## Axiomatizing causal reasoning (1998)

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Venue: | Uncertainty in Artificial Intelligence |

Citations: | 69 - 5 self |

### BibTeX

@INPROCEEDINGS{Halpern98axiomatizingcausal,

author = {Joseph Y. Halpern},

title = {Axiomatizing causal reasoning},

booktitle = {Uncertainty in Artificial Intelligence},

year = {1998},

pages = {202--210},

publisher = {Morgan Kaufmann}

}

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### Abstract

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 (1997, 1998). In addition, the complexity of the decision procedures is characterized for all the languages and classes of models considered. 1.

### Citations

11366 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...positively in the disjunction. To make matters simpler, assume that ' is a formula in 3-CNF. This suffices for NPhardness, since the satisfiability problem for 3-CNF formulas is also NP-hard (Garey & =-=Johnson, 1979-=-). Suppose ' is of the form c 1 : : :c m , where each c l is a clause consisting of a disjunction of three primitive propositions and their negations. Suppose that the primitive propositions that appe... |

873 | Dynamic logic
- Harel
- 1984
(Show Context)
Citation Context ... solution to the original set of equations. 2.2 Syntax I focus here on two languages. Both languages are parameterized by a signature S. The first language, L + (S), borrows ideas from dynamic logic (=-=Harel, 1979-=-). Again, I often write L + rather than L + (S) (and similarly for the other languages defined below) to simplify the notation. A basic causal formula is one of the form [Y 1 / y 1 ; : : : ; Y k / y k... |

530 | Causation, Prediction and Search - Spirtes, Glymour, et al. - 2001 |

213 | A theory of inferred causation
- Pearl, Verma
- 1991
(Show Context)
Citation Context ...tual reasoning—has been argued eloquently in a number of recent papers and books (Chajewska & Halpern, 1997; Heckerman & Shachter, 1995; Henrion & Druzdzel, 1990; Druzdzel & Simon, 1993; Pearl, 1995; =-=Pearl & Verma, 1991-=-; Spirtes, Glymour, & Scheines, 1993). If we are to reason about causality, then it is certainly useful to find axioms that characterize such reasoning. The way we go about axiomatizing causal reasoni... |

153 | Causality - Pearl - 2000 |

54 | Decision-theoretic foundations for causal reasoning
- Heckerman, Shachter
- 1995
(Show Context)
Citation Context ...roduction The important role of causal reasoning—in prediction, explanation, and counterfactual reasoning—has been argued eloquently in a number of recent papers and books (Chajewska & Halpern, 1997; =-=Heckerman & Shachter, 1995-=-; Henrion & Druzdzel, 1990; Druzdzel & Simon, 1993; Pearl, 1995; Pearl & Verma, 1991; Spirtes, Glymour, & Scheines, 1993). If we are to reason about causality, then it is certainly useful to find axio... |

52 | Axioms of Causal Relevance - Galles, Pearl - 1997 |

46 | Causality in Bayesian belief networks
- Druzdzel, Simon
- 1993
(Show Context)
Citation Context ...ediction, explanation, and counterfactual reasoning—has been argued eloquently in a number of recent papers and books (Chajewska & Halpern, 1997; Heckerman & Shachter, 1995; Henrion & Druzdzel, 1990; =-=Druzdzel & Simon, 1993-=-; Pearl, 1995; Pearl & Verma, 1991; Spirtes, Glymour, & Scheines, 1993). If we are to reason about causality, then it is certainly useful to find axioms that characterize such reasoning. The way we go... |

46 | An Axiomatic Characterization of Causal Counterfactuals
- Galles, Pearl
- 1998
(Show Context)
Citation Context ...k; they also provide a complete axiomatic characterization of reasoning about causality in this framework, under the strong assumption that there is a fixed, given causal ordering ≺ of the equations (=-=Galles & Pearl, 1998-=-). Roughly speaking, this means there is a way of ordering the variables that appear in the equations and we have explicit axioms that say Xj has no influence of Xi if Xi ≺ Xj in this causal ordering.... |

43 |
Qualitative Propagation and Scenario-based Approaches to Explanation
- Henrion, Druzdzel
- 1991
(Show Context)
Citation Context ...soning---in prediction, explanation, and counterfactual reasoning---has been argued eloquently in a number of recent papers and books (Chajewska & Halpern, 1997; Heckerman & Shachter, 1995; Henrion & =-=Druzdzel, 1990-=-; Druzdzel & Simon, 1993; Pearl, 1995; Pearl & Verma, 1991; Spirtes, Glymour, & Scheines, 1993). If we are to reason about causality, then it is certainly useful to find axioms that characterize such ... |

42 |
Structural Equation Methods in the Social Sciences." Econometrica 40
- Goldberger
- 1972
(Show Context)
Citation Context ...ausality, using structural equations. The use of structural equations as a model for causality 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 I use here is due to Pearl (1995). Galles and Pearl (1997) introduce some axioms for causal reasoning in this framework; they also provide a complete... |

23 | Defining explanation in probabilistic systems
- Chajewska, Halpern
- 1997
(Show Context)
Citation Context ...red. 1. Introduction The important role of causal reasoning---in prediction, explanation, and counterfactual reasoning---has been argued eloquently in a number of recent papers and books (Chajewska & =-=Halpern, 1997-=-; Heckerman & Shachter, 1995; Henrion & Druzdzel, 1990; Druzdzel & Simon, 1993; Pearl, 1995; Pearl & Verma, 1991; Spirtes, Glymour, & Scheines, 1993). If we are to reason about causality, then it is c... |

1 |
Axiomatizing Causal Structures
- Pearl
- 1995
(Show Context)
Citation Context ... counterfactual reasoning---has been argued eloquently in a number of recent papers and books (Chajewska & Halpern, 1997; Heckerman & Shachter, 1995; Henrion & Druzdzel, 1990; Druzdzel & Simon, 1993; =-=Pearl, 1995-=-; Pearl & Verma, 1991; Spirtes, Glymour, & Scheines, 1993). If we are to reason about causality, then it is certainly useful to find axioms that characterize such reasoning. The way we go about axioma... |

1 |
to appear
- Galles, Pearl
- 1998
(Show Context)
Citation Context ...0s (see [Goldberger 1972] for a discussion); the particular framework that I use here is due to Pearl [1995]. Galles and Pearl [1997] introduce some axioms for causal reasoning in this framework; in [=-=Galles and Pearl 1998-=-], they provide a complete axiomatic characterization of reasoning about causality in this framework, under the strong assumption that there is a fixed, given causal ordering of the equations. Roughly... |