Is Causal Reasoning Harder than Probabilistic Reasoning?

TL;DR

This paper investigates whether causal reasoning is computationally harder than probabilistic reasoning, finding that causal entailment problems can be reduced to probabilistic ones, thus sharing similar complexity levels.

Contribution

It demonstrates that causal entailment problems are computationally no harder than probabilistic ones by providing systematic reductions, and resolves open questions about the complexity of certain probability logics.

Findings

Causal entailment reduces to probabilistic problems without increased complexity.

Proves the $orall ext{R}$-completeness of a polynomial probability calculus.

Shows the logic of comparative conditional probability is computationally simpler.

Abstract

Many tasks in statistical and causal inference can be construed as problems of \emph{entailment} in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for \emph{causal} probabilistic languages than for pure probabilistic (or "associational") languages. Despite several senses in which causal reasoning is indeed more complex -- both expressively and inferentially -- we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. Along the way we answer several open problems concerning the complexity of well known probability logics, in particular demonstrating the $\exists\mathbb{R}$-completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability.