Graph Surgery and the Do-Operator: A Precise Correspondence for Acyclic Structural Causal Models

The $\operatorname{do}$-operator is described graphically by deleting arrows into its targets and functionally by replacing their mechanisms with constants. To call these operations equivalent is not yet a mathematical statement: one returns a graph and remembers only the targets, whereas the other returns mechanisms and also remembers the imposed values. We make a dependency-level comparison precise for deterministic acyclic structural causal models with finitely many endogenous variables. If $\operatorname{Graph}(F)$ extracts the dependencies of a mechanism family $F$, our main theorem is $\operatorname{Graph}(F^ι)=\operatorname{Surg}(\operatorname{Graph}(F),T_ι)$. Thus replacing target mechanisms removes exactly the dependencies removed by graph surgery. For a model $M=(G,F)$ whose graph may contain unused arrows, we characterize when the same equality holds with $G$ in place of $\operatorname{Graph}(F)$; it holds for every intervention exactly when $G$ records the dependencies of $F$ exactly. We then define the intervened model, characterize its run, show how sequential interventions combine, and prove that an outcome depends only on interventions at its actual dependency ancestors.

From probability to causality in probabilistic logic programming

Probabilistic logic programming is a formalism of statistical relational artificial intelligence that supports causal queries, including interventions from outside the system. When the structure of a probabilistic logic program is learned from data, however, only probabilistic information is used, and a single probability distribution may be compatible with several causal orders. This leads to ambiguity in interventional reasoning, raising the question of when the causal order is uniquely determined by the distribution. Exploiting the relationship between acyclic probabilistic logic programs and Bayesian networks, we derive conditions under which the probabilistic information encoded in a program determines a unique causal order. We also incorporate constraints arising from relational structure by taking into account prescribed sets of causal symmetries induced by the underlying relational vocabulary. The result is a method for verifying when a learned probabilistic logic program supports well-defined intervention semantics.

CausalSmith: A Formally Grounded, Self-Improving Agentic Framework for Automated Research in Causal Inference

Automating theoretical research is constrained not only by the generation of candidate results, but also by their reliable evaluation. A common approach is to close the research loop with a large language model (LLM) reviewer. However, such reviewers remain empirically unreliable: they may accept fabricated papers and detect them at rates close to chance (Bad Scientist, 2025). We present CausalSmith, a framework for automated theoretical research in causal inference grounded in the Lean proof assistant. CausalSmith combines Causalean, a foundational Lean library for causal inference containing 7,035 machine-checked declarations developed with language-model assistance under human design and review, with CausalSmith, a self-improving agentic pipeline that selects research topics, proposes results, formalizes statements, constructs proofs, and presents the resulting artifacts for human inspection. Because a machine-checked proof establishes only that a formal statement follows from its assumptions, not that the statement faithfully captures the intended scientific claim, the pipeline augments kernel verification with a statement audit that compares each formal theorem against the informal claim it is intended to express. We evaluate the system using artifacts produced by completed autonomous research runs. The source code, formal library, and run records are available at https://github.com/Jiyuan-Tan/CausalSmith.

A Causal Markov Condition for Value

This paper proposes a causal independence principle for value -- the value Causal Markov Condition (v-CMC) -- and develops the conceptual and mathematical foundations of a "causal value theory" linking causality and utility. After motivating a local formulation of the v-CMC, we introduce a probability-value duality that translates standard causal-inference results into the value setting. In particular, we formulate local, global, and decomposition versions of the v-CMC and prove their equivalence. We also define v-separation and show that it is sound and complete for conditional value independence. Furthermore, we derive a Bellman-type recursion as a special case of the v-CMC, thereby generalizing standard Bellman recursion from linear chains to causal DAGs. Finally, we show how the v-CMC supports modular transfer and updating of utility information across causal contexts and develop algorithms for causally structured utility elicitation and canonical influence-diagram construction.

A cubical formalisation of topos causal models: intervention, forcing, and a contextuality obstruction

Topos causal models recast causal inference inside a topos: a causal world is a presheaf, an intervention is a sub-model named by a characteristic map into the subobject classifier $\Om$, and reasoning is Kripke-Joyal forcing in an intuitionistic internal language. We give the first axiom-free machine-checked account of this 1-topos core, in Cubical Agda over a previously verified probability monad and do-calculus; the framework is otherwise developed on paper, with central claims stated rather than proved. Three of our results go beyond faithful transcription. We exhibit a contextuality obstruction the programme does not treat: pairwise-consistent local causal data with no global model, detected by a degree-one holonomy class. We delimit the claim that interventions are modelled by the subobject classifier: an intervention and an observation name the same subobject, so $\Om$ fixes the target of a do-operation but not the operation itself, which is surgery on the kernels --- where, on a confounder, the interventional and observational laws differ. And we settle the modal unit --- inflationarity is derivable from $j\top = \top$ and naturality, not a fourth axiom. We also machine-check the classifier of sieves with its classification theorem, the pullback collating local mechanisms, and the Kripke-Joyal forcing clauses. The development assumes no axioms and typechecks under Agda's \texttt{--safe} flag, with the ordered field discharged at $\mathbb{Q}$; type-level sheafification and a directed do-calculus are future work.

Verifying formulas for interventional distributions

We formalize verification in causal graphical models: deciding whether a given observational formula identifies a target interventional distribution. This opens a problem complementary to identification, asking not whether any identifying formula exists, but whether the given formula is identifying. We show that even sound and complete solutions to identification do not solve verification. We propose a falsifier as a first practical route forward, prove that it induces an almost-surely correct verifier for regular exponential-family models, and use the resulting verifier to develop the gateway test, which finds all sets admissible for use in a front-door formula.

Unveiling the Structure of Do-Calculus Reasoning via Derivation Graphs

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.