable of determining the cause of quantified effects in a given action history. While this approach successfully handles action preconditions and quantified effects—limitations of earlier Structural Equation Model (SEM) approaches like Halpern and Pearl's (HP)—it lacks a definition grounded in a counterfactual perspective. The HP account, while influential, relies on SEMs which often struggle with temporal reasoning, quantifiers, and distinguishing actions from effects in complex domains. The authors identify a gap in defining actual causality within action languages that is both simple and explicitly counterfactual, particularly regarding how to handle disjunctive goals and preemption without over-complicating the definition.

Methodology
The authors propose a new notion of cause based on counterfactual analysis within the logic ES, a modal variant of the Situation Calculus. This logic simplifies the classical formalism by avoiding explicit situation terms while retaining regression and progression capabilities.

Formal Framework: The paper utilizes a Basic Action Theory (BAT) $\Sigma$ , comprising initial state axioms, action preconditions, and successor state axioms (Reiter's solution to the frame problem).
Minimal Cause Definition: The authors first define a "minimal cause" generally as an action sequence $z$ that achieves a goal $\phi$ (where $\phi$ was initially false) such that $z$ is minimal according to specific criteria (length, fluent set size, or a combination). This establishes a baseline counterfactual: if $z$ had not occurred, $\phi$ would not hold.
Achievement Cause in Narratives: The core contribution is the definition of an Achievement Cause for a specific causal setting $C = \langle z, \phi \rangle$ $C = ⟨ z, ϕ ⟩$ , where $z$ $z$ is a known action history (narrative) leading to $\phi$ $ϕ$ .
- A prefix $z' \subseteq z$ $z^{'} \subseteq z$ is a cause if:
  1. $\phi$ holds after any extension of $z'$ within $z$ .
  2. In the counterfactual scenario where $z'$ is removed, the remaining actions in $z$ (filtered to ensure executability) do not lead to $\phi$ .
  3. $z'$ is minimal (no proper subsequence satisfies the above).
- Crucially, the authors introduce a Filter function to handle the counterfactual removal of actions. If removing a prefix $z'$ renders subsequent actions in the narrative inexecutable (e.g., dropping an object one is no longer holding), the Filter function removes those dependent actions from the counterfactual scenario.

Key Contributions

Counterfactual Definition in Action Languages: The paper provides a definition of actual achievement cause that is explicitly counterfactual, contrasting with the recursive, non-counterfactual definitions of Batusov and Soutchanski.
Generalization: The proposed notion naturally generalizes to the concept of achievement cause in the context of action histories.
Handling Disjunctive Goals and Preemption: The authors analyze how their definition handles disjunctive goals (e.g., $A \lor B$ ). They note that in cases of preemption (where multiple competing actions could achieve a goal), their definition treats the joint set of competing events as the cause, rather than isolating a single preempted event. This aligns with the modified HP account where the conjunction of events is the cause.
Formal Comparison: The paper formally relates their account to Batusov and Soutchanski's work (Theorem 1), showing that the set of achievement causal chains in the Batusov-Soutchanski framework is a subset of the prefixes identified by the authors' counterfactual definition.
Comparison with HP: The authors compare their approach to the HP account, highlighting that while HP is powerful, it is sensitive to domain modeling choices. In contrast, the Situation Calculus approach offers a uniform standard (the BAT) but faces challenges in modeling interleaved, non-atomic competing events without "hard-coding" causal relations.

Results and Analysis

Example Analysis: Through examples (e.g., a blocks world with picking, dropping, quenching, and repairing), the authors demonstrate that their definition successfully identifies minimal prefixes that achieve goals while filtering out redundant actions.
Disjunctive Goals: In scenarios with disjunctive goals (e.g., holding object C or object D), the definition correctly identifies that if both actions are required to ensure the goal holds in the counterfactual absence of the prefix, the entire prefix is the cause. If one action alone suffices in the counterfactual, it is not the cause.
Limitations: The authors acknowledge that their definition may appear counterintuitive in cases of preemption involving non-atomic, interleaved events. For instance, if a sequence $z$ achieves a goal, and removing a prefix $z'$ makes the rest of $z$ inexecutable, the definition might include actions in $z'$ that seem redundant for the goal itself but are necessary to enable the rest of the narrative. This is a consequence of the strict counterfactual requirement that the entire remaining narrative must fail to achieve the goal.

Significance and Claims
The authors modestly claim that their account is not strictly "better" than existing formalisms but offers a simpler, counterfactual-based alternative within the Situation Calculus.

Simplicity: The definition relies on the "but-for" intuition (had the action not happened, the goal would not be true) adapted for rich action settings, avoiding the complex recursive structures of previous Situation Calculus causality definitions.
Applicability: The approach is argued to be beneficial for applications requiring causal reasoning in action languages, such as GOLOG-based robot programming.
Nuance: The paper highlights the specific nuances of applying counterfactuals to disjunctive goals, a known difficulty in actual causality definitions, and positions their joint-event approach as a reasonable solution consistent with recent HP modifications.

The paper concludes by suggesting future work in extending this account to epistemic settings (reasoning about knowledge of causes) and adapting the method for practical robot programming tasks.