SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In causal abstraction, one has a set  $\mathbf{V}_{\mathrm{L}}$  of low-level (or concrete, or micro-) variables representing a fine-grained description and a set  $\mathbf{V}_{\mathrm{H}}$  of high-level (or abstract, or macro-) variables representing a coarser-grained description of the same scenario. The correspondence between the two descriptions is given by a partial translation map  $\tau : \operatorname{Val}(\mathbf{V}_{\mathrm{L}}) \to \operatorname{Val}(\mathbf{V}_{\mathrm{H}})$ . Translations extend canonically to maps of partial valuations (e.g., interventions)  $\tau : \bigcup_{\mathbf{X} \subset \mathbf{V}_{\mathrm{L}}} \operatorname{Val}(\mathbf{X}) \to \bigcup_{\mathbf{X} \subset \mathbf{V}_{\mathrm{H}}} \operatorname{Val}(\mathbf{X})$  by setting  $\tau(\mathbf{x}_{\mathrm{L}}) = \mathbf{x}_{\mathrm{H}}$  iff  $\tau\left(\pi_{\mathbf{X}_{\mathrm{L}}}^{-1}(\mathbf{x}_{\mathrm{L}})\right) = \pi_{\mathbf{X}_{\mathrm{H}}}^{-1}(\mathbf{x}_{\mathrm{H}})$ .
We overload  $\tau$  once more so as to cover counterfactuals, defining as follows yet another partial  $\tau: \operatorname{Val}(\mathfrak{D}_{\mathrm{L}}) \to \operatorname{Val}(\mathfrak{D}_{\mathrm{H}})$  for any sets  $\mathfrak{D}_{\mathrm{L}}, \mathfrak{D}_{\mathrm{H}}$  of potential outcomes over  $\mathbf{V}_{\mathrm{L}}$  and  $\mathbf{V}_{\mathrm{H}}$  respectively. Index an element of  $\operatorname{Val}(\mathfrak{D}_{\mathrm{L}})$  as  $\{(\mathbf{y}_{\mathrm{L}}^{i})_{\mathbf{x}_{\mathrm{L}}^{i}}\}_{1 \leq i \leq m}$ , where  $\mathbf{x}_{\mathrm{L}}^{i} \neq \mathbf{x}_{\mathrm{L}}^{j}$  for any  $i \neq j$  and  $\mathbf{y}_{\mathrm{L}}^{i} \in \operatorname{Val}(\{Y \in \mathbf{V}_{\mathrm{L}}: Y_{\mathbf{x}_{\mathrm{L}}^{i}} \in \mathfrak{D}_{\mathrm{L}}\})$  for each  $i$ , and an element of  $\operatorname{Val}(\mathfrak{D}_{\mathrm{H}})$  likewise as  $\{(\mathbf{y}_{\mathrm{H}}^{j})_{\mathbf{x}_{\mathrm{H}}^{j}}\}_{1 \leq j \leq n}$ . Define  $\tau\left(\left(\left(\mathbf{y}_{\mathrm{L}}^{i}\right)_{\mathbf{x}_{\mathrm{L}}^{i}}\right)_{1 \leq i \leq m}\right) = \left(\left(\mathbf{y}_{\mathrm{H}}^{j}\right)_{\mathbf{x}_{\mathrm{H}}^{j}}\right)_{1 \leq j \leq n}$  if  $\tau(\{\mathbf{x}_{\mathrm{L}}^{i}: 1 \leq i \leq m\}) = \{\mathbf{x}_{\mathrm{H}}^{j}: 1 \leq j \leq n\}$  and  $\tau(\mathbf{y}_{\mathrm{L}}^{i}) = \mathbf{y}_{\mathrm{H}}^{j}$  for any pair  $\mathbf{x}_{\mathrm{L}}^{i}, \mathbf{x}_{\mathrm{H}}^{j}$  where  $\tau(\mathbf{x}_{\mathrm{L}}^{i}) = \mathbf{x}_{\mathrm{H}}^{j}$ .
Definition 8. With counterfactual translation in hand, we define an abstraction relation between probabilistic causal models. The model  $\mathcal{H}$  abstracts  $\mathcal{L}$  over the aligned variables (written  $\mathcal{H} \prec_{\tau} \mathcal{L}$ ) if the translation  $\tau$  pushes the latter's counterfactual distribution to the former's, that is,  $P_{\mathrm{cf}}^{\mathcal{H}} = \tau_{*}(P_{\mathrm{cf}}^{\mathcal{L}})$ .
A stricter and typically more useful notion is that of constructive abstraction (e.g., [5]). These arise from translations that can be generated variable-wise, and thus correspond to a coherent "clustering" of variables:
Definition 9. Translation  $\tau: \operatorname{Val}(\mathbf{V}_{\mathrm{L}}) \to \operatorname{Val}(\mathbf{V}_{\mathrm{H}})$  is constructive if there is a partition  $\Pi$  of  $\mathbf{V}_{\mathrm{L}}$  with non-overlapping cells  $\{\Pi_V\}_{V \in \mathbf{V}_{\mathrm{H}} \cup \{\bot\}}$ , each  $\Pi_V \subset \mathbf{V}_{\mathrm{L}}$ , where  $\Pi_V$  is non-empty for all  $V \neq \bot$ , and a collection  $\{\tau_V\}_{V \in \mathbf{V}_{\mathrm{H}}}$  each of which is a partial surjective map  $\tau_V: \operatorname{Val}(\Pi_V) \to \operatorname{Val}(V)$ , such that  $\tau(\mathbf{v}_L) = \left\{\tau_V\left(\pi_{\Pi_V}(\mathbf{v}_L)\right)\right\}_{V \in \mathbf{V}_{\mathrm{H}}} \text{ for any } \mathbf{v}_L \in \operatorname{Val}(\mathbf{V}_L)$ .
A simple abstraction, ubiquitous in the literature (see, e.g., [16, §1.6.2] and [7]), is that of variable treatment levels. Here a higher-level value corresponds to some collection of lower-level specifications that might represent the potency or dosage of the drug administered, the time of administration, etc.: for example, a distinction of whether one took 300, 400, 500, or  $600\mathrm{mg}$  of aspirin is made at the low level, but at the high level, there is only the binary distinction between having taken aspirin and not. Formally, a treatment variable  $T$  is only binary with values c, tr (control, treatment resp.) at the high level but takes on many values c,  $\mathrm{tr}^1,\dots ,\mathrm{tr}^n$  at the low level. The abstraction is made by omitting the fine-grained details; symbolically, one forms a new model by eliding the superscripts, collapsing
all  $\mathrm{tr}^i$  into tr. So long as for any outcomes we have  $Y_{\mathrm{tr}^i}(u) = Y_{\mathrm{tr}^j}(u)$ , the model thus formed will be a constructive probabilistic abstraction of the low-level model.
The next result provides some useful properties of constructive abstraction.
Proposition 2. Suppose  $\mathcal{H} \prec_{\tau} \mathcal{L}$  with  $\tau$  constructive. Then  $\mathcal{H}$  is effective if  $\mathcal{L}$  is effective. Also, for any submodel  $\mathcal{H}'$  of  $\mathcal{H}$  there is a submodel  $\mathcal{L}'$  of  $\mathcal{L}$  such that  $\mathcal{H}' \prec_{\tau} \mathcal{L}'$ .
Thus our general class of effective RCMs closes under constructive translation. The next example shows that this is not the case for the narrower class of representable models.
Example 1. Let  $X, Y, X', Y'$  be variables with  $\operatorname{Val}(X) = \{0, 1, 2\}$  and  $\operatorname{Val}(X') = \operatorname{Val}(Y') = \operatorname{Val}(Y) = \{0, 1\}$ . Consider the RCM  $\mathcal{R}_{\mathrm{L}}$  defined over  $\mathbf{V}_{\mathrm{L}} = \{X, Y\}$  as a conjunction of POs:
$$
X = 1 \wedge Y = 1 \wedge Y _ {X = 2} = 0 \wedge X _ {Y = 0} = 1 \tag {4}
$$
for a single unit (suppressed above for clarity). Consider a second RCM  $\mathcal{R}_{\mathrm{H}}$  over  $\mathbf{V}_{\mathrm{H}} = \{X^{\prime},Y^{\prime}\}$
$$
X ^ {\prime} = 1 \wedge Y ^ {\prime} = 1 \wedge Y _ {X ^ {\prime} = 1} ^ {\prime} = 0 \wedge X _ {Y ^ {\prime} = 0} ^ {\prime} = 1. \tag {5}
$$
Note that  $\mathcal{R}_{\mathrm{H}} \prec_{\tau} \mathcal{R}_{\mathrm{L}}$  where  $\tau$  is a constructive abstraction with  $\Pi_{X'} = \{X\}$ ,  $\Pi_{Y'} = \{Y\}$  and  $\tau(X = 0) = 0$ ,  $\tau(X = 1) = \tau(X = 2) = 1$ ,  $\tau(Y = y) = y$ . Now  $\mathcal{R}_{\mathrm{H}}$  violates both composition and reversibility, while  $\mathcal{R}_{\mathrm{L}}$  is representable.
A second observation is that the analogue of the claim about submodels in Prop. 2 does not hold for extensions:
Example 2. Consider enlarging (4) with the potential outcome  $Y_{X=1} = 1$ . Then there is no high-level abstraction under  $\tau$  that defines the outcome  $Y_{X' = 1}'$ , since  $Y_{X = 2} = 0 \wedge Y_{X = 1} = 1$  translates to  $Y_{X' = 1}' = 0 \wedge Y_{X' = 1}' = 1$ .
The main result of this section is that the phenomenon exhibited by Ex. 1 accounts for all representability failures:
Theorem 1 (Abstract Representation). Let  $\mathcal{R}$  be an RCM. Then there is a representable  $\mathcal{R}_{\mathrm{L}}$  and constructive translation  $\tau$  such that  $\mathcal{R} \prec_{\tau} \mathcal{R}_{\mathrm{L}}$ .
It is worth remarking on the connection between a well-known twofold condition called the Stable Unit Treatment Value Assumption (SUTVA [16, §1.6]) and causal abstraction. The first part of SUTVA is the assumption that "the potential outcomes for any unit do not vary with the treatments assigned to other units"; this is already presumed by our definition of causal model, which does not admit interventions on multiple units (however, see Rmk. A.1 for a way to model this condition within our framework as an application of abstraction). The second part is that "for each unit, there are no different forms or versions of each treatment level, which lead to different potential outcomes." Note that this assumption can be seen as guaranteeing the viability of the variable treatment levels abstraction, as it is simply a restatement of the condition we already identified—that the outcomes  $Y_{\mathrm{tr}^i}(u)$  for any unit  $u$  and treatment level  $\mathrm{tr}^i$  must all agree.[5]
2 Inference
The raison d'etre of a causal inference framework is to provide a language for encoding causal assumptions and showing when and why conclusions follow from available data and appropriate assumptions. In this section, to provide further granularity on the comparison between RCMs and SCMs, we introduce a neutral formal language that is naturally interpreted relative to both of these models. The language systematizes reasoning about the probabilities of counterfactuals. Fixing a set  $\mathfrak{D}$  of potential outcome pairs, we define a formal language  $\mathcal{L}$  in two stages:
Definition 10. The base language  $\mathcal{L}_{\mathrm{base}}$  is given by all Boolean combinations of statements  $Y_{\mathbf{x}} = y$ , alternatively written  $y_{\mathbf{x}}$ , for all  $Y_{\mathbf{x}} \in \mathfrak{D}$ ,  $y \in \operatorname{Val}(Y)$ . Meanwhile,  $\mathcal{L}$  is defined as the set of Boolean combinations of inequalities  $\mathbf{t}_1 \geqslant \mathbf{t}_2$ , where  $\mathbf{t}_1, \mathbf{t}_2$  are generated as sums, products, and additive inverses of probability terms  $\mathbf{P}(\varepsilon)$ , where  $\varepsilon \in \mathcal{L}_{\mathrm{base}}$ .
<sup>5</sup>Imbens and Rubin [16] also mention ways of fulfilling this condition requiring changes to the causal model definition. In the supplement (Rmk. A.2) we show these alternative models can be represented within our framework.
The language  $\mathcal{L}$  is the most expressive in a sequence of three languages introduced in [13, 4] to formalize the "causal hierarchy" [27]. By restricting probability terms to purely "observational" or "interventional" quantities, it is possible to study the inferential limitations of data and assumptions at lower levels of this hierarchy. For present purposes,  $\mathcal{L}$  naturally encodes prominent reasoning patterns in RCMs and in SCMs. Its semantics are straightforward in any  $\mathcal{M}$  or  $\mathcal{R}$  that includes all outcomes  $\mathfrak{D}$ : we generate a mapping of each polynomial term  $\mathbf{t} \mapsto [[\mathbf{t}]] \in \mathbb{R}$  recursively with the crucial case being to map  $\mathbf{P}(\varepsilon)$  to the probability calculable by marginalization of  $p_{\mathrm{cf}}^{\mathcal{R}}$  or  $p_{\mathrm{cf}}^{\mathcal{M}}$ , and then evaluate the atom  $\mathbf{t}_1 \geqslant \mathbf{t}_2$  true iff  $[[[\mathbf{t}_1]] \geq [[\mathbf{t}_2]]$ , recursing to define a semantics for all of  $\mathcal{L}$ . Over the class of all (recursive, possibly infinite) SCMs,  $\mathcal{L}$  has been axiomatized [13] by a set of principles called  $\mathsf{AX}_3$ , and the complexity of its satisfiability problem has been shown complete for the class  $\exists \mathbb{R}$  [24]. The class of simple probability distributions over the atoms of  $\mathcal{L}_{\mathrm{base}}$  is axiomatized by principles known as  $\mathsf{AX}_1$  [13], which we will abbreviate AX.
2.1 Potential Outcomes Assumptions
Reasoning about potential outcomes is often encoded in what we call the base language  $\mathcal{L}_{\mathrm{base}}$ , augmented with (typically implicit universal) quantifiers over units. For instance, the well known monotonicity (or "no defiers" who do the opposite of their prescription) assumption [15, 16] says
$$
\forall u. X _ {z ^ {-}} (u) = 1 \rightarrow X _ {z ^ {+}} (u) = 1, \tag {6}
$$
where  $X$  and  $Z$  are binary variables respectively meaning the treatment (actually taken) and the treatment prescribed, with  $z^{+}, z^{-}$  abbreviating  $Z = 1, Z = 0$  respectively. We will use the same abbreviation for other binary variables, so that the above condition can be written succinctly as  $x_{z^{+}}^{+} \rightarrow x_{z^{-}}^{+}$ . We also adopt this interpretation of  $X, Z$  for the rest of §2.1. We now explain how this and other causal assumptions in the potential outcomes framework can be encoded in  $\mathcal{L}$ :
Definition 11. Let  $\xi$  be a well-formed, closed predicate formula in prenex normal form with a single quantifier over a variable  $\{u\}$  and matrix in  $\mathcal{L}_{\mathrm{base}}$ ; the  $u$  can alternately be included in the atoms, e.g., by writing  $Y_{\mathbf{x}}(u) = y$ . Define its encoding  $\mathrm{T}(\xi)\in \mathcal{L}$  as follows:
$$
\mathrm {T} (\xi) = \left\{ \begin{array}{l l} \mathbf {P} (\neg \mathrm {T} (\zeta)) = 0, & \xi = \forall u. \zeta , \text {w h e r e} \zeta \in \mathcal {L} _ {\text {b a s e}} \\ \mathbf {P} (\mathrm {T} (\zeta)) > 0, & \xi = \exists u. \zeta , \text {w h e r e} \zeta \in \mathcal {L} _ {\text {b a s e}} \end{array} \right..
$$
Note that  $\zeta$  is quantifier-free in both cases. Thus, e.g., the encoding of (6) is  $\mathbf{P}\left[\neg (x_{z^{-}}^{+} \to x_{z^{+}}^{+})\right] = 0$ .
Where  $S$  is a set of  $\mathcal{L}_{\mathrm{base}}$  assumptions let  $\Re (S)$  be the class of RCMs whose potential outcomes obey every assumption in  $S$ , thus obeying  $\forall u.\sigma$  where  $u$  ranges over units for any  $\sigma \in S$ . Also let  $\mathrm{T}(S) = \{\mathrm{T}(\forall u.\sigma)\}_{\sigma \in S}$  be the encoding of  $S$  via Def. 11. Then we have the following:
Theorem 2.  $\mathbf{AX} + \mathrm{T}(S)$  is sound and complete for  $\Re (S)$ .
Corollary 1.  $\mathsf{RCM} = \mathsf{AX} + \mathsf{T}(1)$  is sound and complete for  $\Re_{\mathrm{eff}}$
One consequence of this completeness result is that purely propositional and predicate logic reasoning about potential outcomes can be interweaved with probabilistic reasoning, as in Ex. 3 below. Another consequence is a complete axiomatization of SCMs (which can be seen as a probabilistic lift of [10]):
Corollary 2. Let  $\mathsf{C}$ , Rev be universal statements of (2) and (3) respectively. Then  $\mathsf{SCM} = \mathsf{RCM} + \mathsf{T}(\mathsf{C}) + \mathsf{T}(\mathsf{Rev})$  is sound and complete for  $\mathfrak{M}_{\mathrm{unic}}$  (where every outcome is included in  $\mathcal{L}_{\mathrm{base}}$ ).
Example 3. A seminal result from [15, 1] is that it is possible to estimate the Average Treatment Effect among the population of units who comply with their treatment assignment, a quantity known as the Local Average Treatment Effect (LATE):  $\mathbf{E}\big(Y_{x^{+}} - Y_{x^{-}}|x_{z^{+}}^{+}\wedge x_{z^{-}}^{-}\big)$ , with  $Y$  the outcome, which we assume binary purely for simplicity, and without loss of generality. Thm. 2 implies that this can be verified in our calculus, by appeal to two key assumptions [15, 1]: monotonicity (6) and
$$
\text {E x c l u s i o n} \quad (\mathrm {E R}). \quad \forall u. y _ {z ^ {-}, x} (u) \leftrightarrow y _ {z ^ {+}, x} (u). \tag {7}
$$
The original discovery was that these principles guarantee that  $\mathrm{LATE} = \mathrm{ITT}_1 / \mathrm{ITT}_2$ , where the latter are the average "causal effects of assignment on treatment received and on the outcome of interest" [1], or symbolically:
$$
\mathrm {I T T} _ {2} = \mathbf {E} \left(X _ {z ^ {+}} - X _ {z ^ {-}}\right) = \mathbf {P} \left(x _ {z ^ {+}} ^ {+} \wedge x _ {z ^ {-}} ^ {-}\right) - \mathbf {P} \left(x _ {z ^ {+}} ^ {-} \wedge x _ {z ^ {-}} ^ {+}\right)
$$
$$
\mathrm {I T T} _ {1} = \mathbf {E} \big (Y _ {z ^ {+}, X _ {z ^ {+}}} - Y _ {z ^ {-}, X _ {z ^ {-}}} \big) = \mathbf {P} \big (y _ {z ^ {+}, X _ {z ^ {+}}} ^ {+} \wedge y _ {z ^ {-}, X _ {z ^ {-}}} ^ {-} \big) - \mathbf {P} \big (y _ {z ^ {+}, X _ {z ^ {+}}} ^ {-} \wedge y _ {z ^ {-}, X _ {z ^ {-}}} ^ {+} \big).
$$
Figure 1: A family of RCMs  $\{\mathcal{R}(\varepsilon)\}_{0\leq \varepsilon \leq 1 / 4}$  such that  $\mathcal{R}(\varepsilon)$  is representable iff  $\varepsilon = 0$  (though (2) and (3) are met for any unit). Note that all members of this family meet (6) and (7) (the latter guaranteed since columns  $Y_{x^{+},z}(u),Y_{x^{-},z}(u)$  give the potential outcome for any  $z$ ). Also, any experimentally testable quantities—including, in particular,  $\mathrm{ITT}_1$  and  $\mathrm{ITT}_2$ —agree across the family, with  $\frac{\mathrm{ITT}_1}{\mathrm{ITT}_2} = 1 / 2$ . However the assumption in question (8) holds in  $\mathcal{R}(\varepsilon)$  iff  $\varepsilon = 0$ , and  $\mathrm{LATE} = 1 / 2 + \varepsilon$ , so that  $\mathrm{LATE} = \frac{\mathrm{ITT}_1}{\mathrm{ITT}_2}$  only when this holds, and it is not estimable in general.
where in  $\mathrm{ITT}_1$ , interventions like  $X_{z}$  set  $X$  at the unit level to the value that it would take under the intervention setting  $Z$  to  $z$ ; thus, e.g., we have  $\mathbf{P}(y_{z,X_z}) = \mathbf{P}(y_{z,x^+} \wedge x_z^+) + \mathbf{P}(y_{z,x^-} \wedge x_z^-)$ . Crucially, these two quantities can be estimated, e.g., through randomized experiments [16, Ch. 23].
However, over our most general class  $\Re_{\mathrm{eff}}$  of RCMs, these two assumptions are not in fact sufficient to identify LATE. Fig. 1 illustrates a family of RCMs that satisfy (6) and (7), but disagree on LATE. An additional principle, which Angrist et al. [1] offer as a matter of notation, we dub:
$$
\text {O u t c o m e d e c o m p o s i t i o n .} \quad \forall u. y _ {x} (u) \leftrightarrow y _ {z ^ {+}, x} (u). \tag {8}
$$
It can then be shown that, taken together, (6), (7), and (8) do indeed logically entail LATE  $= \mathrm{ITT}_1 / \mathrm{ITT}_2$ ; see Prop. B.1 in the technical appendix for the derivation.
There has been much discussion of monotonicity and exclusion restrictions (which are closely related to graphical assumptions; see §2.2 below), but what might justify outcome decomposition (8)? One intuition might be that it somehow follows from the exclusion restriction (7): if the effect of  $X$  on  $Y$  is the same no matter the value of  $Z$ , then it would seem that omitting  $z^{+}$  in the intervention should have no impact on the effect of  $X$  on  $Y$ . Of course, the example in Fig. 1 shows that this is too quick.
It turns out that (8) does follow from (7) if we restrict attention to representable RCMs. In fact, (8) is derivable from (7) and the principle of composition (2) in the calculus AX, so long as we can reason along the way about the potential response  $Z_{x}$ . By composition, for any  $x$  and  $y$  we have  $y_{x} \wedge z_{x}^{+} \to y_{z^{+},x}$  and  $y_{x} \wedge z_{x}^{-} \to y_{z^{-},x}$ , and by ER (7) the latter gives  $y_{x} \wedge z_{x}^{-} \to y_{z^{+},x}$ . As  $Z_{x}$  is binary, we have  $z_{x}^{+} \vee z_{x}^{-}$ , and thus by propositional reasoning,  $y_{x} \to y_{z^{+},x}$ . The other direction  $y_{z^{+},x} \to y_{x}$  follows from the same argument by contraposition, as  $Y$  too is binary.
Thus, while the full power of composition is not invoked, it is natural to read this example and much of the literature as implicitly assuming something like representability (thus implying composition). Another source of support for this is that under representability one can show (see Prop. B.2) that  $\mathrm{ITT}_1 = \mathbf{E}(Y_{z^+} - Y_{z^-})$ , an even simpler and manifestly identifiable expression for this average effect.
2.2 Graphical Assumptions
As we saw above (Prop. 1), SCMs can be understood as representations of suitable RCMs. As such, they also suggest further sources of assumptions for deriving causal inferences. In particular, qualitative patterns of functional dependence introduce the possibility of graphical methods:
Definition 12. Let  $\mathcal{M} = \langle \mathbf{U},\mathbf{V},\{f_V\}_{V\in \mathbf{V}},P\rangle$  be an SCM. Then define the causal diagram  $\mathcal{G}(\mathcal{M})$  of  $\mathcal{M}$  as a graph over nodes  $\mathbf{V}$ , with mixed directed edges  $\rightarrow$  and bidirected arcs  $\leftarrow \longrightarrow$ . For any  $V,V^{\prime}\in \mathbf{V}$ , there is a directed edge  $V\to V^{\prime}$  if  $V\in \mathbf{Pa}_{V^{\prime}}$ , and there is a bidirected edge  $V\leftarrow \longrightarrow V^{\prime}$  if  $\mathbf{U}_V,\mathbf{U}_{V'}\subset \mathbf{U}$  are correlated under  $P$  (including if  $\mathbf{U}_V\cap \mathbf{U}_{V'}\neq \emptyset$ ).
Letting  $\mathfrak{M}(\mathcal{G})$  be the set of SCMs with diagram  $\mathcal{G}$ , we have  $\mathfrak{M}(\mathcal{G}) \subset \mathfrak{M}_{\mathrm{uniq}}$  provided the directed edges in  $\mathcal{G}$  form a dag (see Lem. A.3). We thus assume this acyclicity of any  $\mathcal{G}$ . When interpreting over an SCM, we include every possible potential outcome in  $\mathcal{L}$ . Just as we earlier encoded assumptions about the potential outcomes of an RCM into  $\mathcal{L}$ , we may do the same for SCMs regarding their graphs. A first observation is that Def. 11 translates axiom C6 of [10] to ProbRec of [13], thus
rederiving the system  $\mathsf{AX}_3$  for the class of all acyclic SCMs, i.e.  $\bigcup_{\mathcal{G}}\mathfrak{M}(\mathcal{G})$ , from the latter. We now encode the content of (the assumption of having) a particular diagram  $\mathcal{G}$  into  $\mathcal{L}$ . Let  $\mathbf{P}\mathbf{a}_{V}^{\mathcal{G}} = \{V' \in \mathbf{V}: V' \to V \in \mathcal{G}\}$  be the directed parents in a graph  $\mathcal{G}$  of a vertex  $V$ . We encode by way of two schemas, encapsulating what some [28] have called "the two fundamental laws of causal inference":
Definition 13. Let the exclusion restriction schema  $\mathsf{ER}^{\mathcal{G}}$  be the  $\mathcal{L}_{\mathrm{base}}$  principle  $y_{\mathbf{a}} \leftrightarrow y_{\mathbf{p}}$ , for all variables  $Y \in \mathbf{V}$  and sets of variables  $\mathbf{A} \supset \mathbf{Pa}_V^{\mathcal{G}}$ , where  $y \in \operatorname{Val}(Y)$ ,  $\mathbf{a} \in \operatorname{Val}(\mathbf{A})$ ,  $\mathbf{p} = \pi_{\mathbf{Pa}_V^{\mathcal{G}}}(\mathbf{a})$ . Let the counterfactual independence schema  $\mathsf{cf-sep}^{\mathcal{G}}$  be, for all pairs of variable sets  $\{Y_i\}_{1 \leq i \leq n}, \{Y_j'\}_{1 \leq j \leq n'} \subset \mathbf{V}$  such that there are no  $Y_i, Y_j'$  for which  $Y_i = Y_j'$  or  $Y_i \gets -Y_j'$  in  $\mathcal{G}$ ,
$$
\operatorname {c f - s e p} ^ {\mathcal {G}}. \quad \mathbf {P} \left[ \bigwedge_ {1 \leq i \leq n} \left(y _ {i}\right) _ {\mathbf {p} _ {i}} \wedge \bigwedge_ {1 \leq j \leq n ^ {\prime}} \left(y _ {j} ^ {\prime}\right) _ {\mathbf {p} _ {j} ^ {\prime}} \right] = \mathbf {P} \left[ \bigwedge_ {1 \leq i \leq n} \left(y _ {i}\right) _ {\mathbf {p} _ {i}} \right] \cdot \mathbf {P} \left[ \bigwedge_ {1 \leq j \leq n ^ {\prime}} \left(y _ {j} ^ {\prime}\right) _ {\mathbf {p} _ {j}} \right] \tag {9}
$$
where  $y_{i}\in \mathrm{Val}(Y_{i})$ ,  $y_j^\prime \in \mathrm{Val}(Y_j^\prime)$ ,  $\mathbf{p}_i\in \mathrm{Val}(\mathbf{P}\mathbf{a}_{Y_i}^{\mathcal{G}})$ ,  $\mathbf{p}_j^\prime \in \mathrm{Val}(\mathbf{P}\mathbf{a}_{Y_j'}^{\mathcal{G}})$  for each  $Y_{i},Y_{j}^{\prime}$ . Then the translation of  $\mathcal{G}$  is the combination of axioms  $\mathrm{T}(\mathcal{G}) = \mathrm{T}(\mathsf{ER}^{\mathcal{G}}) + \mathsf{cf - sep}^{\mathcal{G}}$
Note that while Ex. 3 in no way relies on graphs, if we accept a  $\mathcal{G}$  where  $Z \nrightarrow Y$ , then  $\mathsf{ER}^{\mathcal{G}}$  yields  $y_{x} \leftrightarrow y_{zx} \leftrightarrow y_{z'x}$  without further ado. Importantly, however, T(6) is not valid over  $\mathfrak{M}(\mathcal{G})$  for any  $\mathcal{G}$  containing the edge  $Z \to X$ , revealing an extra-graphical provenance. On the other hand, cf-sep is inexpressible in  $\mathcal{L}_{\mathrm{base}}$ —inferentially, the two approaches are incomparable.