# On Probabilistic and Causal Reasoning with Summation Operators

arXiv: https://arxiv.org/abs/2405.03069 · HTML: https://mlsanity.com/paper/2405.03069/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Causal reasoning in LLMs, one of 17 directions in LLM reasoning.
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 5 May 2024.

## Placed in

- LLM reasoning / [Causal reasoning in LLMs](https://mlsanity.com/lr/causal-reasoning-in-llms/)

## Builds on (most cited first, 9 of 9)

- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Probabilistic Reasoning across the Causal Hierarchy](https://mlsanity.com/paper/2001.02889/) (2020)
- [Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions](https://mlsanity.com/paper/2306.14351/) (2023)
- [Smoothing the gap between NP and ER](https://mlsanity.com/paper/1912.02278/) (2019)
- [A Topological Perspective on Causal Inference](https://mlsanity.com/paper/2107.08558/) (2021)
- [Is Causal Reasoning Harder than Probabilistic Reasoning?](https://mlsanity.com/paper/2111.13936/) (2021)
- [The Hardness of Reasoning about Probabilities and Causality](https://mlsanity.com/paper/2305.09508/) (2023)
- [Probing the Quantitative-Qualitative Divide in Probabilistic Reasoning](https://mlsanity.com/paper/2307.05659/) (2023)
- [Logics with probabilistic team semantics and the Boolean negation](https://mlsanity.com/paper/2306.00420/) (2023)

## Built on by (most cited first, 2 of 2)

- [Probabilistic and Causal Satisfiability: Constraining the Model](https://mlsanity.com/paper/2504.19944/) (2025)
- [Verifying formulas for interventional distributions](https://mlsanity.com/paper/2607.13883/) (2026)

## Abstract (arXiv)

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).