Axiomatizing Causal Reasoning

By Joseph Y. Halpern. 1998

Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those without feedback), (2) the class of theories where the solutions to the equations are unique, (3) arbitrary theories (where the equations may not have solutions and, if they do, they are not necessarily unique). It is shown that to reason about causality in the most general third class, we must extend the language used by Galles and Pearl. In addition, the complexity of the decision procedures is examined for all the languages and classes of models considered.

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Citations in the corpus

Explanation in Artificial Intelligence: Insights from the Social Sciences (2017, background): “This is a general model that should be accessible to anyone with a computer science background, has been adopted by philosophers and psychologists, and is accompanied by many additional results, such as an axiomatisation [57] and a series articles on complexity analysis [40, 41].”
Causes and explanations: A structural-model approach (2000, methodology): “…is standard in the social sciences, and seems to go back to the work of Sewall Wright in the 1920s (see [Goldberger 1972] for a discussion); the particular framework that we use here is due to Pearl [1995], and is further developed in [Galles and Pearl 1997; Galles and Pearl 1998; Halpern 1998].”
Causal and Counterfactual Inference (2018, background): “Galles & Pearl (1998) and Halpern (1998) have given a complete axiomatization of structural counterfactuals, embracing both recursive and non-recursive models (see also Pearl (2009b, Chapter 7)).”
Complexity results for structure-based causality (2001, background): “In particular, we consider causal relationships between events due to Galles and Pearl [1997]; see also [Pearl, 1999; 2000], and the notions of weak and actual causality by Halpern and Pearl [2000], which are inspired by Pearl’s causal beams [2000].”
A Lewisian Logic of Causal Counterfactuals (2013, unknown): “To make it more precise, I shall use the following language adapted from Halpern (2000), defined over a signature S = hU, V, Ri.”
Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models (2006, background): “The authors have also been made aware of a paper currently in review [Huang & Valtorta, 2006] which shows a modiﬁed version of an algorithm found in [Tian, 2002] is complete for identifying P x ( y ) , where X , Y are sets.”
Probabilities of causation: Bounds and identification (2000, methodology): “This section presents a brief summary of the structural-equation semantics of counterfactuals as defined in [5,17,18,23].”
Settable Systems: An Extension of Pearl's Causal Model with Optimization, Equilibrium, and Learning (2009, background): “In Section 2, we give a succinct statemen of the elements of the PCM and of a generalization due to Halpern (2000) relevant for motivating and developing our settable systems extension.”
Causes of Effects and Effects of Causes (2015, background): “Letting Mx stand for a modified version of M, with the equation/equations of X replaced by X¼ x, the formal definition of the counterfactual Yx(u) reads (Balke and Pearl 1994a,1994b): YxðuÞ ¼D YMxðuÞ: ð1Þ In words, the counterfactual Yx(u) in model M is defined as the solution for Y in the ‘‘surgica”
On Pearl’s Hierarchy and the Foundations of Causal Inference (2022, background): “For a more comprehensive discussion, see Galles and Pearl [1998] and Halpern [1998, 2000].”
Causes and Explanations: A Structural-Model Approach. Part II: Explanations (2001, unknown): “See [Galles and Pearl, 1997; Halpern, 2000; Pearl, 2000] for more details, motivation, and intuition.”
Causes and explanations in the structural-model approach: Tractable cases (2002, background): “Based on a notion of weak causality, they offer appealing definitions of actual causality [ 12] and of causal explana tions [13].”
Causes and Explanations: A Structural-Model Approach. Part I: Causes (2000, methodology): “…for causal relationships is standard in the social sciences, and seems to go back to the work of Sewall Wright in the 1920s (see [Goldberger 1972] for a discussion); the framework we use here is due to Pearl [1995], and is further developed in [Galles and Pearl 1997; Halpern 2000; Pearl 2000].”
Caveats for Causal Reasoning with Equilibrium Models (2001, background): “2 Galles and Pearl [3] and subsequently Halpern [4] prove a theorem which they label”
The Causal Foundations of Structural Equation Modeling (2012, background): “(8)In other words, a complete axiomization of structural counterfactuals in recursive systems consists of (7) and a few non essential details (Halpern, 1998).”
Complete Identification Methods for the Causal Hierarchy (2008, background): “(Galles and Pearl, 1998; Halpern, 2000; Pearl, 2000).”
Foundations of structural causal models with cycles and latent variables (2016, background): “In the case of cycles, however, this property may be violated, and consequently, the SCM may not have a solution at all, or may allow for multiple different probability distributions [23].”
Probabilities Of Causation: Three Counterfactual Interpretations And Their Identification (1999, background): “One advantage of casting aspects of causation in the language of counterfactuals is that the latter enjoys natural and formal semantics in terms of structural models (Galles and Pearl 1997, 1998; Halpern 1998; Pearl 2000), as well as effective procedures for computing probabilities of counterfactual”