# Unveiling the Structure of Do-Calculus Reasoning via Derivation Graphs

arXiv: https://arxiv.org/abs/2606.03719 · HTML: https://mlsanity.com/paper/2606.03719/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 2 Jun 2026.

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

- [On the Logic of Causal Models](https://mlsanity.com/paper/1304.2355/) (2013)
- [Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables](https://mlsanity.com/paper/2102.10324/) (2021)
- [DiscoBAX: Discovery of Optimal Intervention Sets in Genomic Experiment Design](https://mlsanity.com/paper/2312.04064/) (2023)
- [Enhancing Identification of Causal Effects by Pruning](https://mlsanity.com/paper/1806.07085/) (2018)
- [Simplifying Probabilistic Expressions in Causal Inference](https://mlsanity.com/paper/1806.07082/) (2018)
- [Efficient adjustment for complex covariates: Gaining efficiency with DOPE](https://mlsanity.com/paper/2402.12980/) (2024)
- [Causal Inference with the Napkin Graph](https://mlsanity.com/paper/2512.19861/) (2025)
- [Efficient Real-world Testing of Causal Decision Making via Bayesian Experimental Design for Contextual Optimisation](https://mlsanity.com/paper/2207.05250/) (2022)
- [A note on efficient minimum cost adjustment sets in causal graphical models](https://mlsanity.com/paper/2201.02037/) (2022)

## Built on by (most cited first, 3 of 3)

- [Verifying formulas for interventional distributions](https://mlsanity.com/paper/2607.13883/) (2026)
- [A cubical formalisation of topos causal models: intervention, forcing, and a contextuality obstruction](https://mlsanity.com/paper/2607.15629/) (2026)
- [Causal Effect Identifiability in Causal Abstractions: A Strict Hierarchy of Criteria](https://mlsanity.com/paper/2610.13520/) (2026)

## Abstract (arXiv)

The do-calculus defines a general system of inference for interventional queries, allowing causal quantities to be transformed through successive applications of its rules. This process induces a rich space of equivalent interventional expressions, but combining and ordering these rules remains challenging. In this work, we introduce derivation graphs, which represent how do-calculus rules are applied and combined, and characterize the full space of observational and interventional probabilities which are equivalent under the do-calculus. The structure of these graphs yields a simple procedure that uses at most four applications of do-calculus rules. Finally, we show how applying identification algorithms to equivalent causal queries produces multiple valid estimands for the same causal quantity, eventually yielding more efficient estimators.