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.

翻译：统计推断和因果推断中的许多任务可以被理解为在合适的形式语言中的蕴含问题。我们探究从计算角度而言，与纯概率（或“关联”）语言相比，因果概率语言是否使这些问题更加困难。尽管因果推理在表达能力和推论能力上确实更为复杂，但我们证明因果蕴含（或可满足性）问题可以系统且稳健地归约为纯概率问题。因此，计算复杂性上并不存在跃升。在此过程中，我们解答了关于已知概率逻辑复杂性的几个开放问题，特别是证明了多项式概率演算以及一个看似更简单的系统（即比较条件概率逻辑）具有$\exists\mathbb{R}$-完备性。