# Causal Sufficiency and Actual Causation

arXiv: https://arxiv.org/abs/2102.02311 · HTML: https://mlsanity.com/paper/2102.02311/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 27 references that are in the map.
- First posted on arXiv 3 Feb 2021.

## Builds on (most cited first, 2 of 2)

- [A Modification of the Halpern-Pearl Definition of Causality](https://mlsanity.com/paper/1505.00162/) (2015)
- [The Counterfactual NESS Definition of Causation](https://mlsanity.com/paper/2012.05123/) (2020)

## Built on by (most cited first, 29 of 29)

- [Local Explanations via Necessity and Sufficiency: Unifying Theory and Practice](https://mlsanity.com/paper/2103.14651/) (2021)
- [Causal Explanations and XAI](https://mlsanity.com/paper/2201.13169/) (2022)
- [A Causal Analysis of Harm](https://mlsanity.com/paper/2210.05327/) (2022)
- [CIPCaD-Bench: Continuous Industrial Process datasets for benchmarking Causal Discovery methods](https://mlsanity.com/paper/2208.01529/) (2022)
- [Quantifying Harm](https://mlsanity.com/paper/2209.15111/) (2022)
- [Causal Explanations for Image Classifiers](https://mlsanity.com/paper/2411.08875/) (2024)
- [A Measure-Theoretic Axiomatisation of Causality](https://mlsanity.com/paper/2305.17139/) (2023)
- [Explaining Image Classifiers](https://mlsanity.com/paper/2401.13752/) (2024)
- [Null Counterfactual Factor Interactions for Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2505.03172/) (2025)
- [Rethinking Spatio-Temporal Anomaly Detection: A Vision for Causality-Driven Cybersecurity](https://mlsanity.com/paper/2507.08177/) (2025)
- [Multiple Different Black Box Explanations for Image Classifiers](https://mlsanity.com/paper/2309.14309/) (2023)
- [Automated Discovery of Functional Actual Causes in Complex Environments](https://mlsanity.com/paper/2404.10883/) (2024)
- [Sufficient, Necessary and Complete Causal Explanations in Image Classification](https://mlsanity.com/paper/2507.23497/) (2025)
- [Unifying Invariance and Spuriousity for Graph Out-of-Distribution via Probability of Necessity and Sufficiency](https://mlsanity.com/paper/2402.09165/) (2024)
- [Moral Responsibility for AI Systems](https://mlsanity.com/paper/2310.18040/) (2023)
- [Nondeterministic Causal Models](https://mlsanity.com/paper/2405.14001/) (2024)
- [Unifying Invariant and Variant Features for Graph Out-of-Distribution via Probability of Necessity and Sufficiency](https://mlsanity.com/paper/2407.15273/) (2024)
- [Mathematical Explanations](https://mlsanity.com/paper/2402.09413/) (2023)
- [Algorithmic Identification of Essential Exogenous Nodes for Causal Sufficiency in Brain Networks](https://mlsanity.com/paper/2403.05407/) (2024)
- [Searching for Actual Causes: Approximate Algorithms with Adjustable Precision](https://mlsanity.com/paper/2507.07857/) (2025)
- [Rethinking Chronological Causal Discovery with Signal Processing](https://mlsanity.com/paper/2602.19903/) (2026)
- [Integrating Temporality and Causality into Acyclic Argumentation Frameworks using a Transition System](https://mlsanity.com/paper/2303.09197/) (2023)
- [A Dynamical View of the Question of Why](https://mlsanity.com/paper/2402.10240/) (2024)
- [Learning Dynamic Bayesian Networks from Data: Foundations, First Principles and Numerical Comparisons](https://mlsanity.com/paper/2406.17585/) (2024)
- [Actual Causation and Nondeterministic Causal Models](https://mlsanity.com/paper/2503.07849/) (2025)
- [Evaluation of Black-Box XAI Approaches for Predictors of Values of Boolean Formulae](https://mlsanity.com/paper/2509.09982/) (2025)
- [Activation-Deactivation: A General Framework for Robust Post-hoc Explainable AI](https://mlsanity.com/paper/2510.01038/) (2025)
- [Formalizing and falsifying causal pathways of rare events](https://mlsanity.com/paper/2605.31254/) (2026)
- [Difference-Making without Making a Difference](https://mlsanity.com/paper/2606.24832/) (2026)

## Abstract (arXiv)

Pearl opened the door to formally defining actual causation using causal models. His approach rests on two strategies: first, capturing the widespread intuition that X=x causes Y=y iff X=x is a Necessary Element of a Sufficient Set for Y=y, and second, showing that his definition gives intuitive answers on a wide set of problem cases. This inspired dozens of variations of his definition of actual causation, the most prominent of which are due to Halpern & Pearl. Yet all of them ignore Pearl's first strategy, and the second strategy taken by itself is unable to deliver a consensus. This paper offers a way out by going back to the first strategy: it offers six formal definitions of causal sufficiency and two interpretations of necessity. Combining the two gives twelve new definitions of actual causation. Several interesting results about these definitions and their relation to the various Halpern & Pearl definitions are presented. Afterwards the second strategy is evaluated as well. In order to maximize neutrality, the paper relies mostly on the examples and intuitions of Halpern & Pearl. One definition comes out as being superior to all others, and is therefore suggested as a new definition of actual causation.