On Probabilistic and Causal Reasoning with Summation Operators
Duligur Ibeling, Thomas F. Icard, Milan Mossé
TL;DR
This work develops a probabilistic and causal logic framework that explicitly incorporates a summation (marginalization) operator and analyzes the resulting expressivity. It shows that probabilistic and causal reasoning with summation maintain equal computational hardness, extending prior results to a Level 3 fragment and to cases with free variables, where unrestricted ranges yield undecidability. The authors provide infinitary axiomatizations and prove strong completeness for several closed and finite-range fragments, while also proving decidability boundaries via succExistsR reductions and recursive enumerability results. These findings clarify fundamental limits for causal identification and do-calculus-like reasoning in probability logics and lay groundwork for future work on integration and abstraction in causal probabilistic frameworks.
Abstract
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).