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A Modification of the Halpern-Pearl Definition of Causality
Joseph Y. Halpern
TL;DR
Defining actual causality is difficult because counterfactual but-for reasoning fails in cases such as preemption. This paper modifies the Halpern-Pearl definition by restricting contingencies to actual variable values, obtaining a simpler definition that handles standard examples and has lower complexity.
Problem
Finding a good definition of actual causality remains difficult because the but-for test fails in preemption cases where an alternative event would also produce the outcome.
Method
The paper restricts HP contingencies so variables other than putative causes can only retain their actual values or be frozen there.
Results
The modified definition handles standard counterexamples, agrees with the other HP definitions on but-for causes, and has Δp complexity for determining causality.
Takeaways & Limitations
The modified HP definition is conceptually simpler and can give reasonable results on standard causality examples without adding variables to the model in some cases.
Takeaways & Limitations
The definition is not established as the uniquely right account of actual causality, and some cases still benefit from normality considerations.
Abstract
from arXiv · showhide
The original Halpern-Pearl definition of causality [Halpern and Pearl, 2001] was updated in the journal version of the paper [Halpern and Pearl, 2005] to deal with some problems pointed out by Hopkins and Pearl [2003]. Here the definition is modified yet again, in a way that (a) leads to a simpler definition, (b) handles the problems pointed out by Hopkins and Pearl, and many others, (c) gives reasonable answers (that agree with those of the original and updated definition) in the standard problematic examples of causality, and (d) has lower complexity than either the original or updated definitions.
1 Introduction
Actual causality concerns particular events, but defining it rigorously is difficult because the standard but-for test fails in preemption cases. The paper proposes a stricter modification of the HP definition to address these problems more simply and with lower complexity.
- Actual causality focuses on particular events rather than general causal relationships such as smoking causing lung cancer.
- Finding a good definition of actual causality is notoriously difficult despite its importance in science, law, and everyday reasoning.
- The but-for test says A causes B if B would not have occurred without A, but it fails in preemption cases.
- In the Suzy-Billy example, both rocks would have shattered the bottle, yet Suzy’s throw is intended to count as a cause because it arrived first.
- The original HP approach uses structural equations and contingencies, but it can incorrectly classify Billy’s throw as a cause unless additional variables and restrictions are introduced.
- The paper restricts contingencies so variables other than putative causes retain actual values or are frozen there, yielding a simpler definition with Δp causality-computation complexity.
2 The HP definition(s) and the modified definition
The HP framework represents causal relationships with structural equations in recursive models and defines causality through three conditions, with AC2 doing the substantive work. The modification restricts contingencies to actual variable values, making the definition simpler while preserving agreement in but-for cases and implying the original and updated definitions.
- Causal structures: A causal model consists of a signature listing variables and values, together with modifiable structural equations relating those variables.
- Causal structures: The paper restricts attention to recursive, or acyclic, models, which have no feedback and yield a unique solution given an exogenous context.
- Causal formulas: Causal formulas describe what would hold under interventions that set selected endogenous variables to specified values.
- Causality conditions: All three HP variants use AC1, AC2, and AC3; AC2 is the core condition, while AC1 requires actual occurrence and AC3 enforces minimality.
- Causality conditions: The original and updated definitions use contingencies and sufficiency conditions to handle cases where the basic but-for condition alone misclassifies causes.
- Modified definition: The modified AC2 permits only actual values for contingency variables, so AC2(bu) and AC2(b) follow from AC1 and modified AC2(a), eliminating the need to mention Z.
- Properties: But-for causes remain causes under all three HP variants, while any cause component under the modified definition is also a cause under the original and updated definitions.
3 Examples
The examples show how the modified HP definition restricts contingencies to produce simpler and more intuitive causal judgments, while richer models can represent causal mechanisms and distinguish parts of causes.
- Conjunctive and disjunctive causes: In conjunctive scenarios, all three definitions agree that each jointly necessary factor is a but-for cause, allowing multiple causes of one effect.For forest fire, lightning and arson are both causes because each is individually necessary when the other is held fixed.
- Conjunctive and disjunctive causes: The modified definition treats L = 1 and MD = 1 as parts of a joint cause of FF = 1 rather than separate causes.Both variables must change to change the forest-fire outcome under the relevant model.
- Conjunctive and disjunctive causes: The modified definition can distinguish causal structure in conjunctive and disjunctive cases, although the original and modified definitions may also call individual variables causes.The paper presents this distinction as a feature of the modified definition while noting an alternative view about whether parts should count as causes.
- Rock-throwing example: In the rock-throwing example, the modified definition concludes that Suzy’s throw is a cause but Billy’s is not, using a model that represents their asymmetric roles.The naive model does not distinguish the two throws; the richer model introduces hit variables and analyzes a contingency with Billy’s throw fixed.
- Prisoner example: For the prisoner example, the modified definition rejects A = 1 as a cause because no allowed contingency can hold variables fixed while making A = 0 change the death outcome.C = 1 remains a but-for cause, whereas the original definition could make A a cause by changing B and C.
- Bogus prevention: The modified definition rejects Bodyguard’s antidote as a cause of survival in the bogus-prevention example when other variables remain at their actual values.It nevertheless treats the antidote as part of a cause, and normality considerations can exclude the abnormal witness world.
- Lamp example and causal mechanisms: The modified definition rejects A = 1 in the lamp example because the contingency making A causally relevant cannot be considered, while richer models can distinguish alternative causal stories.It still identifies B = −1 and C = −1 as causes and agrees with richer-model judgments when mechanism variables are added.
- Lamp example and causal mechanisms: The modified definition handles Hall’s nonexistent-threat example appropriately without requiring additional variables, according to the paper.This extends the examples in which restricting contingencies avoids judgments produced by the original and updated definitions.
4 Comparison to other approaches
The modified HP definition is compared with causal-beam, H-account, and path-based approaches, emphasizing how restrictions on contingencies affect causal judgments.
- Comparison with HP definitions: The modified definition requires contingencies in which variables retain their initial values, distinguishing it from the original and updated HP definitions.This restriction clarifies when the sufficiency condition is needed.
- Causal beam: The causal-beam definition requires a condition like AC2(am), distinguishing actual causes from contributory causes when the but-for condition fails.The paper notes that causal beam was abandoned because of problems.
- Path-based definitions: The H-account and Hitchcock’s definition use causal paths, but the H-account is too strong in voting examples with an intermediate vote-count variable.Neither individual affirmative vote qualifies as a cause under the H-account in the example.
- Voting example: In the voting example, the original and updated HP definitions treat each affirmative vote as a cause, while the modified definition treats their conjunction as a cause.The causal-beam definition treats neither individual vote as an actual or contributory cause.
- Hitchcock’s definition: Hitchcock’s path reduction replaces off-path variables’ equations with their actual values before evaluating whether the putative cause is a but-for cause.The reduction is defined relative to a causal path from the cause to the outcome.
- Comparison examples: In the conjunctive forest-fire scenario, all three HP definitions agree that lightning and arson are causes because each is individually but-for.This example also permits multiple causes of one effect.
5 The complexity of determining causality
The paper analyzes the computational complexity of causality under the modified HP definition and shows substantial reductions for general and singleton causes.
- Prior complexity results: Under the original HP definition, causes can always be single conjuncts, yielding ΣP_2-complete causality and making AC3 vacuous.The updated definition instead permits nonsingleton causes and has DP-complete complexity.
- General causes: The modified definition’s causality complexity is DP-complete, while its AC2 component drops from ΣP_2 to NP and AC3 drops from ΠP_2 to co-NP.The analysis separates the AC2 and AC3 languages and combines them through intersection.
- Proof strategy: The proof places AC2(am) in NP by guessing contingencies and checking the resulting intervention, while AC3 is placed in co-NP by checking for counterexamples.Hardness follows from reductions from satisfiability for AC2 and unsatisfiability for AC3.
- Singleton causes: For singleton causes, causality is NP-complete because minimality AC3 holds vacuously.The paper states that AC2(am) remains NP-hard even when only singleton causes are considered.
6 Conclusion
The modified HP definition makes a relatively small change that simplifies the framework and performs well on many standard counterexamples, while not establishing a uniquely correct account of causality.
- Conclusion: The modified definition is conceptually and computationally simpler than the original and updated HP definitions.The paper characterizes the change as relatively small but potentially significant in its effects.
- Conclusion: The modified definition gives reasonable results on many standard counterexamples, especially when combined with normality, responsibility, and blame.The paper presents this performance as evidence of usefulness, not proof that the definition is correct.
- Open questions: The persistence of counterexamples suggests that actual causality and its interaction with normality, responsibility, and blame remain open for further study.The conclusion explicitly rejects treating the modified definition as definitively correct.
A Proof of Theorem 2.3
Theorem 2.3 shows that any conjunct of a cause under the modified Halpern-Pearl definition is itself a cause under both the original and updated definitions. The proof constructs witnesses and uses minimality to establish the required conditions.
- Appendix framing: The appendix explicitly identifies its purpose as proving Theorem 2.3 and repeats the theorem statement for convenience.The proof concludes that X = x is a cause according to the original definition after establishing AC2(b).
- Theorem statement: Theorem 2.3 states that if X = x is part of a cause under the modified HP definition, then X = x is a cause under both the original and updated definitions.The proof begins with a cause vector containing X = x and establishes the claim for that conjunct.
- Proof for the original definition: For a singleton cause, the witness for the original definition is immediate; for a conjunct of a larger cause, the proof isolates it as X1 = x1.The remaining components are denoted by X−1 and handled through a constructed witness.
- Proof for the original definition: The constructed witness satisfies AC2(a), while AC3 holds because minimality excludes the possibility that the remaining conjuncts alone satisfy AC2(am).Assuming AC2(b) fails would imply that X−1 alone satisfies AC2(am), contradicting minimality under the modified definition.
- Proof for the original definition: If a subset of variables in Z changes under the intervention and restoring its original values makes ϕ false, then the full conjunction violates AC3 under the modified definition.That subset provides a witness showing that the remaining variables satisfy AC2(am), so the original cause cannot fail AC2(b).
- Proof for the updated definition: The updated-definition argument is similar, additionally checking AC2(bu) for subsets of X−1, W, and Z.The proof notes that AC1 establishes the relevant condition for the empty subset, while failure for a strict nonempty subset contradicts AC3 under the modified definition.