The Computational Complexity of Structure-Based Causality

TL;DR

This paper examines how modifications to the definition of actual causality impact the computational complexity of causality problems, introducing new complexity classes and providing a complete characterization under the updated definition.

Contribution

It introduces the family of complexity classes D_k^P and characterizes the complexity of causality computation as D_2^P-complete under the revised definition.

Findings

Complexity of causality is D_2^P-complete with the new definition.

Introduces the complexity class family D_k^P for causality analysis.

Provides a complete complexity characterization for the updated causality definition.

Abstract

Halpern and Pearl introduced a definition of actual causality; Eiter and Lukasiewicz showed that computing whether X=x is a cause of Y=y is NP-complete in binary models (where all variables can take on only two values) and\ Sigma_2^P-complete in general models. In the final version of their paper, Halpern and Pearl slightly modified the definition of actual cause, in order to deal with problems pointed by Hopkins and Pearl. As we show, this modification has a nontrivial impact on the complexity of computing actual cause. To characterize the complexity, a new family D_k^P, k= 1, 2, 3, ..., of complexity classes is introduced, which generalizes the class DP introduced by Papadimitriou and Yannakakis (DP is just D_1^P). %joe2 %We show that the complexity of computing causality is $\D_2$-complete %under the new definition. Chockler and Halpern \citeyear{CH04} extended the We show that the complexity of computing causality under the updated definition is $D_2^P$-complete. Chockler and Halpern extended the definition of causality by introducing notions of responsibility and blame. The complexity of determining the degree of responsibility and blame using the original definition of causality was completely characterized. Again, we show that changing the definition of causality affects the complexity, and completely characterize it using the updated definition.