  Ibeling et al. (2023). axiomatize increasingly expressive languages of
causation and probability, and Mosse et al. (2024) show that reasoning
(specifically the satisfiability problem) in each causal language is as
difficult, from a computational complexity perspective, as reasoning in its
merely probabilistic or "correlational" counterpart. Introducing a summation
operator to capture common devices that appear in applications -- such as the
$do$-calculus of Pearl (2009) for causal inference, which makes ample use of
marginalization -- van der Zander et al. (2023) partially extend these earlier
complexity results to causal and probabilistic languages with marginalization.
We complete this extension, fully characterizing the complexity of
probabilistic and causal reasoning with summation, demonstrating that these
again remain equally difficult. Surprisingly, allowing free variables for
random variable values results in a system that is undecidable, so long as the
ranges of these random variables are unrestricted. We finally axiomatize these
languages featuring marginalization (or more generally summation), resolving
open questions posed by Ibeling et al. (2023).
