The Hardness of Reasoning about Probabilities and Causality

TL;DR

This paper analyzes the computational complexity of reasoning about probabilities and causality, establishing the exact complexity class for related satisfiability problems and introducing a new complexity class, succ∃R.

Contribution

It introduces the succ∃R complexity class and proves that satisfiability problems in probabilistic and causal reasoning are complete for this class, revealing stronger computational limitations.

Findings

Problems are complete for the new class succ∃R

Results imply stronger limitations than previous complexity bounds

Establishes the exact complexity of probabilistic and causal reasoning satisfiability

Abstract

We study formal languages which are capable of fully expressing quantitative probabilistic reasoning and do-calculus reasoning for causal effects, from a computational complexity perspective. We focus on satisfiability problems whose instance formulas allow expressing many tasks in probabilistic and causal inference. The main contribution of this work is establishing the exact computational complexity of these satisfiability problems. We introduce a new natural complexity class, named succ$\exists$R, which can be viewed as a succinct variant of the well-studied class $\exists$R, and show that the problems we consider are complete for succ$\exists$R. Our results imply even stronger algorithmic limitations than were proven by Fagin, Halpern, and Megiddo (1990) and Moss\'{e}, Ibeling, and Icard (2022) for some variants of the standard languages used commonly in probabilistic and causal inference.