Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for 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. (shrink)

This paper is concerned with a two-sorted probabilistic language, denoted by $\textsf{QPL}$, which contains quantifiers over events and over reals, and can be viewed as an elementary language for reasoning about probability spaces. The fragment of $\textsf{QPL}$ containing only quantifiers over reals is a variant of the well-known ‘polynomial’ language from Fagin et al. (1990, Inform. Comput., 87, 78–128). We shall prove that the $\textsf{QPL}$-theory of the Lebesgue measure on $\left [ 0, 1 \right ]$ is decidable, and moreover, all (...) atomless spaces have the same $\textsf{QPL}$-theory. Also, we shall introduce the notion of elementary invariant for $\textsf{QPL}$ and use it to translate the semantics for $\textsf{QPL}$ into the setting of elementary analysis. This will allow us to obtain further decidability results as well as to provide exact complexity upper bounds for a range of interesting undecidable theories. (shrink)