A Counterfactual Cause in Situation Calculus

Summary

This partial ingest is based on the extracted full PDF text. Liu and Belle propose a counterfactual notion of cause in a modal situation-calculus variant, focusing on action sequences that achieve a goal. The paper contrasts minimal counterfactual causes, narrative-based achievement causes, and Batusov-Soutchanski achievement causes, showing that the new counterfactual achievement-cause sequence contains the Batusov-Soutchanski causal chain while making disjunctive-goal and preemption tradeoffs explicit.

Key Claims

• Existing achievement-cause accounts in situation calculus can identify causes of quantified effects, but are not always presented in a directly counterfactual way.
• A simple counterfactual cause can be defined as a minimal action sequence that changes an initially false goal into a true one.
• In a known narrative, the relevant cause is a minimal prefix subsequence whose removal, after filtering out newly inexecutable actions, prevents the goal.
• This account treats some preemption and disjunctive-goal cases by including all competing events in the cause, aligning with the idea that individual events may be parts of a joint cause.
• The Batusov-Soutchanski achievement causal chain is contained in the prefix chain generated by the paper’s counterfactual achievement cause.

Methods / Formalism

• The paper works in a modal situation-calculus logic ES with formulas [t]phi for “phi after action t” and Box phi for “phi after any action sequence.”
• A basic action theory Sigma contains initial-state axioms, action-precondition axioms, and successor-state axioms.
• Minimal cause for a static goal phi is defined by:

$$\Sigma \models \neg \varphi ,\Sigma \models [z]\varphi ,$$

with z minimal under a chosen ordering such as length, affected fluents, or both.

• For narrative achievement cause, a causal setting is C=<Sigma,z,phi> with Sigma |= exec(z) and [z]phi. A prefix z' is a cause when all extensions from z' to z maintain phi, while Filter(z\z') makes not phi hold, and no smaller subsequence satisfies both conditions.
• Situation Calculus Causal Models collects the shared action-history causal machinery across this paper and Hopkins2005 - Causality and Counterfactuals in the Situation Calculus.

Evidence / Experiments

• The paper is formal and example-driven.
• Blocks-world examples illustrate length-based, fluent-based, and plan-and-effect minimality.
• Narrative examples show how filtering the remainder of an action history prevents removed prefixes from leaving impossible actions in the counterfactual sequence.
• The paper proves that the Batusov-Soutchanski achievement causal chain is a subset of the chain induced by the proposed counterfactual achievement cause.

Connections

• Extends Actual Causality into reasoning about actions and narratives.
• Connects directly to Situation Calculus through regression, executability, and basic action theories.
• Complements Hopkins2005 - Causality and Counterfactuals in the Situation Calculus, which proposed SCCMs as a causal-model framework but did not settle an actual-cause definition.
• Related to Structural Equation Models through comparison with Halpern-Pearl actual causality.

Open Questions

• The paper intentionally drops some features such as indirect cause and preemption-sensitive refinements, so its causes may be coarser than richer actual-cause accounts.
• How should minimality be chosen in domains where action length, affected fluents, and cost are all meaningful?
• Can the narrative cause definition be extended to concurrent actions or partially observed histories?

Citation

Liu, D., and Belle, V. (2026). A Counterfactual Cause in Situation Calculus. arXiv:2501.06857v2.