# Complexity Results for Structure-Based Causality

Artificial Intelligence, no. 1 (2002): 53-89

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Abstract

We analyze the computational complexity of causal relationships in Pearl's structural models, where we focus on causality between variables, event causal- ity, and probabilistic causality. In particular, we an- alyze the complexity of the sophisticated notions of weak and actual causality by Halpern and Pearl. In the course of this, we al...More

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Introduction

- Representing and reasoning with causal knowledge has received much attention in the recent decade.
- The existing approaches to causality in the AI literature can be roughly divided into those that have been developed as modal nonmonotonic logics and those that evolved from the area of Bayesian networks.
- A representative of the former is Geffner’s modal nonmonotonic logic for handling causal knowledge [3, 4], which has been inspired by default reasoning from conditional knowledge bases.
- The evaluation of deterministic and probabilistic counterfactuals has been explored, which is at the core of problems in fault diagnosis, planning, decision making, and determination of liability [1]

Highlights

- Representing and reasoning with causal knowledge has received much attention in the recent decade
- The existing approaches to causality in the AI literature can be roughly divided into those that have been developed as modal nonmonotonic logics and those that evolved from the area of Bayesian networks
- A representative of the former is Geffner’s modal nonmonotonic logic for handling causal knowledge [3, 4], which has been inspired by default reasoning from conditional knowledge bases
- Other more specialized formalisms play an important role in dealing with causal knowledge about actions and change; see especially the work by Turner [13] and the references therein for an overview
- The evaluation of deterministic and probabilistic counterfactuals has been explored, which is at the core of problems in fault diagnosis, planning, decision making, and determination of liability [1]
- Note that the evaluation of probabilistic counterfactuals can be polynomially reduced to standard inference tasks in Bayesian networks, and has similar computational properties

Results

- The authors' complexity results for checking the above notions of causality are summarized in Table 1.
- It is important to point out that for all these causal relationships, hardness holds even if Å is binary and bounded, and is a singleton.
- The authors' first result shows that deciding causal irrelevance is Ó-ÆÈ-complete.
- Theorem 3.2 Given ÅÍ Î μ and.
- Deciding whether is causally irrelevant to given is Ó-ÆÈ-complete.
- Hardness holds even if (1) Å is binary and bounded, (2) is empty,

Conclusion

**Conclusion and Outlook**

The authors analyzed the complexity of causal relationships in Pearl’s structural models.- Similar to independencies [10], deterministic and probabilistic causal relationships might be used to simplify the evaluation of probabilistic counterfactuals.
- By the results, this seems reasonable, as the complexity of testing simple causal relationships is much lower than the complexity of evaluating probabilistic counterfactuals ( È-hard)

- Table1: Complexity of Causality between Variables is causally irrelevant to given
- Table2: Complexity of Event Causality

Funding

- This work has been partially supported by the Austrian Science Fund under project N Z29-INF and a DFG grant

Reference

- A. Balke and J. Pearl. Probabilistic evaluation of counterfactual queries. In Proceedings AAAI-94, pages 230–237, 1994.
- D. Galles and J. Pearl. Axioms of causal relevance. Artif. Intell., 97:9–43, 1997.
- H. Geffner. Causal theories for nonmonotonic reasoning. In Proc. AAAI-90, pages 524–530, 1990.
- H. Geffner. Default Reasoning: Causal and Conditional Theories. MIT Press, 1992.
- F. Green. On the power of deterministic reductions to È. Math. Syst. Theory, 26(2):215–233, 1993.
- J. Y. Halpern. Axiomatizing causal reasoning. J. Artif. Intell. Res., 12:317–337, 2000.
- J. Y. Halpern and J. Pearl. Causes and explanations: A structural-model approach. Technical Report R-266, UCLA Cognitive Systems Lab, 2000.
- C. H. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, MA, 1994.
- J. Pearl. Reasoning with cause and effect. In Proceedings IJCAI-99, pages 1437–1449, 1999.
- J. Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, 2000.
- D. Roth. On the hardness of approximate reasoning. Artif. Intell., 82(1–2):273–302, 1996.
- J. Toran. Complexity classes defined by counting quantifiers. J. ACM, 38(3):753–774, 1991.
- H. Turner. A logic of universal causation. Artif. Intell., 113:87–123, 1999.

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