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arxiv: 2606.03719 · v1 · pith:CUVSMCO6new · submitted 2026-06-02 · 💻 cs.AI

Unveiling the Structure of Do-Calculus Reasoning via Derivation Graphs

Cl\'ement Yvernes , Emilie Devijver , Marianne Clausel , Eric Gaussier This is my paper

Pith reviewed 2026-06-28 09:58 UTC · model grok-4.3

classification 💻 cs.AI
keywords do-calculuscausal identificationderivation graphsinterventional queriesestimandsobservational equivalencecausal inference
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The pith

Derivation graphs show that do-calculus can transform any interventional query into an equivalent observational expression with at most four rule applications.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines derivation graphs as structures that track every possible way do-calculus rules can be sequenced and combined to rewrite interventional probabilities as observational ones. These graphs map the entire space of expressions that are equivalent under the do-calculus for a given causal query. Their structure produces a bounded procedure that never needs more than four rule applications. The same graphs also generate multiple distinct but valid estimands for one causal quantity when identification algorithms are run on the equivalent expressions.

Core claim

Derivation graphs represent how do-calculus rules are applied and combined, and they 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. Applying identification algorithms to equivalent causal queries produces multiple valid estimands for the same causal quantity, eventually yielding more efficient estimators.

What carries the argument

Derivation graphs, which enumerate sequences of do-calculus rule applications to rewrite interventional expressions as observational ones.

If this is right

  • Any identification of an interventional query can be completed in four or fewer rule steps.
  • Running identification on each equivalent expression produces a set of distinct but correct estimands for the same target quantity.
  • The collection of estimands can be combined to produce a single estimator with lower variance than any one of them alone.
  • The graphs give a complete map of all observational expressions that the do-calculus can derive from a given interventional query.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Automated causal-identification software could limit its search to depth four without losing completeness for the queries covered by the graphs.
  • Having several independent estimands for one quantity supplies a practical way to cross-check numerical results or to detect model misspecification.
  • The same graph-construction idea might be applied to other rule-based causal systems to obtain similar bounded rewriting procedures.

Load-bearing premise

The graphs list every valid sequence of rule applications without missing any or including redundant ones, so the four-application bound holds for all cases considered.

What would settle it

A concrete causal query for which every valid identification path requires five or more do-calculus rule applications, or for which the graphs omit at least one valid sequence.

Figures

Figures reproduced from arXiv: 2606.03719 by Cl\'ement Yvernes, Emilie Devijver, Eric Gaussier, Marianne Clausel.

Figure 1
Figure 1. Figure 1: A causal diagram with 4 variables used in Example 1. 1 arXiv:2606.03719v1 [cs.AI] 2 Jun 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Derivation graph for the ADMG G0 over variables A, B, and C with no edges. Blue nodes denote expressions involving interventions, while orange nodes denote observational (do-free) expressions. Edges represent atomic applications of the do-calculus rules (one variable at a time): gray solid lines for Rule R1, orange dashed lines for Rule R2, and blue dotted lines for Rule R3. Interventions, Do-Operator and … view at source ↗
Figure 3
Figure 3. Figure 3: Derivation graph for the ADMG G : A → B → C. Blue nodes denote expressions involving interventions, while orange nodes denote observational (do-free) expressions. Edges represent atomic applications of the do-calculus rules (one variable at a time): gray solid lines for Rule R1, orange dashed lines for Rule R2, and blue dotted lines for Rule R3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The Napkin Graph (top) and its derivation graph re￾stricted to the connected component of P(y | do(x)) (bottom). In the derivation graph, blue nodes denote expressions involving in￾terventions. Edges represent atomic applications of the do-calculus rules (one variable at a time): gray solid lines for Rule R1, orange dashed lines for Rule R2, and blue dotted lines for Rule R3. paths. This highlights the nee… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of two estimators associated with two equiv￾alent identification formulae over 1000 runs. We then estimate E(Y | do(w,z)): E(Y | do(w,z)) = Z y y Z x f(y | w,z, x)f(x | w,z)f(w)dxdy = Z x f(x | w,z)f(w)(eγ + eαx + eβw +eδz)dx = eγ + eαE(X) + eβE(W) +eδ(γ ′ + α ′X + β ′w), where we fit two regression models, E(Y | x,w,z) = eγ+eαx+ eβw +eδz and E(Z | x,w) = γ ′ + α ′ x + β ′w. The total effect of … view at source ↗
Figure 6
Figure 6. Figure 6: Graphs in the Proof of Theorem A. G ⋆ is shown in black. G △ is obtained by adding the path πZ in red. Wavy edges denote arbitrary paths and wavy directed edges denote directed paths. Each Ci is a collider on π. The paths πCi are pairwise distinct, intersect π only at Ci , and are disjoint from πZ. Moreover, πZ intersects π only at Z. Third step. Check for each rule that Q1M , Q2M. When working in the matr… view at source ↗
Figure 7
Figure 7. Figure 7: Portion of the derivation graph of the Napkin graph. B.2. Redundancy of R1 The three rules R1, R2, R3 have a simple visualization on the derivation graph, meaning that redundancy of R1 can be visualised on the triangle in [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Graphical conditions of Rule 1 applications. Each node represents an expression of the form P(Y | do(X), do(·),W, ·), and each edge corresponds to a valid application of a do-calculus rule under the indicated independence condition. Theorem 2. Let G = (VG, EG) be an ADMG, and let Y, X, W, Z be pairwise disjoint subsets of VG, with Y , ∅. Consider an expression of the form P(y | do(x), w). Then the followin… view at source ↗
Figure 9
Figure 9. Figure 9: represents all the possible rules. Definition A (Mutilated Graphs for Rule 2 Transitions). Let G be an ADMG over V and let Y, X,W,Z,Ze be pairwise disjoint subsets of V. We define the following mutilated graphs used to apply Rule 2 of the do-calculus: Ga B GXZZe, Gb B GX Z, Gc B GX Ze, Gd B GXZ Ze , Ge B GX ZZe. P(y | do(x), do(z,ez), w) P(y | do(x), do(z), w,ez) P(y | do(x), do(ez), w, z) P(y | do(x), w, … view at source ↗
Figure 10
Figure 10. Figure 10: represents all the possible rules. Definition B (Mutilated Graphs for Rule 3 Transitions). Let G = (VG, EG) be an ADMG, and let Y, X, W, Z,Ze be pairwise disjoint subsets of VG, with Y , ∅. We define the following mutilated graphs used to apply Rule 3 of the do-calculus: Ga B GXZZe\Anc(W,GXZ) , Gb B GX Z\Anc(W,GX ) , Gc B GX Ze\Anc(W,GX ) , Gd B GXZe Z\Anc(W,G XZe ) , Ge B GX (ZZe)\Anc(W,GX ) . P(y | do(x… view at source ↗
Figure 11
Figure 11. Figure 11: represents all the possible rules. Definition C (Mutilated Graphs for Rule 2 and Rule 3 Transitions). Let G be an ADMG over V and let Y, X,W,Z,Ze be pairwise disjoint subsets of V. We define the following mutilated graphs used to apply Rule 2 and Rule 3 of the do-calculus: Ga B GXZZe\Anc(W,GXZ) , Gb B GX Z, Gc B GX Ze\Anc(W∪Z,GX ) , Gd B GXZ Ze . P(y | do(x), do(z,ez), w) P(y | do(x), do(z), w) P(y | do(x… view at source ↗
Figure 12
Figure 12. Figure 12: The causal graph (left) and its derivation graph restricted to the connected component of P(a | do(b)) (right). Blue nodes denote expressions involving interventions, while orange nodes denote observational (do-free) expressions. Edges represent atomic applications of the do-calculus rules (one variable at a time): gray solid lines for Rule R1, orange dashed lines for Rule R2, and blue dotted lines for Ru… view at source ↗
Figure 13
Figure 13. Figure 13: A causal diagram over four variables, used in Example 1, and the connected component of the expression P(y | do(z)) in the derivation graph. We generate data with the following process: the vector (W, Z, X, Y) is Gaussian, and the relations are linear, with Latent confounder: U ∼ N(0, 1) W = aUW U + ϵW , ϵW ∼ N(0, 1), aUW = 1.0 Mediator 1: Z = aWZ W + ϵZ, ϵZ ∼ N(0, 1), aWZ = 1.5 Mediator 2: X = aZX Z + ϵX… view at source ↗
Figure 14
Figure 14. Figure 14: The DAG in Sachs et al. (2005a) biologically validated [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Causal graph (top) and connected component of P(y | do(x)) in its derivation graph (bottom). We list all identification formulae obtained by applying the ID algorithm (via the R package causaleffect) to all equivalent interventional queries within the same connected component of the derivation graph. This yields eight distinct formulae, each exhibiting different statistical properties. P(y | do(x)) = P(y … view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces derivation graphs to represent sequences of do-calculus rule applications, claims these graphs fully characterize the space of equivalent interventional and observational expressions under the do-calculus, derives from their structure a simple procedure requiring at most four rule applications, and shows that identification algorithms applied to equivalent queries can produce multiple valid estimands for the same causal quantity, potentially yielding more efficient estimators.

Significance. If the derivation graphs are shown to be exhaustive and the four-application bound is tight and complete, the work would offer a structural tool for systematically enumerating do-calculus derivations and exploiting equivalence classes for estimator efficiency in causal inference.

major comments (2)
  1. [Abstract] The central claim of a procedure using at most four applications rests on the assertion that derivation graphs exhaustively enumerate all valid sequences of the three do-calculus rules without omission or duplication across equivalence classes. The abstract provides no indication of a completeness argument (e.g., induction on rule applications or exhaustive enumeration of sequences up to length 4), leaving the bound dependent on an unstated assumption.
  2. [Abstract (central claim on derivation graphs)] The weakest assumption identified—that the graph-construction procedure enumerates every admissible ordering—directly undermines the 'simple procedure' and 'full space' characterization if even one sequence is missed; this must be addressed with a formal proof or verification before the bound can be accepted as general.
minor comments (2)
  1. Define 'derivation graph' and its construction algorithm with explicit pseudocode or a small worked example in the main text rather than deferring to supplementary material.
  2. Clarify whether the four-application bound applies only to certain equivalence classes or holds for arbitrary interventional queries; state any restrictions explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need to make the completeness argument more visible. The manuscript contains a formal proof of exhaustiveness for derivation graphs (via induction on rule applications), but we agree the abstract does not reference it explicitly. We will revise the abstract accordingly.

read point-by-point responses

  1. Referee: [Abstract] The central claim of a procedure using at most four applications rests on the assertion that derivation graphs exhaustively enumerate all valid sequences of the three do-calculus rules without omission or duplication across equivalence classes. The abstract provides no indication of a completeness argument (e.g., induction on rule applications or exhaustive enumeration of sequences up to length 4), leaving the bound dependent on an unstated assumption.

    Authors: The full manuscript (Section 3) proves completeness by induction on the number of rule applications, establishing that the graph-construction procedure enumerates every admissible sequence without omission or duplication. The abstract summarizes the resulting bound but omits reference to this proof. We will revise the abstract to state that the four-application bound is supported by the completeness theorem. revision: yes

  2. Referee: [Abstract (central claim on derivation graphs)] The weakest assumption identified—that the graph-construction procedure enumerates every admissible ordering—directly undermines the 'simple procedure' and 'full space' characterization if even one sequence is missed; this must be addressed with a formal proof or verification before the bound can be accepted as general.

    Authors: The manuscript directly addresses this concern with a formal completeness proof (Section 3) showing that the construction enumerates all admissible orderings. We will update the abstract to explicitly reference this proof, making the support for the 'full space' characterization and the simple procedure clear. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces derivation graphs as a new representational tool to enumerate and structure do-calculus rule applications, then derives from their structure a bound of at most four applications and a method for multiple estimands. No quoted equations or definitions reduce the central claims to self-referential inputs, fitted parameters, or load-bearing self-citations; the exhaustiveness claim is presented as following from the graph construction itself rather than being presupposed by it. This is the normal case of an independent structural contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on the standard soundness and completeness properties of the three do-calculus rules and introduces derivation graphs as a new modeling device whose claimed properties are not independently evidenced outside the paper.

axioms (1)
  • domain assumption The three rules of do-calculus are sound and complete for transforming interventional queries into observational ones under the usual causal assumptions.
    Invoked as the foundation for defining equivalence classes of expressions.
invented entities (1)
  • derivation graph no independent evidence
    purpose: To represent and enumerate all sequences of do-calculus rule applications for a given causal query.
    Newly defined construct whose structure is used to derive the four-application bound and multiple-estimand result.

pith-pipeline@v0.9.1-grok · 5649 in / 1266 out tokens · 24739 ms · 2026-06-28T09:58:50.119771+00:00 · methodology

discussion (0)

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Reference graph

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