{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}C$ (along with
$I_{A}\longrightarrow A$ of course). The d-separation
$(A\perp^{d}C)_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}}$ implies the
independence $(A\upmodels C)_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}}$
(Definition IV.1) and also that $A$ does not affect $C$ (Corollary IV.2).
Similarly, we can derive $C$ does not affect $A$, $A$ does not affect $BC$,
$C$ does not affect $AB$ and $AC$ does not affect $B$. Combined with the lack
of any zeroth-order affects relations between any two of the RVs, this implies
that there are no affects relations of order $\geq 1$ (by the contrapositive
of Lemma IV.4). Further, using Lemma IV.1 and the exogeneity of $B$, we can
derive $AB$ does not affect $C$ as follows. In the causal structure
$\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(AB)}$, $A$ is d-separated from $B$
and $C$, while $B$ and $C$ are independent of each other due to the exogeneity
of $B$ and the dashed arrow connecting them. Using the lemma, this gives
$(AB\upmodels C)_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(AB)}}$ which can be
explicitly written as
$P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(AB)}}(C|A,B)=P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(AB)}}(C)$.
The right hand side can be simplified in the following two steps. Firstly as
$P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(AB)}}(C)=P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}}(C)$
noting that $\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(AB)}$ and
$\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}$ are effectively the same graph
due to the exogeneity of $B$. Then the d-separation
$(A\perp^{d}C)_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}}$ implies the
independence
$P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}}(C|A)=P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}}(C)$,
which along with $A$ does not affect $C$ (as noted earlier) gives
$P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(A)}}(C)=P_{\mathcal{G}}^{\mathrm{jam}}(C)$.
Putting this together, we get
$P_{\mathcal{G}^{\mathrm{jam}}_{\mathrm{do}(AB)}}(C|A,B)=P_{\mathcal{G}}^{\mathrm{jam}}(C)$
i.e., $\\{A.B\\}$ does not affect $C$. Similarly, one can obtain $BC$ does not
affect $A$. All these are summarised in Figure 13.

###  A.2 Fine-tuned collider (Figure 14a)

In the causal structure of Figure 14a, the independence $A\upmodels C$ follows
from the d-separation condition (Definition IV.1), while $A\upmodels B$ and
$C\upmodels B$ follow from the dashed arrow structure. These are the same
independences as the case in the previous example of jamming with unobserved
$\Lambda$ (where $A\upmodels C$ was an additional independence in the jamming
example but follows from d-separation in this case). Thus the distribution
$P_{ABC}$ from Example IV.6 is compatible with both the jamming (Figure 9a) as
well the fine-tuned collider (Figure 14a) causal structures.212121Note that
this is essentially the one-time pad example from earlier. However
interventions on the two causal structures yield different results. We have
$AC$ affects $B$ for the fine-tuned collider (since $AC$ consists of exogenous
nodes and is correlated with $B$) but not for the jamming case. We also have
$A$ affects $BC$ and $C$ affects $AB$ for the fine-tuned collider even though
$A$ and $C$ do not individually affect $B$ due to the dashed arrow structure.
This follows from the exogeneity of $A$ and $C$ and the joint correlations
$A=B\oplus C$. Further, $AB$ does not affect $C$ since these sets become
d-separated upon intervention on $AB$ and by a similar reasoning, $BC$ does
not affect $A$ and $B$ does not affect $AC$ (in contrast with the jamming case
where $B$ affects $AC$). As for higher-order affects relations, $A$ affects
$B$ given do$(C)$ and $C$ affects $B$ given do$(A)$ are the only ones, and
these follow by applying Lemma A.2 to this example. Again, these conclusions
are summarised in Figure 13.

(a)  (b)

(c)

Figure 14: Some fine-tuned and/or cyclic causal structures:  (a) A fine-tuned
collider (b) A Type 4 affects causal loop (c) A Type 1 affects causal loop

###  A.3 A Type 4 affects causal loop (VilasiniColbeckPRL , Figure 14b)

Example VI.2 outlined a cyclic causal model proposed in VilasiniColbeckPRL
for demonstrating the mathematical possibility of compatibly embedding affects
causal loops in Minkowski space-time. Here, we reproduce further details of
this model and discuss its properties it in the context of the more general
framework developed in the present paper. Consider the cyclic causal structure
$\mathcal{G}^{\mathrm{ACL}4}$ of Figure 14b along with the following classical
causal mechanisms where all 4 variables are taken to be binary: $A=\Lambda$,
$B=A\oplus C$, $C=B\oplus\Lambda$, where the exogenous variable $\Lambda$ is
uniformly distributed. One can check that the distribution $P_{ABC}$ obtained
through these mechanisms would be the same as that of the jamming as well as
the fine-tuned collider examples above, but the affects relations differ from
those of these examples and instead correspond to those of Example VI.2 which
is an affects causal loop of Type 4. To obtain these affects relations,
firstly note that in the causal model of
$\mathcal{G}^{\mathrm{ACL}4}_{\mathrm{do}(A)}$, $\Lambda$ is no longer a
parent of $A$, but using the remaining causal mechanisms $B=A\oplus C$ and
$C=B\oplus\Lambda$ (which remain the same), we can still obtain $A=\Lambda$.
Therefore the intervention on $A$ does not change the observed distribution
and $A$ and $B$ continue to be independent in $\mathcal{G}^{\mathrm{ACL}4}$ as
well as $\mathcal{G}^{\mathrm{ACL}4}_{\mathrm{do}(A)}$, and in both graphs the
marginal distributions over $A$, $B$ and $C$ are uniform, which gives $A$ does
not affect $B$. On the other hand, $B$ does not affect $A$ can be established
simply from the d-separation
$(B\perp^{d}A)_{\mathcal{G}^{\mathrm{ACL}4}_{\mathrm{do}(B)}}$. In the causal
model of $\mathcal{G}^{\mathrm{ACL}4}_{\mathrm{do}(C)}$, neither $B$ nor
$\Lambda$ are parents of $C$ but the remaining mechanisms $A=\Lambda$ and
$B=A\oplus C$ give $C=B\oplus\Lambda$. Again, the observed distribution here
is the same as the pre-intervention distribution, which gives $C$ does not
affect $B$. By a similar argument, $B$ does not affect $C$ can also be
established. Further, we have both $B$ affects $AC$ (as in the jamming case)
_and_ $AC$ affects $B$ (as in the fine-tuned collider) since
$P_{\mathcal{G}^{\mathrm{ACL}4}_{\mathrm{do}}(AC)}(B|A,C)$ and
$P_{\mathcal{G}^{\mathrm{ACL}4}_{\mathrm{do}}(B)}(A,C|B)$ are deterministic
while $P_{\mathcal{G}^{\mathrm{ACL}4}}(B)$ and
$P_{\mathcal{G}^{\mathrm{ACL}4}}(A,C)$ are uniform. We also have $A$ affects
$BC$ and $C$ affects $AB$ as in the fine-tuned collider, which can be verified
using the causal mechanisms given.222222Note that in the absence of the causal
mechanisms, many of the affects/non-affects relations may not be identifiable.
For example, to deduce that $AB$ does not affect $C$ in the jamming case, we
used Lemma IV.1 along with the fact that $B$ was exogenous in
$\mathcal{G}^{\mathrm{jam}}$. But the same argument cannot be applied here
since $B$ is not exogenous in $\mathcal{G}^{\mathrm{ACL}4}$. As in the jamming
case, we also get $AB$ does not affect $C$ and $BC$ does not affect $A$. The
higher-order affects relations here are identical to the previous example, and
obtained in a similar manner and these are given in Figure 13. Furthermore,
even though this corresponds to an affects causal loop the existence of which
can be operationally certified and this causal model admits a non-trivial and
compatible embedding in Minkowski space-time as explained i.e., it does not
lead to superluminal signalling VilasiniColbeckPRL  (see also Example VI.2).

###  A.4 A Type 1 affects causal loop (Figure 14c)

Consider a faithful causal model associated with the cyclic causal structure
of Figure 14c. Here all 4 nodes are observed and hence classical. The
faithfulness implies that the only conditional independences are those implied
by d-separation. Since $B\perp^{d}D|AC$ and $A\perp^{d}C|BD$ are the only
d-separations, $B\upmodels D|AC$ and $A\upmodels C|BD$ are the only
conditional independences. Further, the faithfulness also implies that every
causal arrow is associated with an affects relation, which also reflected in
the fact that all arrows are solid arrows, and we have $A$ affects $B$, $B$
affects $C$, $C$ affects $D$ and $D$ affects $A$, which forms an affects
causal loop of Type 1 (there is an affects relation is both direction between
every pair of RVs). One can easily check that the only irreducible affects
relations are the zeroth-order affects relations between single RVs, as one
would expect for faithful causal models. To further illustrate the kind of
causal loops allowed in this framework, consider the pairwise correlations
$A=B$, $B=C$, $C=D$ and $D\neq A$. Since this system of equations has no
solutions, there exists no joint distribution $P_{ABCD}$ from which the
pairwise marginals producing these correlations can be obtained. Such examples
correspond to grandfather type paradoxes and cannot be modelled in frameworks
that demand the existence of a valid joint probability distribution over all
variables involved in a causal loop. On the other hand, examples of solid
arrow directed cycles where the functional dependences of the loop variables
admit solutions such as $A=B=C=D$ (with any probability) or the examples
considered in Pearl2013  for other cyclic causal structures can be modelled in
our framework. Additionally, there can also be Type 1 and Type 2 affects
causal loops that do not involve any solid arrows, for example through a
concatenation of structures such as that of Figure 9b. We discuss causal loops
in more detail in Appendix C, also in the case of quantum causal structures.

##  Appendix B Further classes of affects causal loops and their space-time
embeddings

As motivated in the main text (see paragraph after Definition VI.8), we can
consider further classes of affects causal loops that are distinct from ACL1,
$\ldots$, ACL6. The intuition is that the chain of irreducible affects
relations involved in these previous definitions is such that for any two
adjacent affects relations in the chain the second set of the first is
contained in the first set of the second. Relaxing this containment condition
can lead to a violation of Theorem VI.1, as also explained in the main text.
So we can consider relaxing this condition and only requiring a non-trivial
intersection between the sets (which would make the chain “incomplete”), as
long as we include additional conditions on the affects relations that will
again guarantee cyclicity of the causal structure. Here we propose four more
classes of affects causal loops ACL7, ACL8, ACL9 and ACL10 based on this idea,
illustrate them with examples which also show that there can in general be
more classes of affects causal loops even beyond these.

######  Definition B.1 (Affects causal loops, Type 7 (ACL7)).

A set of affects relations $\mathscr{A}$ is said to contain a Type 7 affects
causal loop if the following conditions are satisfied

  1. 1.

There exist disjoint sets of RVs $S_{1}$ and $S_{2}$ such that $S_{1}$ affects
$S_{2}$ belongs to $\mathscr{A}$ and is irreducible.

  2. 2.

There exists a chain of irreducible affects relations (possibly incomplete)
$\mathscr{C}_{s_{2}}$ from some subset $s_{2}\subseteq S_{2}$ to $S_{1}$ i.e.,
there exists sets of RVs $s_{2}\subseteq S^{\prime}_{2}$, $S_{3}$,
$S^{\prime}_{3}$, $\ldots$ $S_{n}$, $S^{\prime}_{n}$, $s_{1}\subseteq S_{1}$
such that $\\{S^{\prime}_{2}$ affects $S_{3}$, $S^{\prime}_{3}$ affects
$S_{4}$, $\ldots$ ,$S^{\prime}_{n-1}$ affects $S_{n}$, $S^{\prime}_{n}$
affects $s_{1}\\}\subseteq\mathscr{A}$, where all the affects relations are
irreducible, every pair of sets connected by an affects relation is disjoint,
$S_{i}\bigcap S^{\prime}_{i}\neq\emptyset$ for all $i\in\\{3,\ldots,n\\}$ and
$S_{2}\bigcap S_{2}^{\prime}=s_{2}$. Each pair $(S_{i},S_{i}^{\prime})$ such
that $S_{i}\not\subseteq S^{\prime}_{i}$ for $i\in\\{2,\ldots,n\\}$ is called
an incomplete node of the affects chain $\mathscr{C}_{s_{2}}$, a complete
affects chain has no incomplete nodes.

  3. 3.

For each affects chain $\mathscr{C}_{s_{2}}$ that connects the subset $s_{2}$
of $S_{2}$ back to $S_{1}$ as above, and each incomplete node
$(S_{i},S_{i}^{\prime})$ in $\mathscr{C}_{s_{2}}$ (for
$i\in\\{2,\ldots,n\\}$), there exists a complete affects chain
$\mathscr{D}^{\mathscr{C}}_{s_{2}}$ in $\mathscr{A}$ from
$S_{i}\backslash(S_{i}\bigcap S_{i}^{\prime})$ to $S_{i}$.

######  Definition B.2 (Affects causal loops, Type 8 (ACL8)).

A set of affects relations $\mathscr{A}$ is said to contain a Type 8 affects
causal loop if the following conditions are satisfied

  1. 1.

There exist disjoint sets of RVs $S_{1}$ and $S_{2}$ such that $S_{1}$ affects
$S_{2}$ belongs to $\mathscr{A}$ and is irreducible.

  2. 2.

For each element $e_{2}\in S_{2}$, there exists a chain of irreducible affects
relations (possibly incomplete) $\mathscr{C}_{e_{2}}$ that connects it back to
$S_{1}$ i.e., for each $e_{2}$, there exists sets of RVs $e_{2}\in
S^{\prime}_{2}$, $S_{3}$, $S^{\prime}_{3}$, $\ldots$ $S_{n}$ $S^{\prime}_{n}$,
$s_{1}\subseteq S_{1}$ such that $\\{S^{\prime}_{2}$ affects $S_{3}$,
$S^{\prime}_{3}$ affects $S_{4}$, $\ldots$ ,$S^{\prime}_{n-1}$ affects
$S_{n}$, $S^{\prime}_{n}$ affects$s_{1}\\}\subseteq\mathscr{A}$, where all the
affects relations are irreducible, every pair of sets connected by an affects
relation is disjoint, $S_{i}\bigcap S^{\prime}_{i}\neq\emptyset$ for all
$i\in\\{3,\ldots,n\\}$ and $S_{2}\bigcap S_{2}^{\prime}=e_{2}$.

  3. 3.

For each element $e_{2}\in S_{2}$, each affects chain $\mathscr{C}_{e_{2}}$
that connects it back to $S_{1}$ as above, and each incomplete node
$(S_{i},S_{i}^{\prime})$ in $\mathscr{C}_{e_{2}}$ (for
$i\in\\{2,\ldots,n\\}$), there exists a complete affects chain
$\mathscr{D}^{\mathscr{C}}_{e_{2}}$ in $\mathscr{A}$ from
$S_{i}\backslash(S_{i}\bigcap S_{i}^{\prime})$ to $S_{i}$.

The following Theorem (proven in Appendix D.5) generalises Theorem VI.1 to
ACL7 and ACL8, and justifies categorising them as affects causal loops.

######  Theorem B.1.

Any set of affects relations $\mathscr{A}$ containing an affects causal loops
of Type 7 or Type 8 can only arise from a causal model over a cyclic causal
structure.

More generally, for a given affects chain $\mathscr{C}_{s_{2}}$ in Definition
B.1, and an incomplete node $(S_{i},S_{i}^{\prime})$ in $\mathscr{C}_{s_{2}}$,
instead of a single complete affects chain $\mathscr{D}^{\mathscr{C}}_{s_{2}}$
we could consider a set of incomplete affects chains that serve the same
purpose and for which an analogous theorem holds. For example, for each
incomplete node $(S_{i},S_{i}^{\prime})$ of $\mathscr{C}_{s_{2}}$, there can
exist an incomplete affects chain $\mathscr{D}^{\mathscr{C}}_{s_{2}}$ in
$\mathscr{A}$ from $S_{i}\backslash(S_{i}\bigcap S_{i}^{\prime})$ to $S_{i}$,
such that for each incomplete node $(R_{j},R_{j}^{\prime})$ of
$\mathscr{D}^{\mathscr{C}}_{s_{2}}$, there exists another complete affects
chain in $\mathscr{A}$ from $R_{j}\backslash(R_{j}\bigcap R_{j}^{\prime})$ to
$R_{j}$. This could go on recursively for arbitrarily many chains depending on
the number of RVs appearing in $\mathscr{A}$. This recursive definition
defines yet another class ACL9, and an analogous recursive version of ACL8
would define another class ACL10. Theorem B.1 for ACL9 and ACL10 follows
through similar arguments, so we note this point without proof. We illustrate
these new classes with some examples, along with an example to show that these
(ACL1-ACL10) do not cover all possible affects causal loops.

######  Example B.1 (A Type 7 affects causal loop).

Consider the set of irreducible affects relations $\mathscr{A}=\\{X$ affects
$Y$, $Y$ affects $AB$, $A$ affects $X$, $C$ affects $AB$, $B$ affects $C\\}$.
One can check that $\mathscr{A}$ does not contain affects causal loops of
Types 1 to 6, since no affects relation in $\mathscr{A}$ is such that every
element of the second set has a complete affects chain leading it back to the
first set. It however contains at least one Type 7 affects loop. For the
affects relation $Y$ affects $AB$, we have the incomplete chain
$\mathscr{C}_{A}=\\{A$ affects $X$, $X$ affects $Y\\}$ that connects $A$ to
$Y$ with the incomplete node $(S_{2}=AB,S_{2}^{\prime}=A)$. Then
$S_{2}\backslash(S_{2}\bigcap S_{2}^{\prime})=\\{B\\}$ and we have the
complete affects chain $\mathscr{D}^{\mathscr{C}}_{A}=\\{B$ affects $C$, $C$
affects $AB\\}$ that connects $S_{2}\backslash(S_{2}\bigcap S_{2}^{\prime})$
to $S_{2}$ as required.

######  Example B.2 (A Type 9 affects causal loop).

Consider the set of irreducible affects relations $\mathscr{A}=\\{X$ affects
$Y$, $Y$ affects $AB$, $A$ affects $X$, $C$ affects $AB$, $B$ affects $CD$,
$D$ affects $E$, $E$ affects $CD\\}$. This set is similar to the previous
example, but does not contain a Type 7 causal loop (or ACLs of any lower
types). It does contain a Type 9. For the affects relation $Y$ affects $AB$,
there is an incomplete chain $\mathscr{C}_{A}=\\{A$ affects $X$, $X$ affects
$Y\\}$ that connects $A$ to $Y$ as before. But we have no complete chains from
$S_{2}\backslash(S_{2}\bigcap S_{2}^{\prime})=\\{B\\}$ to $S_{2}=AB$ as
before, only the incomplete chain $\mathscr{D}^{\mathscr{C}}_{A}=\\{B$ affects
$CD$, $C$ affects $AB\\}$. The incomplete node $(R_{j},R_{j}^{\prime})$ of
$\mathscr{D}^{\mathscr{C}}_{A}$ is $(R_{j}=CD,R^{\prime}_{j}=C)$ and we have a
complete affects chain $\\{D$ affects $E$, $E$ affects $CD\\}$ from
$R_{j}\backslash(R_{j}\bigcap R_{j}^{\prime})=\\{D\\}$ to $R_{j}$.

######  Example B.3 (An affects causal loop not covered by Types 1 to 10).

Consider the set of irreducible affects relations $\mathscr{A}=\\{X$
affects$Y$, $Y$ affects $AB$, $A$ affects $X$, $C$ affects $AB$, $B$ affects
$CD$, $BD$ affects $AC\\}$. With some effort, one can convince themselves that
$\mathscr{A}$ does not contain affects causal loops of Types 1-10. It
nevertheless implies cyclicity, as follows. Applying Corollary IV.3, we have
that $X$ is a cause of $Y$, $Y$ is either a cause of $A$ or of $B$ and $A$ is
a cause of $X$. If $Y$ is a cause of $A$, we already have a directed cycle, so
consider the case where $Y$ is a cause of $B$. Using the remaining affects
relations, we have $C$ is a cause of either $A$ or $B$, $B$ is a cause of
either $C$ or $D$. Irrespective of whether $C$ is a cause of either $A$ or
$B$, if $B$ is a cause of $C$, we would have a directed cycle, so we must take
$B$ to be a cause of $D$ to avoid this. The last affects relation implies that
$B$ is either a cause of $A$ or of $C$. Irrespective of the choice here and
the choice for whether $C$ is a cause of $A$ or of $B$, one can verify that we
will always have a directed cycle. Hence this set of affects relations is an
affects causal loop that is not of a previously defined Type.

Consider now, the space-time embedding for the affects relations of Example
B.1 in Minkowski space-time. Imposing compat (Definition V.7) on the affects
relations $\mathscr{A}=\\{X$ affects $Y$, $Y$ affects $AB$, $A$ affects $X$,
$C$ affects $AB$, $B$ affects $C\\}$ implies that $\mathcal{Y}$ must contain
the joint inclusive future of $\mathcal{A}$ and $\mathcal{B}$ but
$\mathcal{A}$ is in the past of $\mathcal{X}$ which is in the past of
$\mathcal{Y}$. The only way this can be satisfied is if $\mathcal{B}$ is in
the future of $\mathcal{A}$ such that the joint inclusive future of
$\mathcal{A}$ and $\mathcal{B}$ coincides with the inclusive future of
$\mathcal{B}$. The last two affects relations then imply that $\mathcal{B}$
and $\mathcal{C}$ must be embedded at the same location and since we have $B$
affects $C$, this embedding is trivial. This implies that there is no non-
trivial and compatible embedding of these affects relations in Minkowski
space-time. In other words, the absence of affects causal loops of Types 1-6
does not guarantee the existence of a non-trivial and compatible space-time
embedding. The presence of Type 3 and above ACLs does not rule out the
existence of such an embedding as we have seen in Example VI.2, in contrast to
the case of Type 1 and 2 ACLs (Lemma VI.2). This suggests that for each
individual Type $i$ of affects causal loops other than Types 1 and 2, the
existence of a non-trivial and compatible space-time embedding is neither
necessary nor sufficient for there to be no affects causal loops of that Type.
By Lemma VI.3, for Type 3, the existence of a non-degenerate and compatible
space-time embedding is sufficient but not necessary to rule out ACL3.

##  Appendix C Do-conditionals from causal mechanisms in quantum cyclic causal
models

In Section IV we have outlined how interventions and do-conditionals (i.e.,
the post intervention distribution) are defined in our framework, and Theorem
IV.1 provides some conditions under which the post and pre intervention
distributions can be related. Ideally though, one would expect that it should
be possible to fully specify the post-intervention distribution if we are
given all the underlying causal mechanisms involved in the causal structure.
For example, in the classical case, the structural equations of the causal
model Pearl2009  provide these causal mechanisms. Here for each node $X$ in
the causal structure, the dependence of $X$ on its parents par$(X)$
corresponds to a stochastic map, which can be written in terms of a
deterministic function $X=f_{X}(\text{par}(X),E_{X})$ by including an
additional exogenous random variable $E_{X}$ for each node $X$. This is called
a _structural equation_. If the structural equations for all the nodes and the
distributions of the parentless nodes are known, then the complete post-
intervention distribution can be calculated. This has been shown to be the
case for classical cyclic causal models in Forre2017 . An intervention
$\mathrm{do}(x)$ on $X$, corresponds to updating the structural equation for
$X$ to $X=x$ while keeping the remaining structural equations the same.
Another important result for the classical case derived in Forre2017  is that
the d-separation property or the _global directed Markov condition_ of
Definition IV.1 is recovered whenever all the random variables are discrete
and the structural equations of the causal model satisfy a property known as
_ancestrally unique solvability_ (anSEP). Roughly, this property demands that
the structural equation for each node must admit a unique solution given the
values of the node’s ancestors. We need not define this concept formally for
our purposes here.

Ideally we would like to extend these ideas to quantum and post-quantum cyclic
causal structures, where the causal mechanisms involve measurements and
transformations on non-classical systems, which cannot be expressed using
deterministic structural equations. In the non-classical case, it is unclear
what conditions allow for the d-separation condition to be recovered. Even to
make this question precise in the non-classical case, one would need to
specify the analog of structural equations for such causal models which is an
open problem. Here, we present a possible method for achieving this by
explaining it using the following example and sketching how it might
generalise to a larger class of causal models.

######  Example C.1 (A quantum cyclic causal model).

Consider the cyclic variation of the bipartite Bell causal structure
illustrated in Figure 15a. Let the common cause $\Lambda$ correspond to the
Bell state $|\psi_{\Lambda}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$.
Suppose that $A$ and $B$ are the settings of local measurements on the two
subsystems such that when these variables take the value $0$, it denotes a
measurement in the $\\{|0\rangle,|1\rangle\\}$ basis on the associated
subsystem, and the value $1$ denotes a measurement in the
$\\{|+\rangle,|-\rangle\\}$ basis. $X$ and $Y$ are the binary outcomes of
these measurements where $|0\rangle$ or $|+\rangle$ correspond to outcome 0
and $|1\rangle$ or $|-\rangle$ correspond to 1. The additional constraints
coming from the causal loop are that $B=X$ and $A=Y$. This specifies all the
causal mechanisms, how do we calculate the observed distribution $P_{XYAB}$?

#### A method based on post-selection:

One method is to first calculate the observed correlations for the specified
state and measurements in the original Bell scenario (Figure 1b), and then
post-select on the observations that obey the loop conditions $B=X$ and $A=Y$.
More formally, this corresponds to transforming the original cyclic causal
structure of Figure 15a to the acyclic causal structure of Figure 15b by
cutting off the edges $A\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}X$ and
$B\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ and replacing them
with the edges $A^{*}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}X$ and
$B^{*}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Y$ by introducing two
exogenous nodes $A^{*}$ and $B^{*}$. Then the inputs $A^{*}$ and $B^{*}$ and
outputs $X$ and $Y$ along with the shared system $\Lambda$ define a Bell
scenario, while the variables $A=Y$ and $B=X$ can simply be seen as local
post-processings of the outcomes. We can then calculate the observed
probabilities for this acyclic causal structure using the Born rule, and post-
select on $A^{*}=A$ and $B^{*}=B$, which effectively achieves the post-
selection $A=Y$ and $B=X$ in the original Bell scenario (Figure 1b). This
distribution needs to be renormalised to obtain the observed distribution
$P_{XYAB}$. This is calculated in Figure 15 and can be used to find all the
affects relations. An intervention on $A$ would cut off the arrow from $Y$ to
$A$. This means $A$ does not affect $X$ since $A$ is effectively exogenous in
the post-intervention causal structure and will be uncorrelated with $X$ since
$\Lambda$ is the maximally entangled state. Similarly $B$ does not affect $Y$.
However, $AB$ affects $XY$ since a joint intervention on $A$ and $B$ takes us
back to the original Bell scenario in which these sets are correlated, and
correlation in the post-intervention causal structure implies an affects
relation (cf. Lemma IV.2) and it can be checked that this affects relation is
irreducible. We also have $X$ affects $B$ and $Y$ affects $A$ due to the loop
conditions $A=Y$ and $B=X$. And naturally, also $XY$ affects $AB$, which is
also irreducible. With a bit more effort, one can also check that we have $A$
affects $Y$ and $B$ affects $X$. Therefore we have two Type 1 affects causal
loops (Definition VI.3) $\\{A$ affects $Y,$ $Y$ affects $A\\}$, and similarly
for $B$ and $X$. We also have a Type 4 affects causal loop (Definition VI.6)
formed by the irreducible relations $\\{AB$ affects $XY$, $XY$ affects
$AB\\}$. In this example, $\mathcal{G}_{\mathrm{do}(A,B)}$ corresponds to a
quantum causal structure (the Bell scenario) while
$\mathcal{G}_{\mathrm{do}(X,Y)}$ is a simple classical causal structure. Then
the (observed) arrows  of $\mathcal{G}$ can be classified into dashed and
solid arrows as:
$A\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}X$,
$B\mathrel{\mathchoice{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa
04B}{}}{\mathrel{}\mathrel{\mathop{\dabar@\dabar@}\limits}\mathrel{\mathchar
0\relax\symAMSa 04B}{}}}Y$, $X\longrightarrow B$ and $Y\longrightarrow A$. The
post-intervention distribution is fully specified here because, all
interventions (except that on $\Lambda$ alone) are associated with acyclic
post-intervention graphs and for interventions on the exogenous $\Lambda$, the
post and pre-intervention distributions coincide.

#### Applying the method to fine-tuned explanations of non-classical
correlations:

It is known that certain non-classical correlations arising in the bipartite
Bell causal structure cannot be obtained in the same causal structure if the
common cause $\Lambda$ was classical. However, these correlations can be
easily generated in the classical, fine-tuned causal structure of Figure 15c,
which differs from the original causal structure by the inclusion of fine-
tuned causal influences from each party’s input to the other party’s output.
We now explain how this is achieved and then apply the post-selection method
explained above to create a causal loop in Figure 15c by adding
$X\longrightarrow B$ and $Y\longrightarrow A$. This will demonstrate that,
even though the same non-classical correlations and affects relations can be
obtained in the original Bell causal structure and its fine-tuned classical
counterpart 15c, the two causal structures behave differently in the presence
of causal loops.

First consider the PR box, which is one of the maximally non-classical
correlations of the Bell causal structure. It is defined by the condition
$X\oplus Y=A.B$ where all the variables are binary. This is easily generated
in the classical causal structure of Figure 15c by the structural equations
$\Lambda=E$, $Y=E$ and $X=E\oplus A.B$ (where $E$ is binary and uniformly
distributed). Other non-classical correlations can be obtained by adding some
“noise” to this PR box example. Let $\Lambda=(E,F)$ correspond to two
variables $E$ and $F$ both binary, and the former distributed uniformly. Then
the structural equations $Y=E$ and $X=E\oplus F\oplus A.B$ for different
distributions over the exogenous variable $F$ correspond to the PR box mixed
with different levels of noise.232323Note that the model can be symmetrised by
including an additional, uniformly distributed binary variable $G$ in the
description of $\Lambda=(E,F,G)$ and using the structural equations
$X=E\oplus(G\oplus 1)(A.B\oplus F)$ and $Y=E\oplus G(A.B\oplus F)$.

$\displaystyle\begin{split}X&=A.B\oplus E\oplus F,\\\ Y&=E.\end{split}$ (10)

Therefore, the causal mechanisms that allow us to produce non-classical
correlations $P_{XYAB}$ in the acyclic causal structure 15c are the functional
dependences (10) along with a specification of the distributions over the
exogenous variables $E$ and $F$ that constitute $\Lambda$. $E$ is uniform
while $F$ can vary depending on the correlation to be generated. We now
construct the causal loop by including the additional arrows
$X\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}B$ and
$Y\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}A$ and by effectively
post-selecting on the loop condition $A=Y$ and $B=X$. These, along with the
causal mechanisms (10) of the acyclic case define the mechanisms for the
cyclic causal structure. We will now see that these causal mechanisms are
incompatible with each other. We have $Y=E$, $X=E\oplus F\oplus A.B$, $A=Y$
and $B=X$, which gives $X=E.X\oplus E\oplus F$ and $Y=E$. Therefore for
$(E,F)=(0,0)$, we have $(X,Y)=(0,0)$ and for $(E,F)=(0,1)$ we have
$(X,Y)=(1,0)$. However for $(E,F)=(1,0)$ we get $X=X\oplus 1$ which does not
have a solution. For $(E,F)=(1,1)$ we get $X=X$ which is not a unique
solution. Therefore if we demand unique solvability, we must require $E=0$
deterministically which contradicts the initial assumption that $E$ is
uniform. Even if we do not require uniqueness, we can not have $(E,F)=(1,0)$
and forbidding this would make $E$ and $F$ correlated and non-uniform.

Therefore, in the classical, fine-tuned explanation of the Bell correlations,
adding the loop is not consistent with the causal mechanisms that generate the
non-classical correlations in the absence of the loop— in particular, they are
in conflict with the preparation of the exogenous variable $\Lambda$. If we
have a consistent loop, then intervention on $A$ and $B$ will no longer
recover the original non-classical correlations. This is in contrast to the
faithful case analysed in Figure 15 (and explained previously in the text),
when $\mathrm{do}(A,B)$ gives back the non-classical correlations of the Bell
scenario. This suggests that certain (non-local) hidden variable explanations
for quantum correlations (in a Bell experiment) can in principle be
distinguished from the explanation provided by standard quantum mechanics in
the presence of causal loops. We have only shown this for a particular set of
functions or causal mechanisms for generating the former and it would be
interesting to consider if this generalises, in particular to causal
mechanisms provided by Bohmian mechanisms Bohm1952 , a non-local hidden
variable theory.

(a)

(b)

(c)

(d)

Figure 15: A cyclic quantum causal model:  (a) A cyclic variation of the
bipartite Bell causal structure (Figure 1b). (b) A method to calculate the
observed distribution of (a) when $\Lambda$ is non-classical involves this
intermediate causal structure. This is obtained from (a) by copying the nodes
$A$ and $B$ and removing the directed cycle as shown. This gives an acyclic
causal structure for which the distribution $P_{XYABA^{*}B^{*}}$ can be
calculated using known methods. Then, post-selecting on $A=A^{*}$ and
$B=B^{*}$ gives the distribution $P_{XYAB}$ for the original cyclic causal
structure of (a). (c) A classical causal, fine-tuned structure that can
generate, all non-classical correlations of the bipartite Bell causal
structure. Creating a causal loop in this case by adding the arrows
$X\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}B$ and
$Y\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}A$ does not lead to
the same predictions as (a), which corresponds to adding these arrows to the
original Bell causal structure. This method explained in the main text. (d)
The table provides the observed distribution for Example C.1 calculated using
the proposed method. The only values of $A$, $B$, $X$ and $Y$ that are
compatible with the loop conditions $A=Y$ and $B=X$ are those listed here, and
the fifth column lists the measurements and outcomes that these values
correspond to, according to Example C.1. $P_{XYAB}^{QM}$ denotes the
probabilities of the measurements and outcomes listed in the fifth column
calculated using the Born rule. These values are sub-normalised, and upon
renormalisation, the observed distribution $P_{XYAB}$ for the cyclic causal
structure (a) is obtained. Note that the d-separation condition IV.1 is
satisfied in this case.

#### Generalising to other causal structures:

The idea behind the post-selection method employed for Example C.1 above can
in principle be generalised to other non-classical, cyclic causal structures
where every directed cycle includes at least one edge $W\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z$ connecting
classical nodes $W$ and $Z$. The intuition is that cutting off such an edge in
every directed cycle and replacing it with an edge $W^{*}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z$, by introducing an
additional, exogenous variable $W^{*}$ would result in a directed acyclic
graph (DAG). One can then apply the generalised causal model framework of
Henson2014  to obtain the observed distribution in this DAG and then post-
select on $W=W^{*}$ for all the edges that were cut off. Then a way to recover
the d-separation condition (using the result of Forre2017 ) would be to check
whether these exists a classical causal model for the same cyclic causal
structure that produces identical observed correlations and satisfies the
anSEP property. Note that this classical causal model need not yield the same
post-intervention distributions. In the example of Figure 15a, an intervention
on $A$ and $B$ gives the Bell scenario, which as we know produces non-
classical correlations that cannot be obtained in the corresponding classical
causal model Wood2015 . Finally, it would be interesting to compare this
method with the framework of post-selected closed time-like curves Lloyd2011 .

######  Remark C.1.

We note that assuming the d-separation condition of Definition IV.1 as a
primitive property of the framework rules out certain cyclic causal structures
from being described in our current framework. In the classical case, these
are precisely those cyclic causal models that do not satisfy anSEP or those
involving continuous random variables (due to the result of Forre2017 ). An
example of such a causal model is given in Neal2000 , and Forre2017  proposes
a generalisation of d-separation called $\sigma-$separation through which they
derive a _generalised global directed Markov_ condition that applies to
classical causal models involving continuous variables and/or do not satisfy
anSEP. This reduces to d-separation in the acyclic case. Therefore, one option
would be to replace d-separation with $\sigma$-separation in Definition IV.1
to generalise our framework for cyclic causal models. Doing so would not
affect the results of the main paper, but would only enlarge the class of
causal models to which they can be applied.

##  Appendix D Proofs of all results

###  D.1 Proofs of Lemma IV.1 and Theorem IV.1

See IV.1

###### Proof.

The conditional independendence $S_{1}\upmodels S_{2}|S_{3}$ stands for
$P_{S_{1}S_{2}|S_{3}}=P_{S_{1}|S_{3}}P_{S_{2}|S_{3}}$, which implies

$P_{S_{1}|S_{2}S_{3}}=P_{S_{1}|S_{3}}.$ (11)

The three d-separation relations $S\perp^{d}S_{i}$ for $i\in\\{1,2,3\\}$ imply
that $S$ is d-separated from every subset of the union $S_{1}S_{2}S_{3}$. This
implies the following independences by Definition IV.1 of compatibility of the
distribution $P$ with the causal model represented by $\mathcal{G}$,

$P_{S|S^{\prime}}=P_{S}\quad\forall S^{\prime}\subseteq S_{1}S_{2}S_{3}.$ (12)

Now consider the conditional distribution $P_{S_{2}|SS_{1}S_{3}}$. We have,

$\displaystyle\begin{split}P_{S_{2}|SS_{1}S_{3}}&=\frac{P_{S_{2}SS_{1}S_{3}}}{P_{SS_{1}S_{3}}}\\\
&=\frac{P_{S_{3}}P_{S_{2}|S_{3}}P_{S_{1}|S_{2}S_{3}}P_{S|S_{1}S_{2}S_{3}}}{P_{SS_{1}S_{3}}}\\\
&=\frac{P_{S_{3}}P_{S_{2}|S_{3}}P_{S_{1}|S_{3}}P_{S}}{P_{S}P_{S_{1}S_{3}}}\\\
&=P_{S_{2}|S_{3}},\end{split}$ (13)

where we have used Equations (11) and (12) in the third line, noting that
$P_{S|S_{1}S_{3}}=P_{S}\Rightarrow P_{SS_{1}S_{3}}=P_{S}P_{S_{1}S_{3}}$.
Equation (13) is equivalent to
$P_{SS_{1}S_{2}|S_{3}}=P_{SS_{1}|S_{3}}P_{S_{2}|S_{3}}$ which denotes the
conditional independence $SS_{1}\upmodels S_{2}|S_{3}$. The conditional
independence $S_{1}\upmodels SS_{2}|S_{3}$ can be derived analogously due to
the symmetry between $S_{1}$ and $S_{2}$.

Finally, we have

$\displaystyle
P_{S_{2}|SS_{3}}=\frac{P_{S_{2}SS_{3}}}{P_{SS_{3}}}=\frac{P_{S_{3}}P_{S_{2}|S_{3}}P_{S|S_{2}S_{3}}}{P_{S}P_{S_{3}}}=P_{S_{2}|S_{3}},$
(14)

and similarly $P_{S_{1}|SS_{3}}=P_{S_{1}|S_{3}}$. Together with Equation (13),
this implies $P_{S_{2}|SS_{1}S_{3}}=P_{S_{2}|SS_{3}}$. This is equivalent to
$P_{S_{1}S_{2}|SS_{3}}=P_{S_{1}|SS_{3}}P_{S_{2}|SS_{3}}$ which denotes the
final conditional independence $S_{1}\upmodels S_{2}|SS_{3}$. ∎

See IV.1

###### Proof.

Rule 1: We first note that the graph $\mathcal{G}_{\mathrm{do}(X)}$ differs
from $\mathcal{G}_{\overline{X}}$ only by the inclusion of the additional
nodes $I_{X_{i}}$ and corresponding edge $I_{X_{i}}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}X_{i}$ for each
$X_{i}\in X$. Therefore, the d-separation relation
$(Y\perp^{d}Z|XW)_{\mathcal{G}_{\overline{X}}}$ for the latter implies the
same relation $(Y\perp^{d}Z|XW)_{\mathcal{G}_{\mathrm{do}(X)}}$ for the
former. Using the compatibility condition of Definition IV.1 for the graph
$\mathcal{G}_{\mathrm{do}(X)}$, this implies the conditional independence of
$Y$ and $Z$ given $XW$ for the distribution $P_{\mathcal{G}_{\mathrm{do}(X)}}$
i.e.,
$P_{\mathcal{G}_{\mathrm{do}(X)}}(y,z|x,w)=P_{\mathcal{G}_{\mathrm{do}(X)}}(y|x,w)P_{\mathcal{G}_{\mathrm{do}(X)}}(z|x,w)$.
This conditional independence is equivalently expressed by the required
Equation (5).

Rule 2: $\mathcal{G}_{\overline{X},\underline{Z}}$ is the graph where all
incoming arrows to $X$ and outgoing arrows from $Z$ are removed in
$\mathcal{G}$. Hence, the d-separation condition
$(Y\perp^{d}Z|XW)_{\mathcal{G}_{\overline{X},\underline{Z}}}$ implies that the
only paths between $Y$ and $Z$ in the graph $\mathcal{G}_{\overline{X}}$ that
are not blocked by $X$ and $W$ are paths involving an outgoing arrow from $Z$.
These are precisely the paths that get removed in going from
$\mathcal{G}_{\overline{X}}$ to $\mathcal{G}_{\overline{X},\underline{Z}}$,
resulting in the required d-separation there. The same statement holds for the
graph $\mathcal{G}_{\mathrm{do}(X)}$ (by the argument used in the proof of
Rule 1), and also for the graph $\mathcal{G}_{\mathrm{do}(X),I_{Z}}$ which
corresponds to adding the nodes $I_{Z_{i}}$ and edges
$I_{Z_{i}}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{i}$ to
$\mathcal{G}_{\mathrm{do}(X)}$ for each $Z_{i}\in Z$. The latter holds true
since the addition of the $I_{Z_{i}}$ nodes and $I_{Z_{i}}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{i}$ edges cannot
create any additional paths between $Z$ and $Y$ that are left unblocked by $X$
and $W$. This implies that the only paths between $Y$ and the set
$I_{Z}:=\\{I_{Z_{i}}\\}_{i}$ not blocked by $X$ and $W$ in
$\mathcal{G}_{\mathrm{do}(X),I_{Z}}$ are paths from $I_{Z}$, going through $Z$
and involving an outgoing arrow from $Z$ i.e., paths involving the subgraph
$I_{Z}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots$. All these
paths would get blocked when conditioning additionally on $Z$. This gives
$(Y\perp^{d}I_{Z}|XWZ)_{\mathcal{G}_{\mathrm{do}(X),I_{Z}}}$, which through
the compatibility condition (Definition IV.1) implies the conditional
independence $(Y\upmodels I_{Z}|XWZ)_{\mathcal{G}_{\mathrm{do}(X),I_{Z}}}$,
equivalently expressed as

$P_{\mathcal{G}_{\mathrm{do}(X),I_{Z}}}(y|x,w,z,I_{Z}=\mathrm{idle})=P_{\mathcal{G}_{\mathrm{do}(X),I_{Z}}}(y|x,w,z,I_{Z}=\mathrm{do}(z))\quad\forall
y,x,w,z.$ (15)

Using Equations 4a and 4b, we have
$P_{\mathcal{G}_{\mathrm{do}(X),I_{Z}}}(y|x,w,z,I_{Z}=\mathrm{idle})=P_{\mathcal{G}_{\mathrm{do}(X)}}(y|x,w,z)$
and
$P_{\mathcal{G}_{\mathrm{do}(X),I_{Z}}}(y|x,w,z,I_{Z}=\mathrm{do}(z))=P_{\mathcal{G}_{\mathrm{do}(XZ)}}(y|x,w,z)$
respectively $\forall y,x,w,z$. Along with Equation (15), this gives the
required Equation 6. In other words, once $X$, $W$ and $Z$ are given, $Y$ does
not depend on whether the given value $z$ of $Z$ was obtained through an
intervention ($I_{Z}=\mathrm{do}(z)$) or passive observation (i.e., where
$I_{Z_{i}}=\mathrm{idle}$ for all $i$, which is the causal model where no
interventions are made on elements of $Z$).

Rule 3: Consider the graph $\mathcal{G}_{\mathrm{do}(X)I_{Z}}$ which is the
post-intervention graph with respect to the nodes $X$ augmented with
$I_{Z_{i}}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{i}$ for all
$Z_{i}\in Z$. In this graph, suppose we had the d-separation relation
$(Y\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$. By Definition
IV.1, this would result in the conditional independence $(Y\upmodels
I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$ which can be expressed as

$P_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}(Y|W,X,I_{Z}=\mathrm{idle})=P_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}(Y|W,X,I_{Z}=\mathrm{do}(z))\quad\forall
z$

Using the defining rules (4a) and (4b) then gives
$P_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}(Y|W,X,I_{Z}=\mathrm{idle})=P_{\mathcal{G}_{\mathrm{do}(X)}}(Y|W,X)$
and
$P_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}(Y|W,X,I_{Z}=\mathrm{do}(z))=P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|W,X,Z=z)$
$\forall z$, and consequently
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|W,X,Z)=P_{\mathcal{G}_{\mathrm{do}(X)}}(Y|W,X)$
which is the required Equation (7). Therefore, showing that the d-separation
condition $(Y\perp^{d}Z|XW)_{\mathcal{G}_{\overline{X},\overline{Z(W)}}}$
implies the d-separation relation
$(Y\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$ would complete the
proof. This is shown by contradiction. Suppose that
$(Y\perp^{d}Z|XW)_{\mathcal{G}_{\overline{X},\overline{Z(W)}}}$ and
$(Y\not\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$. Then there
must exist a path from a member $I_{Z_{i}}$ of $I_{Z}$ to a member $Y_{j}$ of
$Y$ in $\mathcal{G}_{\mathrm{do}(X)I_{Z}}$ that is unblocked by $X$ and $W$.
There are two possibilities for such a path: either it contains the subgraph
$I_{Z_{i}}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{i}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots Y_{j}$ or the
subgraph $I_{Z_{i}}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{i}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.20001pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{{}}{{}}{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}} {{}} } {{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{17.07181pt}{2.84544pt}\pgfsys@lineto{17.07181pt}{2.84544pt}\pgfsys@lineto{15.57181pt}{1.89545pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{0.39996pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{1.65997pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots Y_{j}$.
Denoting these possibilities as cases 1 and 2 respectively, let $\mathscr{P}$
be the shortest such path. We will show that a contraction arises in each
case.

_Case 1:_ Consider the first case where $\mathscr{P}$ contains the subgraph
$I_{Z_{i}}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{i}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots Y_{j}$. Note
$(Y\not\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$ (which we have
assumed) implies $(Y\not\perp^{d}Z_{i}|XW)_{\mathcal{G}_{\mathrm{do}(X)}}$.
Along with the assumption that
$(Y\perp^{d}Z_{i}|XW)_{\mathcal{G}_{\overline{X},\overline{Z(W)}}}$, this
implies that there exists a path from $Z_{i}$ to $Y$ in
$\mathcal{G}_{\mathrm{do}(X)}$ unblocked by $X$ and $W$ that passes through
some member $Z_{k}$ of $Z(W)$ which would blocked when the incoming arrows to
$Z_{k}$ are removed. This leads to the following subcases where the path from
$Z_{i}$ to $Y_{j}$ in $\mathcal{G}_{\mathrm{do}(X)}$ contains the following
subgraphs:

  * •

_Case 1a:_ $Z_{i}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{k}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.20001pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{{}}{{}}{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}} {{}} } {{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{17.07181pt}{2.84544pt}\pgfsys@lineto{17.07181pt}{2.84544pt}\pgfsys@lineto{15.57181pt}{1.89545pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{0.39996pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{1.65997pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots Y_{j}$ or

  * •

_Case 1b:_ $Z_{i}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.20001pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{{}}{{}}{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}} {{}} } {{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{17.07181pt}{2.84544pt}\pgfsys@lineto{17.07181pt}{2.84544pt}\pgfsys@lineto{15.57181pt}{1.89545pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{0.39996pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{1.65997pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{k}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.20001pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{{}}{{}}{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}} {{}} } {{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{17.07181pt}{2.84544pt}\pgfsys@lineto{17.07181pt}{2.84544pt}\pgfsys@lineto{15.57181pt}{1.89545pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{0.39996pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{1.65997pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots Y_{j}$ or

  * •

_Case 1c:_ $Z_{i}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{k}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots Y_{j}$

Each of these cases are not possible due to the following reasons. In _Case
1a_ , some descendant of $Z_{k}$ must be in $W$ for the path to be unblocked
in $\mathcal{G}_{\mathrm{do}(X)}$ but by definition, $Z(W)$ (which contains
$Z_{k}$) is the set of all nodes in $Z$ that do not have descendants in $W$.
In _Case 1b_ , the path between $Z_{i}$ and $Z_{k}$ must contain a collider.
For this path to be unblocked by $X$ and $W$ in $\mathcal{G}_{\mathrm{do}(X)}$
the collider node must have a descendant in $W$ but the other requirement that
this path must be blocked in $\mathcal{G}_{\overline{X},\overline{Z(W)}}$
implies that the same collider node must be a member of $Z(W)$ which by
definition does not have any descendants in $W$, yielding a contradiction. In
_Case 1c_ , there is either a directed path from $Z_{k}\in Z$ to $Y_{j}\in Y$
in $\mathcal{G}_{\mathrm{do}(X)}$ or a collider in the path between $Z_{k}$
and $Y_{j}$. The latter is ruled out by the same argument used in Case 1b. If
there is a directed path from $Z_{k}$ to $Y_{j}$, then there is a directed
path from $I_{Z_{k}}$ to $Y_{j}$ in $\mathcal{G}_{\mathrm{do}(X)I_{Z}}$ i.e.,
there is a path from a member of $Z$ to $Y$ that is unblocked by $X$ and $W$
in $\mathcal{G}_{\mathrm{do}(X)I_{Z}}$ and that is shorter than the shortest
path $\mathscr{P}$, which is not possible.

Finally, consider _Case 2_ where the path $\mathscr{P}$ contains the subgraph
$I_{Z_{i}}\leavevmode\hbox to17.47pt{\vbox
to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {}
}{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{}} }
{{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@lineto{1.5pt}{3.79544pt}\pgfsys@lineto{4.5pt}{1.89545pt}\pgfsys@lineto{7.5pt}{3.79544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{9.0pt}{2.84544pt}\pgfsys@lineto{16.67184pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.41183pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Z_{i}\leavevmode\hbox
to17.47pt{\vbox to4.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.20001pt\lower
0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{{}}{{}}{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}} {{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}{{}}}
{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}} {} }{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}} {{}} } {{{{}{}{{}} }}{{}} }{{{{}{}{{}} }}{{}}{{}}}
{}{}{}{}{{}}\pgfsys@moveto{17.07181pt}{2.84544pt}\pgfsys@lineto{17.07181pt}{2.84544pt}\pgfsys@lineto{15.57181pt}{1.89545pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{14.07181pt}{2.84544pt}\pgfsys@lineto{0.39996pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{1.65997pt}{2.84544pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ldots Y_{j}$. The
initial assumption that
$(Y\not\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$ implies that
the collider node $Z_{i}$ must have descendants in the conditioning set $W$
i.e., $Z_{i}\not\in Z(W)$. However, in this case we will violate the
assumption that
$(Y\perp^{d}Z|XW)_{\mathcal{G}_{\overline{X},\overline{Z(W)}}}$. On the other
hand, to satisfy this d-separation, we would require $Z_{i}\in Z(W)$ but this
would violate $(Y\not\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$.
Hence we have shown that
$(Y\perp^{d}Z|XW)_{\mathcal{G}_{\overline{X},\overline{Z(W)}}}$ and
$(Y\not\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$ can never be
simultaneously satisfied and hence that
$(Y\perp^{d}Z|XW)_{\mathcal{G}_{\overline{X},\overline{Z(W)}}}$ implies
$(Y\not\perp^{d}I_{Z}|XW)_{\mathcal{G}_{\mathrm{do}(X)I_{Z}}}$ which in turn
implies the required Equation (7).

∎

###  D.2 Proofs of Lemmas IV.3, IV.4, IV.5, IV.6, IV.7, IV.8 and Corollary
IV.3

See IV.3

###### Proof.

  1. 1.

We prove this by contradiction. $X$ is not a cause of $Y$ is equivalent to the
absence of any directed paths from $X$ to $Y$ in $\mathcal{G}$ i.e.,
$(X\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(X)}}$ and consequently
$(X\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$, for any subset $Z$ of
observed nodes, pairwise disjoint to $X$ and $Y$. Since $Z$ is effectively
exogenous in $\mathcal{G}_{\mathrm{do}(XZ)}$,
$(X\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ implies
$(X\perp^{d}Y|Z)_{\mathcal{G}_{\mathrm{do}(XZ)}}$. Applying Rule 3 of Theorem
IV.1 (noting the relation between $\mathcal{G}_{\bar{Z}\bar{X}}$ and
$\mathcal{G}_{\mathrm{do}(XZ)}$) to the latter implies that
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z)$
which is equivalent to $X$ does not affect $Y$ given do$(Z)$.

  2. 2.

This follows from the first part of Lemma IV.8 (proven later in this appendix)
and the first part of this lemma. By Lemma IV.8, $X$ affects $Y$ given
$\\{\mathrm{do}(Z),W\\}$ implies $X$ affects $YW$ given do$(Z)$, which in turn
implies that $X$ must either be a cause of $Y$ or of $W$, by the first part,
proven above.

∎

See IV.4

###### Proof.

To establish the lemma, we show that it is not possible to have $Z$ does not
affect $Y$ given $W$ _and_ $XZ$ does not affect $Y$ given $W$ whenever $X$
affects $Y$ given $\\{\mathrm{do}(Z),W\\}$. Writing these three conditions out
we have

$P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)=P_{\mathcal{G}}(Y|W),$ (16a)
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}}(Y|W),$ (16b)
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)\neq
P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W).$ (16c)

Equations (16a) and (16b) imply
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)$
in contradiction with Equation (16c). ∎

See IV.5

###### Proof.

By Lemma IV.4, if $X$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$ then there
are three only possibilities 1) $Z$ affects $Y$ given $W$ and $XZ$ does not
affect $Y$ given $W$, 2) $Z$ does not affect $Y$ given $W$ and $XZ$ affects
$Y$ given $W$, and 3) $Z$ affects $Y$ given $W$ and $XZ$ affects $Y$ given $W$
i.e., the only case where the required conclusion does not follow is 1). Then
the proof will be complete if we show that whenever $X$ consists only of
exogenous nodes the undesired case does not arise. We show this by
establishing that for exogenous $X$, $Z$ affects $Y$ given $W$ implies $XZ$
affects $Y$ given $W$. Suppose by contradiction that $XZ$ does not affect $Y$
given $W$ i.e.,
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}}(Y|W)$. By the
exogeneity of $X$, this becomes
$P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|X,Z,W)=P_{\mathcal{G}}(Y|W)$ or
equivalently,
$P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y,X,Z,W)=P_{\mathcal{G}}(Y|W)P_{\mathcal{G}_{\mathrm{do}(Z)}}(X,Z,W)$.
Summing over values of $X$ and rearranging gives
$P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)=P_{\mathcal{G}}(Y|W)$ which is
equivalent to $Z$ does not affect $Y$ given $W$. Therefore $Z$ affects $Y$
given $W$ implies $XZ$ affects $Y$ given $W$ whenever $X$ is exogenous. ∎

See IV.6

###### Proof.

By definition, if $X$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$ is reducible,
then there exists a proper subset $s_{X}$ of $X$ such that $s_{X}$ does not
affect $Y$ given $\\{\mathrm{do}(Z\tilde{s}_{X}),W\\}$. We now show that for
every such $s_{X}$, its complement $\tilde{s_{X}}:=X\backslash s_{X}$ is such
that $\tilde{s}_{X}$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$. We show this
by contradiction. Assume that $\tilde{s}_{X}$ does not affect $Y$ given
$\\{\mathrm{do}(Z),W\\}$ while $X$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$
and $s_{X}$ does not affect $Y$ given $\\{\mathrm{do}(Z\tilde{s}_{X}),W\\}$.
Explicitly, these correspond to the following conditions, noting that
$s_{X}\tilde{s}_{X}=X$:

$P_{\mathcal{G}_{\mathrm{do}(\tilde{s}_{X}Z)}}(Y|\tilde{s}_{X},Z,W)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W),$
(17a) $P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)\neq
P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W),$ (17b)
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}_{\mathrm{do}(\tilde{s}_{X}Z)}}(Y|\tilde{s}_{X},Z,W).$
(17c)

Equations (17a) and (17c) imply that
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)$,
which contradicts Equation (17b). ∎

See IV.7

###### Proof.

The proof is similar to that of Lemma IV.6. $X_{1}$ affects $Y$ given
$\\{\mathrm{do}(Z),W\\}$ and $X_{2}$ does not affect $Y$ given
$\\{\mathrm{do}(ZX_{1}),W\\}$ are equivalent to

$P_{\mathcal{G}_{\mathrm{do}(X_{1}Z)}}(Y|X_{1},Z,W)\neq
P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W),$ (18a)
$P_{\mathcal{G}_{\mathrm{do}(X_{1}X_{2}Z)}}(Y|X_{1},X_{2},Z,W)=P_{\mathcal{G}_{\mathrm{do}(X_{1}Z)}}(Y|X_{1},Z,W).$
(18b)

These yield $P_{\mathcal{G}_{\mathrm{do}(X_{1}X_{2}Z)}}(Y|X_{1},X_{2},Z,W)\neq
P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)$ which is equivalent to $X_{1}X_{2}$
affects $Y$ given $\\{\mathrm{do}(Z),W\\}$. ∎

See IV.8

###### Proof.

  1. 1.

We prove this through the contrapositive. Suppose that $X$ does not affect
$YW$ given do$(Z)$ i.e.,

$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y,W|X,Z)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y,W|Z)$
(19)

Summing over values of $Y$ on both sides, we have
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(W|X,Z)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(W|Z)$
i.e., $X$ does not affect $W$ given do$(Z)$. Hence

$\begin{split}P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y,W|X,Z)/P_{\mathcal{G}_{\mathrm{do}(XZ)}}(W|X,Z)&=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y,W|Z)/P_{\mathcal{G}_{\mathrm{do}(Z)}}(W|Z)\\\
\Rightarrow
P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)&=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W),\end{split}$
(20)

which is equivalent to $X$ does not affect $Y$ given $\\{\mathrm{do}(Z),W\\}$.

  2. 2.

Suppose that $X$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$ is irreducible
i.e.,

$\forall s_{X}\subset X,\qquad P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)\neq
P_{\mathcal{G}_{\mathrm{do}(\tilde{s}_{X}Z)}}(Y|\tilde{s}_{X},Z,W),$ (21)

where $s_{X}\tilde{s}_{X}:=X$. If $X$ affects $YW$ given $\mathrm{do}(Z)$ is
reducible, then there exists a partition of $X=s_{X}\tilde{s}_{X}$ such that

$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y,W|X,Z)=P_{\mathcal{G}_{\mathrm{do}(\tilde{s}_{X}Z)}}(Y,W|\tilde{s}_{X},Z).$
(22)

As in the proof of part 1., this implies that
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}_{\mathrm{do}(\tilde{s}_{X}Z)}}(Y|\tilde{s}_{X},Z,W)$,
which contradicts the first equation. Therefore $X$ affects $Y$ given
$\\{\mathrm{do}(Z),W\\}$ is irreducible implies $X$ affects $YW$ given
$\mathrm{do}(Z)$ is irreducible.

  3. 3.

For the forward direction, it is again convenient to use the contrapositive,
i.e., to show that $X$ does not affect $Y$ given $\\{\mathrm{do}(Z),W\\}$ and
$X$ does not affect $W$ given $\mathrm{do}(Z)$ imply $X$ does not affect $YW$
given $\mathrm{do}(Z)$. The first two statements are

$\displaystyle P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)$
$\displaystyle=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)\quad\text{and}$
$\displaystyle P_{\mathcal{G}_{\mathrm{do}(XZ)}}(W|X,Z)$
$\displaystyle=P_{\mathcal{G}_{\mathrm{do}(Z)}}(W|Z).$

Multiplying these gives

$\displaystyle
P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)P_{\mathcal{G}_{\mathrm{do}(XZ)}}(W|X,Z)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)P_{\mathcal{G}_{\mathrm{do}(Z)}}(W|Z)$

which rearranges to

$\displaystyle
P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y,W|X,Z)=P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y,W|Z),$

which is $X$ does not affect $YW$ given $\mathrm{do}(Z)$.

For the reverse direction, we note that we have shown $X$ affects $W$ given
do$(Z)$ implies $X$ affects $YW$ given do$(Z)$ in the proof of part 1 of this
lemma. From the main statement of part 1. we also have $X$ affects $Y$ given
$\\{\mathrm{do}(Z),W\\}$ implies $X$ affects $YW$ given do$(Z)$. Therefore we
have $X$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$ or $X$ affects $W$ given
do$(Z)$ implies $X$ affects $YW$ given do$(Z)$. ∎

See IV.3

###### Proof.

  1. 1.

Given that $X$ affects $Y$ given do$(Z)$ is irreducible, we know that for
every $s_{X}\subset X$, $s_{X}$ affects $Y$ given do$(Z\tilde{s}_{X})$, where
$s_{X}\tilde{s}_{X}:=X$. In particular, this means that for every element
$e_{X}\in X$, $e_{X}$ affects $Y$ given do$(Z\tilde{e}_{X})$. Then by using
Lemma IV.3, we know that $e_{X}$ is a cause of $Y$, which by Definition II.1
means that there exists a directed path from $e_{X}$ to at least one element
$e_{Y}\in Y$ which in turn means that $e_{X}$ is a cause of $e_{Y}$.

  2. 2.

By parts 1 and 2 of Lemma IV.8, $X$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$
implies $X$ affects $YW$ given do$(Z)$ and the irreducibility of the former
implies the irreducibility of the latter which in turn implies (by the first
part of the current lemma) that for every $e_{X}\in X$, there exists
$e_{YW}\in YW$ such that $e_{X}$ is a cause of $e_{YW}$.

∎

###  D.3 Proofs of Theorems V.1, VI.1 and Lemma VI.1

See V.1

###### Proof.

_1._ If compat$(\mathcal{S},\mathscr{A})$ holds then
$\mathcal{R}_{\mathcal{X}}=\overline{\mathcal{F}}(\mathcal{X})$ for all
$\mathcal{X}\in\mathcal{S}$. Hence by Definition V.5 of accessible regions for
sets of ORVs, we have compat1′($\mathcal{S},\mathscr{A}$). The remaining
affects relations in $\mathscr{A}^{\prime}$ are all of the form $\mathcal{X}$
affects $\mathcal{X}^{\prime}$ where $\mathcal{X}^{\prime}$ is a copy of
$\mathcal{X}$, and so, since the location of $\mathcal{X}^{\prime}$ is in
$\mathcal{R}_{\mathcal{X}}=\overline{\mathcal{F}}(\mathcal{X})$,
compat1′($\mathcal{S}^{\prime},\mathscr{A}^{\prime}$) also holds.

_2._ compat1′($\mathcal{S}^{\prime},\mathscr{A}^{\prime}$) when applied to the
affects relations of the form $\mathcal{X}$ affects $\mathcal{X}^{\prime}$
when $\mathcal{X}^{\prime}$ is a copy of $\mathcal{X}$ tells us that
$\overline{\mathcal{F}}(\mathcal{X}^{\prime})\subseteq\overline{\mathcal{F}}(\mathcal{X})$
for every copy $\mathcal{X}^{\prime}$ of $\mathcal{X}$, while Definition V.4
tells us that
$\overline{\mathcal{F}}(\mathcal{X}^{\prime})\subseteq\mathcal{R}_{\mathcal{X}}$
for every copy $\mathcal{X}^{\prime}$ of $\mathcal{X}$. If
$\mathcal{R}_{\mathcal{X}}\nsubseteq\overline{\mathcal{F}}(\mathcal{X})$ then
it would be possible for a copy of $X$ to be accessible outside its future,
and hence that
$\overline{\mathcal{F}}(\mathcal{X}^{\prime})\nsubseteq\overline{\mathcal{F}}(\mathcal{X})$,
contradicting compat1′($\mathcal{S}^{\prime},\mathscr{A}^{\prime}$). Therefore
$\mathcal{R}_{\mathcal{X}}\subseteq\overline{\mathcal{F}}(\mathcal{X})$ must
hold.

∎

See VI.1

###### Proof.

Noting that all affects causal loops of Types 1, 2, 3 and 4 are also affects
causal loops of Type 5, proving the theorem for ACL5 and ACL6 would imply the
required result for ACL1, $\ldots$, ACL6.

  1. 1.

_Proof for ACL5_ Applying Corollary IV.3 to all affects relations in
$S_{i}\subseteq\hat{S}_{i}$, $i=1,\ldots,n$ such that $\\{\hat{S}_{1}$ affects
$S_{2}$, $\hat{S}_{2}$ affects $S_{3}$, $\ldots$,$\hat{S}_{n-1}$ affects
$S_{n}$, $\hat{S}_{n}$ affects $S_{1}\\}\subseteq\mathscr{A}$, we know that
each element of $\hat{S}_{i}$ must be a cause of some element of
$S_{i+1\\!\\!\mod n}$. Following the chain, this implies that each element
$e^{1}\in S_{1}\subseteq\hat{S}_{1}$ is a cause of some element $e^{2}\in
S_{1}$. If $e^{2}=e^{1}$ we are done. If not, we can continue the chain from
$e^{2}$ until we return to an element $e^{3}\in S_{1}$. If $e^{3}=e^{1}$ or
$e^{3}=e^{2}$ we are done; otherwise we continue. Since $S_{1}$ is finite, we
must eventually return to an element of $S_{1}$ we already considered,
establishing a causal loop.

  2. 2.

_Proof for ACL6_ Applying Corollary IV.3 to the first condition of ACL6
(Definition VI.8) we have that for every RV $e^{1}\in s_{1}$, there exists an
RV $e_{2}\in S_{2}$ such that $e^{1}$ is a cause of $e_{2}$. Applying the
Corollary IV.3 to the second condition, we have that $e_{2}\in
S_{2}\subseteq\hat{S}_{2}$ must be a cause of some element $e^{2}\in s_{1}$.
Either $e^{1}=e^{2}$ and we are done or we continue the chain as in the proof
for ACL5.

∎

See VI.1

###### Proof.

  1. 1.

By definition, any set of affects relations that does not contain an affects
causal loop is such that the cyclicity of the underlying causal structure is
not guaranteed by the affects relations. In other words, it is possible to
have the same set of affects relations in a causal model with an acyclic
causal structure $\mathcal{G}$. Every causal model over an acyclic causal
structure admits a non-trivial space-time embedding since an acyclic causal
structure is a directed acyclic graph (DAG) and every DAG implies a partial
order. This embedding would be such that the causal arrows  of $\mathcal{G}$
flow from past to future in the embedded space-time, which ensures no
signalling outside the space-time’s future.

  2. 2.

In faithful causal models, any RV $X$ is a cause of an RV $Y$ if and only if
$X$ affects $Y$. [This follows because if $X$ is a cause of $Y$ then $X\nperp
Y$ in $\mathcal{G}_{\mathrm{do}(X)}$. Since faithful, $X\not\upmodels Y$ in
$\mathcal{G}_{\mathrm{do}(X)}$ and then Lemma IV.2 gives $X$ affects $Y$. The
converse is Lemma IV.3 (which does not rely on faithfulness).] The existence
of a causal loop between $X$ and $Y$ corresponds to $X$ being a cause of $Y$
and $Y$ being a cause of $X$ which is equivalent to $X$ affects $Y$ and $Y$
affects $X$. The latter is the definition of a Type 1 affects causal loop
(Definition VI.3). Hence, under the faithfulness assumption, the absence of a
Type 1 ACL is equivalent to the acyclicity of the underlying causal structure.
As argued in part 1 above, any acyclic causal structure can be non-trivially
and compatibly embedded in any space-time structure.

∎

###  D.4 Proofs of Lemmas A.1 and A.2

See A.1

###### Proof.

  1. 1.

We use Definition IV.1 on the d-separation relation
$(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ to obtain the conditional
independence

$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y).$
(23)

Then noting that $(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ implies
$(XZ\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$, we can apply Corollary IV.2
to the latter d-separation relation to obtain
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y)=P_{\mathcal{G}}(Y)$. Combined with the
above equation, this gives

$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)=P_{\mathcal{G}}(Y).$ (24)

Now, we show that $P_{\mathcal{G}}(Y)=P_{\mathcal{G}}(Y|W)$ must hold in this
case, which would (using the above equation) imply that $XZ$ does not affect
$Y$ given $W$. Suppose by contradiction that $P_{\mathcal{G}}(Y)\neq
P_{\mathcal{G}}(Y|W)$, which would imply that $(Y\nperp^{d}W)_{\mathcal{G}}$.
The assumed d-separation $(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$
implies that $(W\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$. The only way
that we could have d-connection between $Y$ and $W$ in $\mathcal{G}$ but not
in $\mathcal{G}_{\mathrm{do}(XZ)}$ is through the existence of a directed path
between $XZ$ and $Y$ in $\mathcal{G}$ which gives
$(XZ\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$, contradicting our assumption
$(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$. This establishes the first
part.

  2. 2.

We show that $(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ implies
$(ZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$, which in turn implies that
$Z$ does not affect $Y$ given $W$. Then along with the first part of the
lemma, this gives us $(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$
$\Rightarrow$ $XZ$ does not affect $Y$ given $W$ and $Z$ does not affect $Y$
given $W$. Then using Lemma IV.4, this implies that $X$ _does not affect_ $Y$
given $\\{\mathrm{do}(Z),W\\}$, which is the required conclusion.

Suppose that $(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ but
$(ZW\nperp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$. There are two ways that this
is possible

    1. (i)

$(Z\nperp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$ : By assumption, we have
$(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$, which implies
$(Z\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$. The only way we can then have
$(Z\nperp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$ is through the existence of a
directed path from $X$ to $Y$ in $\mathcal{G}_{\mathrm{do}(Z)}$. This gives
$(X\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$, which contradicts our
assumption.

    2. (ii)

$(W\nperp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$ : The assumption
$(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ implies
$(W\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$. If the d-connection
$(W\nperp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$ is due to a directed path from
$W$ to $Y$ in $\mathcal{G}_{\mathrm{do}(Z)}$, this path must go through $X$ in
order to ensure that $(W\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$. But this
would violate the original assumption
$(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ as it would lead to a
directed path from $X$ to $Y$ in $\mathcal{G}_{\mathrm{do}(XZ)}$. On the other
hand, if the d-connection $(W\nperp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$ is
due to a common cause, it is not possible to have the d-connection
$(W\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$, which also contradicts the
assumed d-separation.

The above establishes that $(XZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$
implies $(ZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$, and
$(ZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$ implies $Z$ does not affect
$Y$ given $W$ follows from the first part of the proof (with $Z$ playing the
role of $XZ$).

∎

See A.2

###### Proof.

  1. 1.

The given dependence $(XZW\not\upmodels Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ is
equivalent to

$\exists x,x^{\prime},y,z,z^{\prime},w,w^{\prime}\qquad
P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y=y|X=x,Z=z,W=w)\neq
P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y=y|X=x^{\prime},Z=z^{\prime},W=w^{\prime})$
(25)

Suppose that $XZ$ does not affect $Y$ given $W$ i.e.,

$\forall x,y,z,w\qquad
P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y=y|X=x,Z=z,W=w)=P_{\mathcal{G}}(Y=y|W=w)$
(26)

Clearly it is not possible to satisfy both of the above equations and
$(XZW\not\upmodels Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ must imply $XZ$ affects
$Y$ given $W$.

  2. 2.

Firstly, the d-separation $(ZW\perp^{d}Y)_{\mathcal{G}_{\mathrm{do}(Z)}}$
implies that $Z$ does not affect $Y$ given $W$, which follows from part 1. of
Lemma A.1. From part 1. above, we have $(XZW\not\upmodels
Y)_{\mathcal{G}_{\mathrm{do}(XZ)}}$ implies $XZ$ affects $Y$ given $W$. We now
show that $Z$ does not affect $Y$ given $W$ and $XZ$ affects $Y$ given $W$
implies that $X$ affects $Y$ given $\\{\mathrm{do}(Z),W\\}$, which would
complete the proof. Writing out the first two conditions, we have

$P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)=P_{\mathcal{G}}(Y|W),$ (27)
$P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)\neq P_{\mathcal{G}}(Y|W).$ (28)

Together, these imply that $P_{\mathcal{G}_{\mathrm{do}(XZ)}}(Y|X,Z,W)\neq
P_{\mathcal{G}_{\mathrm{do}(Z)}}(Y|Z,W)$ i.e., $X$ affects $Y$ given
$\\{\mathrm{do}(Z),W\\}$.

∎

###  D.5 Proof of Theorem B.1

See B.1

###### Proof.

The proofs for ACL7 and ACL8 are similar. We describe the proof for ACL8 here,
and at the end explain how it also applies to ACL7. Applying the Corollary
IV.3 to the affects relations $\\{S^{\prime}_{2}$ affects $S_{3}$,
$S^{\prime}_{3}$ affects $S_{4}$, $\ldots$ ,$S^{\prime}_{n-1}$ affects
$S_{n}$, $S^{\prime}_{n}$ affects $s_{1}\\}\subseteq\mathscr{A}$ in the second
condition of ACL8 (Definition B.2) we have that for each element
$e^{\prime}_{2}\in S_{2}$ there exists an element $e_{3}\in S_{3}$ of which it
is a cause, for each element $e^{\prime}_{3}\in S_{3}$ there exist an element
$e_{4}\in S_{4}$ of which it is a cause, $\ldots$ , for each element
$e^{\prime}_{n}\in S_{n}$ there exist an element $e_{1}\in s_{1}\subseteq
S_{1}$ of which it is a cause. This does not immediately imply that there is a
directed path from $S^{\prime}_{2}$ to $s_{1}$, since for example the element
$e^{\prime}_{3}\in S_{3}$ of which $e^{\prime}_{2}\in S_{2}$ is a cause might
not belong to the next set $S^{\prime}_{3}$ in the chain, i.e., we could have
$e_{3}\in S_{3}\backslash(S_{3}\bigcap S^{\prime}_{3})$ if
$(S_{3},S_{3}^{\prime})$ forms an incomplete node of $\mathscr{C}_{e_{2}}$. In
this case, the third condition of Definition B.2 tells us that there is
another complete affects chain $\mathscr{D}^{\mathscr{C}}_{e_{2}}$ that
connects $S_{3}\backslash(S_{3}\bigcap S^{\prime}_{3})$ to $S_{3}$. Since this
is a complete affects chain, we can apply the same argument as in the proof of
Theorem VI.1 to conclude that for each element in
$S_{3}\backslash(S_{3}\bigcap S^{\prime}_{3})$, there exists an element
$e^{*}_{3}\in S_{3}$ of which it is a cause. We consider two cases depending
on whether we have $e^{*}_{3}\in S_{3}\backslash(S_{3}\bigcap S^{\prime}_{3})$
or $e^{*}_{3}\in S_{3}\bigcap S^{\prime}_{3}$. We will show that in the former
case, the affects relations in the secondary chain
$\mathscr{D}^{\mathscr{C}}_{e_{2}}$ already guarantees cyclicity while the
latter case, these (the set formed by such secondary chains, one for every
incomplete node) guarantee cyclicity when taken together with those in the
primary chain $\mathscr{C}_{e_{2}}$.

In the first case, $\mathscr{D}^{\mathscr{C}}_{e_{2}}$ corresponds to a Type 5
affects causal loop since it involves a complete chain of irreducible affects
relations from a set $S_{3}\backslash(S_{3}\bigcap S^{\prime}_{3})$ on to
itself. The cyclicity claim for this case then follows from Theorem VI.1.
Therefore, we now consider the case where for each incomplete node
$(S_{i},S_{i}^{\prime})$ of $\mathscr{C}_{e_{2}}$, the corresponding element
$e^{*}_{i}\in S_{i}$ belongs to the intersection of the sets $S_{i}\bigcap
S^{\prime}_{i}$. Then, applying Corollary IV.3 repeatedly to each pair of
affects relations in $\\{S^{\prime}_{2}$ affects $S_{3}$, $S^{\prime}_{3}$
affects $S_{4}$, $\ldots$ ,$S^{\prime}_{n-1}$ affects $S_{n}$,
$S^{\prime}_{n}$ affects $s_{1}\\}\subseteq\mathscr{A}$, we can conclude that
for every element $e^{\prime}_{2}\in S_{2}^{\prime}$, there exists an element
$e_{1}\in s_{1}\subseteq S_{1}$ such that $e_{2}^{\prime}$ is a cause of
$e_{1}$. By Definition B.2 (second condition), we considered such a set
$\mathscr{A}$ of affects relations for every element $e_{2}\in S_{2}$,
defining $S_{2}^{\prime}$ such that it contains $e_{2}$. Since the above
argument holds for all sets of affects relations $\mathscr{A}$ defined as
above and for all elements of $S^{\prime}_{2}$, this implies that for every
element $e_{2}\in S_{2}$, there exists a corresponding element $e_{1}\in
S_{1}$ of which it is a cause. Applying Corollary IV.3 to the first condition
of Definition B.2 i.e., the irreducible affects relation $S_{1}$ affects
$S_{2}$, we have that for every element $e_{1}\in S_{1}$, there exists a
corresponding element $e_{2}\in S_{2}$ of which it is a cause. This was also
the case for ACL1-6 as shown in Theorem VI.1, so the statement of the present
theorem then follows from the proof of Theorem VI.1.

For ACL7, the first condition says that there is an irreducible affects
relation $S_{1}$ affects $S_{2}$ in $\mathscr{A}$ and the second condition of
Definition B.1 guarantees the existence of an affects chain from
$s_{2}\subseteq S_{2}$ to $S_{1}$. The subtlety here is to note that if
$s_{2}\subset S_{2}$, then $(S_{2},S^{\prime}_{2})$ will be an incomplete node
of $\mathscr{C}_{s_{2}}$. By the above proof for ACL8, we have concluded that
the affects relations $\\{S^{\prime}_{2}$ affects $S_{3}$, $S^{\prime}_{3}$
affects $S_{4}$, $\ldots$ ,$S^{\prime}_{n-1}$ affects $S_{n}$,
$S^{\prime}_{n}$ affects $s_{1}\\}\subseteq\mathscr{A}$ along with the third
condition of ACL8 (which is similar for ACL7) either imply cyclicity of the
causal structure or that for every element $e^{\prime}_{2}\in S_{2}^{\prime}$,
there exists an element $e_{1}\in s_{1}\subseteq S_{1}$ such that
$e_{2}^{\prime}$ is a cause of $e_{1}$. If the node $(S_{2},S^{\prime}_{2})$
is also incomplete as noted above, one can extend the same arguments using the
third condition to conclude that either the causal structure is cyclic or for
every element $e_{2}\in S_{2}$, there exists an element $e_{1}\in
s_{1}\subseteq S_{1}$ such that $e_{2}$ is a cause of $e_{1}$. The same
condition was obtained at the end of the previous paragraph, in the proof for
ACL8, and shown to imply cyclicity. Therefore this establishes the theorem
also for ACL7. ∎

## References

  * (1) Vilasini, V. & Colbeck, R.  Impossibility of superluminal signalling in Minkowski space-time does not rule out causal loops.  arXiv:2206.12887 (2021).  link.
  * (2) Pearl, J.  Causality: Models, reasoning, and inference.  _Second edition, Cambridge University Press_ (2009).  link.
  * (3) Spirtes, P., Glymour, C. N. & Scheines, R.  _Causation, prediction, and search_ (The MIT Press, 2nd ed., 2001).  link.
  * (4) Kleinberg, S. & Hripcsak, G.  A review of causal inference for biomedical informatics.  _Journal of Biomedical Informatics_ 44, 1102–1112 (2011).  link.
  * (5) Raita, Y., Camargo, C. A., Liang, L. & Hasegawa, K.  Big data, data science, and causal inference: A primer for clinicians.  _Frontiers in Medicine_ 8, 998 (2021).  link.
  * (6) Spirtes, P.  Graphical models, causal inference, and econometric models.  _J. Econ. Methodol._ 12, 3–34 (2005).  link.
  * (7) Petersen, M. J., Maya L.; van der Laan.  Causal models and learning from data.  _Epidemiology_ 25, 418–426 (2014).  link.
  * (8) Arti, S., Hidayah, I. & Kusumawardani, S. S.  Research trend of causal machine learning method: A literature review.  _IJID (International Journal on Informatics for Development)_ 9, 111–118 (2020).  link.
  * (9) Liu, T., Ungar, L. & Kording, K.  Quantifying causality in data science with quasi-experiments.  _Nature Computational Science_ 1, 24–32 (2021).  link.
  * (10) Bell, J. S.  _Speakable and unspeakable in quantum mechanics_ (Cambridge University Press, 1987).
  * (11) Wood, C. J. & Spekkens, R. W.  The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning.  _New Journal of Physics_ 17, 33002 (2015).  link.
  * (12) Tucci, R. R.  Quantum Bayesian nets.  _International Journal of Modern Physics B_ 09, 295–337 (1995).  link.
  * (13) Leifer, M. S.  Quantum dynamics as an analog of conditional probability.  _Physical Review A_ 74 (2006).  link.
  * (14) Laskey, K. B.  Quantum causal networks.  arXiv:0710.1200 (2007).  link.
  * (15) Leifer, M. & Poulin, D.  Quantum graphical models and belief propagation.  _Annals of Physics_ 323, 1899–1946 (2008).  link.
  * (16) Leifer, M. S. & Spekkens, R. W.  Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference.  _Physical Review A_ 88, 052130 (2013).  link.
  * (17) Henson, J., Lal, R. & Pusey, M.  Theory-independent limits on correlations from generalized Bayesian networks.  _New Journal of Physics_ 16, 113043 (2014).  link.
  * (18) Pienaar, J. & Brukner, Č.  A graph-separation theorem for quantum causal models.  _New Journal of Physics_ 17, 73020 (2015).  link.
  * (19) Ried, K. _et al._ A quantum advantage for inferring causal structure.  _Nature Physics_ 11, 414–420 (2015).  link.
  * (20) Costa, F. & Shrapnel, S.  Quantum causal modelling.  _New Journal of Physics_ 18, 63032 (2016).  link.
  * (21) Fritz, T.  Beyond Bell’s theorem II: Scenarios with arbitrary causal structure.  _Communications in Mathematical Physics_ 341, 391–434 (2015).  link.
  * (22) Allen, J.-M. A., Barrett, J., Horsman, D. C., Lee, C. M. & Spekkens, R. W.  Quantum common causes and quantum causal models.  _Physical Review X_ 7, 031021 (2017).  link.
  * (23) Barrett, J., Lorenz, R. & Oreshkov, O.  Quantum causal models.  arXiv:1906.10726 (2020).  link.
  * (24) Pienaar, J.  Quantum causal models via quantum Bayesianism.  _Physical Review A_ 101 (2020).  link.
  * (25) Forré, P. & Mooij, J. M.  Markov properties for graphical models with cycles and latent variables.  arXiv:1710.08775 (2017).  link.
  * (26) Bongers, S., Forré, P., Peters, J. & Mooij, J. M.  Foundations of structural causal models with cycles and latent variables.  _The Annals of Statistics_ 49 (2021).  link.
  * (27) Araújo, M., Guérin, P. A. & Baumeler, A.  Quantum computation with indefinite causal structures.  _Physical Review A_ 96 (2017).  link.
  * (28) Barrett, J., Lorenz, R. & Oreshkov, O.  Cyclic quantum causal models.  arXiv:2002.12157 (2020).  link.
  * (29) Vilasini, V. & Renner, R.  Embedding cyclic causal structures in acyclic spacetimes: no-go results for process matrices.  arXiv:2203.11245 (2022).  link.
  * (30) Grunhaus, J., Popescu, S. & Rohrlich, D.  Jamming nonlocal quantum correlations.  _Phys. Rev. A_ 53, 3781–3784 (1996).  link.
  * (31) Horodecki, P. & Ramanathan, R.  The relativistic causality versus no-signaling paradigm for multi-party correlations.  _Nature Communications_ 10, 1701 (2019).  link.
  * (32) Oreshkov, O., Costa, F. & Brukner, Č.  Quantum correlations with no causal order.  _Nature Communications_ 3, 1092 (2012).  link.
  * (33) Zych, M., Costa, F., Pikovski, I. & Brukner, Č.  Bell’s theorem for temporal order.  _Nature Communications_ 10, 3772 (2019).  link.
  * (34) Agresti, I. _et al._ Experimental test of quantum causal influences.  arXiv:2108.08926 (2021).  link.
  * (35) Kent, A.  Secure classical bit commitment over finite channels.  _Journal of Cryptology_ 18, 313–335 (2005).  link.
  * (36) Colbeck, R. & Kent, A.  Variable bias coin tossing.  _Physical Review A_ 73, 032320 (2006).
  * (37) Bohm, D.  A suggested interpretation of the quantum theory in terms of “hidden” variables. I.  _Physical Review_ 85, 166–179 (1952).  link.
  * (38) Vilasini, V. & Colbeck, R.  In preparation (2022).
  * (39) Barrett, J.  Information processing in generalized probabilistic theories.  _Phys. Rev. A_ 75, 032304 (2007).  link.
  * (40) Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. & Wehner, S.  Bell nonlocality.  _Reviews of Modern Physics_ 86, 419–478 (2014).  link.
  * (41) Geiger, D.  Towards the formalization of informational dependencies.  _Tech. rep. 880053. UCLA Computer Science_ (1987).  link.
  * (42) Verma, T. & Pearl, J.  Causal networks: Semantics and expressiveness.  In _Proceedings of the Fourth Annual Conference on Uncertainty in Artificial Intelligence_ , UAI ’88, 69–78 (North-Holland Publishing Co., Amsterdam, The Netherlands, The Netherlands, 1990).  link.
  * (43) Verma, T. & Pearl, J.  Causal networks: Semantics and expressiveness.  _Proceedings of the Fourth Annual Conference on Uncertainty in Artificial Intelligence (UAI ’88)_ 69–78 (1990).  link.
  * (44) Pearl, J. & Dechter, R.  Identifying independencies in causal graphs with feedback.  arXiv:1302.3595 (2013).  link.
  * (45) Friedman, M.  The fed’s thermostat.  _The Wall Street Journal_ (2003).  link.
  * (46) Rowe, N.  Why there’s so little good evidence that fiscal (or monetary) policy works (online) (2009).  link.
  * (47) Brulé, J.  A causation coefficient and taxonomy of correlation/causation relationships.  arXiv:1708.05069 (2017).  link.
  * (48) Colbeck, R. & Renner, R.  No extension of quantum theory can have improved predictive power.  _Nature Communications_ 2, 411 (2011).  link.
  * (49) Colbeck, R. & Renner, R.  A short note on the concept of free choice.  arXiv:1302.4446 (2013).  link.
  * (50) Salazar, R. _et al._ A no-go theorem for device-independent security in relativistic causal theories.  arXiv:1712.01030 (2020).  link.
  * (51) Reichenbach, H.  _The direction of time_ (Univ. of California Press, Berkeley - Los Angeles, 1956).
  * (52) Pearl, J.  Causal diagrams for empirical research.  _Biometrika_ 82, 669–688 (1995).  link.
  * (53) Kempf, A.  Replacing the notion of spacetime distance by the notion of correlation.  _Frontiers in Physics_ 9 (2021).  link.
  * (54) Bombelli, L., Lee, J., Meyer, D. & Sorkin, R. D.  Space-time as a causal set.  _Phys. Rev. Lett._ 59, 521–524 (1987).  link.
  * (55) Surya, S.  The causal set approach to quantum gravity.  _Living Reviews in Relativity_ 22, 5 (2019).  link.
  * (56) Dukovski, I.  Causal structure of spacetime and geometric algebra for quantum gravity.  _Phys. Rev. D_ 87, 064022 (2013).  link.
  * (57) Coecke, B. & Kissinger, A.  _Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning_ (Cambridge University Press, 2017).  link.
  * (58) Giacomini, F., Castro-Ruiz, E. & Brukner, Č.  Quantum mechanics and the covariance of physical laws in quantum reference frames.  _Nature Communications_ 10, 494 (2019).  link.
  * (59) Castro-Ruiz, E., Giacomini, F., Belenchia, A. & Brukner, Č.  Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems.  _Nature Communications_ 11 (2020).  link.
  * (60) Deutsch, D.  Quantum mechanics near closed timelike lines.  _Phys. Rev. D_ 44, 3197–3217 (1991).  link.
  * (61) Bennett, C. & Schumacher, B.  Talk at QUPON Wien.  link.
  * (62) Svetlichny, G.  Time travel: Deutsch vs. teleportation.  _International Journal of Theoretical Physics_ 50, 3903–3914 (2011).  link.
  * (63) Lloyd, S., Maccone, L., Garcia-Patron, R., Giovannetti, V. & Shikano, Y.  Quantum mechanics of time travel through post-selected teleportation.  _Phys. Rev. D_ 84, 025007 (2011).  link.
  * (64) Lloyd, S. _et al._ Closed timelike curves via postselection: Theory and experimental test of consistency.  _Phys. Rev. Lett._ 106, 040403 (2011).  link.
  * (65) Aaronson, S. & Watrous, J.  Closed timelike curves make quantum and classical computing equivalent.  _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 465, 631–647 (2008).  link.
  * (66) Aharonov, Y., Bergmann, P. G. & Lebowitz, J. L.  Time symmetry in the quantum process of measurement.  _Phys. Rev._ 134, B1410–B1416 (1964).  link.