Topological Conditional Separation on Dgs

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2 Alexandrov topology on a graph

In §2.1, we provide background on binary relations and graphs. In §2.2, we present Alexan- drov topologies induced by binary relations.

2.1 Background on binary relations, graphs and topologies We use the notation Jr, sK = {r, r +1,...,s − 1,s} for any two integers r, s such that r ≤ s.

2.1.1 Binary relations Let V be a nonempty set (ﬁnite or not). We recall that a (binary) relation R on V is a subset R⊂V×V and that γ R λ means (γ,λ) ∈ R. For any subset Γ ⊂ V, the (sub)diagonal relation is ∆Γ = (γ,λ) ∈V×V γ = λ ∈ Γ and the diagonal relation is ∆ = ∆V . A foreset of a relation R is any set of the form R λ = γ ∈V γ R λ , where λ ∈V, or, by extension, of the form R Λ = γ ∈V ∃λ ∈ Λ , γ R λ, where Λ ⊂ V. An afterset of a relation R is any set of the form γ R = λ ∈V γ R λ , where γ ∈ V, or, by extension, of the form c Γ R = λ ∈V ∃γ ∈ Γ , γ R λ , where Γ ⊂ V. The opposite or complementary R of a c c binary relation R is the relation R = V×V\R, that is, deﬁned by γ R λ ⇐⇒ ¬(γ R λ).

The converse R−1 of a binary relation R is defined by γ R−1 λ ⇐⇒ λ R γ. A relation R is symmetric if R−1 = R, and is anti-symmetric if R−1 ∩ R ⊂ ∆. The composition RR′ of two binary relations R, R′ on V is defined by γ(RR′)λ ⇐⇒ ∃δ ∈ V, γ R δ and δ R′ λ; then, by induction we define Rn+1 = RRn for n ∈ N∗. The + ∞ k + transitive closure of a binary relation R is R = ∪k=1R (and R is transitive if R = R) and ∗ + ∞ k 0 the reflexive and transitive closure is R = R ∪ ∆= ∪k=0R with the convention R = ∆. A partial equivalence relation is a symmetric and transitive binary relation (generally denoted by ∼ or ≡). An equivalence relation is a reflexive, symmetric and transitive binary relation.

2.1.2 Preorders A preorder (or “quasi-ordering”) on V is a reﬂexive and transitive binary relation (generally denoted by ), whereas an order is an anti-symmetric preorder (generally denoted by ≤). For a preorder, the foreset (resp. afterset) of a subset Γ ⊂ V is called the downset (resp. upset) of Γ and is denoted by ↓Γ (resp. by ↑Γ): ↓Γ= α ∈V ∃γ ∈ Γ , α γ , ↑Γ= α ∈V ∃γ ∈ Γ , γ α . Then, a subset Γ ⊂V is called an upper set (resp. a lower set) — oralso an upward closed set (resp. downward closed set) — with respect to the preorder if ↓ V ⊂ V (resp. ↑ V ⊂ V ) or, equivalently, if ↓V = V (resp. ↑V = V ).

2 2.1.3 Graphs Let V be a nonempty set (finite or not), whose elements are called vertices. Let E⊂V×V be a relation on V, whose elements are ordered pairs (that is, couples) of vertices called edges. The first element of an edge is the tail of the edge, whereas the second one is the head of the edge. Both tail and head are called endpoints of the edge, and we say that the edge connects its endpoints. We define a loop as an element of ∆ ∩E, that is, a loop is an edge that connects a vertex to itself. A graph, as we use it throughout this paper, is a couple (V, E) where E⊂V×V. This definition is very basic and we now stress proximities and differences with classic notions in graph theory. As we define a graph, it may hold a finite or infinite number of vertices; there is at most one edge that has a couple of ordered vertices as single endpoints, hence a graph (in our sense) is not a multigraph (in graph theory); loops are not excluded (since we do not impose ∆ ∩E = ∅). Hence, what we call a graph would be called a directed simple graph permitting loops in graph theory.

2.1.4 Topologies We refer the reader to [4, Chapter 4] for notions in topology. Let V be a nonempty set. The set T ⊂ 2V is said to be a topology on V if T contains both ∅, V and is stable under the union and ﬁnite intersection operations. The space (V, T ) is called topological space. Any element O ∈ T is called an open set (more precisely a T -open set), and any element in

T ′ = C ⊂V Cc ∈ T (1) is called a closed set (more precisely a T -closed set). For any subset V ⊂V, the intersection of all the closed sets that contain V is a closed set called topological closure and denoted T by V (or, when needed, V ). A clopen set (more precisely a T -clopen set) is a subset of V which is both closed and open, that is, an element of T ∩ T ′. A topological space (V, T ) is said to be disconnected if it is the union of two disjoint nonempty open sets; otherwise, it is said to be connected. A subset V ⊂V of V is said to be connected (more precisely T -connected) if it is connected under its subspace topology T ∩ V = O ∩ V ∈ T O ∈ T (also called trace topology or relative topology). A connected component of the topological space (V, T ) (also called a T -

connected component) is a maximal (for the inclusion order) connected subset. A connected component is necessarily closed and the connected components of (V, T ) form a partition of V [4, Exercise 4.11.13]. Any clopen set is a union of (possibly inﬁnitely many) connected components. Let (Vi, Ti), i =1, 2 be two topological spaces. The product topology T1 ⊗T2 is the smallest V1×V2 subset T ⊂ 2 which is a topology on the product set V1 ×V2 and which contains all the ﬁnite rectangles O1 × O2 Oi ∈ Ti, i =1, 2 .

Specialization preorder. With any topology T on V, one associates the so-called specialization (or canonical) preorder as the binary relation T on V deﬁned by [4, § 4.2.1,

3 Lemma 4.2.7] T γ T λ ⇐⇒ γ ∈ λ ∀γ,λ ∈V . (2) The relation T is reﬂexive and transitive, hence is a preorder (hence the notation). Follow- ing the notation in §2.1.1 — with the notation ↓T for a downset and ↑T for an upset — we have that T ↓T λ = λ , ∀λ ∈V , (3) it is readily shown (and well-known [4, Lemmas 4.2.6 and 4.2.7]) that every open set is an upper set and every closed set is a lower set.

Preorder topology. It can be shown that, for any preorder on V, the set

T = O ⊂V ↑O ⊂ O (4) is a topology and that it is the ﬁnest topology T that has as specialization order (that is, such that T =) [4, Proposition 4.2.11]. The topology T is an Alexandrov topology as follows.

Alexandrov topology. The set T ⊂ 2V is said to be an Alexandrov topology on V if T contains both ∅, V and is stable under the union and (not necessarily ﬁnite) intersection operations. If T is an Alexandrov topology, then T ′ in (1) also is an Alexandrov topology, that we call the dual (Alexandrov) topology [1].

It is established that a topology T is an Alexandrov topology if and only if T = TT , where T is the specialization preorder of T , that is, if and only if the open sets (in T ) are ′ exactly the upper sets (with respect to T ) — or, equivalently, the closed sets (in T ) are exactly the lower sets (with respect to T ) [4, Proposition 4.2.11, Exercise 4.2.13]. Thus, if T is an Alexandrov topology, we have that

′ O ∈T ⇐⇒↓T O ⊂ O and C ∈ T ⇐⇒ ↑T C ⊂ C . (5)

In an Alexandrov topology, it can be shown that, for any family {Γs}s∈S of subsets Γs ⊂V, the topological closure satisﬁes

Γs = Γs , ∀Γs ⊂V , s ∈ S . (6) s[∈S s[∈S

Indeed, as Γs ⊂ s′∈S Γs′ , we get that s∈S Γs ⊂ s∈S Γs ⊂ s∈S Γs. As the set s∈S Γs is closed, by deﬁnitionS of Alexandrov topology,S we conclude.S ByS (6), it is readily deducedS that

Γ = ↓T Γ , ∀Γ ⊂V . (7)

4 2.2 Alexandrov topology induced by a binary relation As recalled, a topology is an Alexandrov topology if and only if it is the topology of a preorder [4, Exercise 4.2.13]. In fact, one can associate a topology with any binary relation (see (8) below) and prove that a topology is an Alexandrov topology if and only if it is the topology of a binary relation [1, Théorème 1.2]. In Proposition 1, we analyze the Alexandrov topology induced by a binary relation; we recover known results [1, Théorème 1.2] and we add some new results. Additional results are provided in Appendix A.

Proposition 1 Let (V, E) be a graph, that is, V is a set and E⊂V×V. The following set

TE = O ⊂V OE ⊂ O (8) is an Alexandrov topology on V with the property that open subsets are characterized by

+ ∗ ∗ O ∈ TE ⇐⇒ OE ⊂ O ⇐⇒ OE ⊂ O ⇐⇒ OE ⊂ O ⇐⇒ OE = O. (9)

1 E In the Alexandrov topology TE , the topological closure Γ of a subset Γ ⊂V is given by

E Γ = E ∗Γ , ∀Γ ⊂V , (10)

that is, is the E ∗-foreset; the closed subsets are characterized by

′ + ∗ ∗ E C ∈ TE ⇐⇒ EC ⊂ C ⇐⇒ E C ⊂ C ⇐⇒ E C ⊂ C ⇐⇒ E C = C ⇐⇒ C = C. (11)

The Alexandrov topology TE satisﬁes

TE = TE+ = TE∗ , (12a)

′ and the dual Alexandrov topology TE satisﬁes

′ −1 −1 + −1 ∗ TE = TE = T(E ) = T(E ) . (12b)

Regarding the specialization preorder T , it is well-known that, for any preorder on V, we have that T = [4, Proposition 4.2.11]. More generally, it holds that

∗ 2 TE = E , ∀E ⊂V . (13)

Proof. We prove that the set TE in (8) contains both ∅, V and is stable under the union and intersection operations, be they ﬁnite or inﬁnite, which is what is required for an Alexandrov topology. Indeed, both ∅, V ∈TE as ∅E = ∅ and VE ⊂V. Let {Os}s∈S be a family in TE , that is, OsE ⊂ Os for all s ∈ S. We deduce that (∪s∈SOs)E = ∪s∈SOsE⊂∪s∈SOs, hence stability by union, and also that (∩s∈SOs)E⊂∩s∈SOsE⊂∩s∈SOs, hence stability by intersection.

E T 1To alleviate the notation, we have denoted the topological closure by Γ instead of Γ E .

5 We establish the useful equivalences:

OE∗ = O ⇐⇒ OE∗ ⊂ O (because O ⊂ OE∗ since ∆ ⊂E∗ = E+ ∪ ∆) ⇐⇒ OE+ ⊂ O (because E∗ = E+ ∪ ∆) + ∞ k ⇐⇒ OE ⊂ O (because E = ∪k=1E and then by induction) c c ⇐⇒ EO ⊂ O indeed, suppose by contradiction that OE ⊂ O but that there exists α ∈ EOc such that α 6∈ Oc, that is, α ∈ O; as a consequence, there exists γ ∈ Oc such that αEγ, hence that γ ∈ αE; now, as α ∈ O, we get that γ ∈ αE ⊂ OE ⊂ O by assumption; but this contradicts that γ ∈ Oc; the reverse implication is proved in the same way ; the rest of the equivalences below are proved as above

c c ⇐⇒ E+O ⊂ O c c ⇐⇒ E∗O ⊂ O c c ⇐⇒ E∗O = O .

∗ We deduce that (9) holds true, hence also that TE = TE+ = TE by (8), and that (11) holds true,

′ −1 −1 + −1 ∗ hence also that TE = TE = T(E ) = T(E ) by (8) and by (1). E Finally, we consider a subset Γ ⊂V and we characterize its topological closure Γ , the smallest closed subset that contains Γ. On the one hand, we have that Γ ⊂ E∗Γ since E∗ = E+ ∪ ∆. On the other hand, the set E∗Γ is closed since E∗(E∗Γ) = (E∗)2Γ = E∗Γ, because the relation E∗ is E E transitive. By deﬁnition of the topological closure Γ , we deduce that Γ ⊂E∗Γ. Now, let Λ ⊂V be a closed subset such that Γ ⊂ Λ. We necessarily have that E∗Γ ⊂E∗Λ = Λ, where the last equality is by (11) as Λ is closed. E As a consequence, the topological closure Γ will always contain the closed set E∗Λ, from which E E we get that E∗Λ ⊂ Γ . We conclude that Γ = E∗Γ. This ends the proof.

3 Equivalence between d-separation and t-separation

In §3.1, we recall the (extended) deﬁnition of d-separation, then introduce a suitable topology on the set of vertices, and deﬁne a new notion of conditional topological separation (t-separation). Then, we show that d-separation and t-separation between vertices (and between subsets of vertices) are equivalent. In §3.2, we put forward a practical characterization of t-separation between subsets of vertices.

3.1 d- and t-separation between vertices We first recall the (extended) definition of d-separation, second define a new notion of conditional topological separation (t-separation) and third prove their equivalence.

6 3.1.1 d-separation between vertices In the companion paper [2] we generalize Pearl’s d-separation beyond acyclic graphs as follows.

Definition 2 ([2, Definition 3]) Let (V, E) be a graph, that is, V is a set and E⊂V×V, and let W ⊂V be a subset of vertices. We define the conditional parental relation E W as

W W c E = ∆W c E that is, γE λ ⇐⇒ γ ∈ W and γEλ ∀γ,λ ∈V , (15a) the conditional ascendent relation BW as

W ∗ W ∗ W ∗ W ∗ B = E(∆W c E) = EE where E =(E ) (15b)

which relates a descendent with an ascendent by means of elements in W c. We deﬁne their converses E −W and B−W as

−W W −1 −1 E =(E ) = E ∆W c , (15c) −W W −1 −1 ∗ −1 −W ∗ −1 −W ∗ −W ∗ B = B =(E ∆W c ) E = E E where E =(E ) . (15d) With these elementary binary relations, we deﬁne the conditional common cause relation KW as the symmetric relation

W −W W −W + W + K = B ∆W c B = E E , (15e)

the conditional cousinhood relation CW as the partial equivalence relation

W W + C = ∆W K ∆W ∪ ∆W , (15f) and the conditional active relation AW as the symmetric relation

AW = ∆ ∪ BW ∪ B−W ∪KW ∪ BW ∪KW CW B−W ∪KW . (15g) With the conditional active relation AW , we can now deﬁne the notion of d-separation between vertices (which can readily be extended to d-separation between subsets of vertices).

Deﬁnition 3 (d-separation between vertices, [2, Deﬁnition 2]) Let (V, E) be a graph, that is, V is a set and E⊂V×V, and let W ⊂ V be a subset of vertices. Let γ, λ ∈ V be two vertices. We denote γ k λ | W ⇐⇒ ¬(γAW λ) , (16) d and we say that the vertices γ and λ are d-separated (w.r.t. W ).

7 3.1.2 t-separation between vertices We introduce a suitable topology on the set of vertices, and we deﬁne a new notion of conditional topological separation. Let (V, E) be a graph, W ⊂V be a subset of vertices, and E W in (15a) be the correspond- W ing conditional parental relation. To alleviate the notation, in the Alexandrov topology TE W in (8), we use the following. For any subset Γ ⊂V, the topological closure is denoted2 by Γ , 3 and the downset is denoted by ↓W Γ. By (7) and (10), we get that

W W ∗ Γ = E Γ= ↓W Γ . (17)

c W W Notice that the subset W is TE -open, that is, W ∈ TE . Indeed, the complementary set W W ∗ c W + c c c W c W is closed as it satisfies E W =(E ) W ∪ W ⊂ W , as E V ⊂ W because E = ∆W c E by (15a) and by definition of the subdiagonal relation ∆W c . With the conditional ascendent relation BW , the conditional common cause relation KW W W the conditional cousinhood relation C and the TE -topological closure, we can now define the notion of t-separation between vertices (which can readily be extended to t-separation between subsets of vertices).

Deﬁnition 4 (Conditional topological separation between vertices, t-separation) Let (V, E) be a graph, and W ⊂V be a subset of vertices. We set

SW = ∆ ∪ CW B−W ∪KW . (18) Let γ, λ ∈V be two vertices. We denote

W W γ k λ | W ⇐⇒ SW γ ∩ SW λ = ∅ , (19) t and we say that the vertices γ and λ are conditionally topologically separated (w.r.t. W ) or, shortly, t-separated.

With the above deﬁnitions, we now show that the notions of d- and t-separation are equivalent on the complementary set W c.

Theorem 5 Let (V, E) be a graph, that is, V is a set and E⊂V×V, and let W ⊂ V be a subset of vertices. We have the equivalence

γ k λ | W ⇐⇒ γ k λ | W ∀γ,λ ∈ W c . (20) t d Proof. To prove (20), it is equivalent, by Deﬁnition 3, to prove the equivalence

c γ k λ | W ⇐⇒ ¬(γAW λ) ∀γ, λ ∈ W . (21) t W E T W 2Instead of Γ or even of Γ E . 3 Instead of ↓T W . E

8 For this purpose, we set S−W =∆ ∪ BW ∪ KW CW = (SW )−1 . (22) c Let γ, λ ∈V be two vertices such that γ, λ ∈ W . We have

W W SW γ ∩ SW λ 6= ∅ ⇐⇒ E W ∗SW γ ∩ E W ∗SW λ 6= ∅ W because the topological closure of a subset Γ ⊂V is given by Γ = E W ∗Γ by (17)

⇐⇒ γS−W (E W ∗)−1 ∩ E W ∗SW λ 6= ∅ (by deﬁnition of the converse relation) ⇐⇒ γS−W E −W ∗E W ∗SW λ by deﬁnition of relation composition and by (E W ∗)−1 = E −W ∗

−W −W ∗ W ∗ W c ⇐⇒ γ∆W c S E E S ∆W c λ ( because γ, λ ∈ W by assumption) W ⇐⇒ γ∆W c A ∆W c λ (by (36) in Appendix B) ⇐⇒ γAW λ . ( because γ, λ ∈ W c by assumption)

Thus, by taking the negation, we have obtained (21), hence (20) by Deﬁnition 3.

3.2 Characterization of t-separation between subsets We put forward a practical characterization of t-separation between subsets of vertices. For this purpose, we introduce the notion of splitting, which slightly generalizes the notion of partition.

For any subset Γ ⊂V and for any family {Γs}s∈S of subsets Γs ⊂V, we write ⊔s∈SΓs =Γ ′ when we have, on the one hand, s =6 s =⇒ Γs ∩ Γs′ = ∅ and, on the other hand, s∈S Γs = Γ. We will also say that {Γs}s∈S is a splitting of Γ (we do not use the vocable ofS partition because it is not required that the subsets Γs be nonempty).

Proposition 6 (Topological separation between subsets) Let (V, E) be a graph, and W ⊂V be a subset of vertices. Let Γ, Λ ⊂V be two subsets of vertices such that

Γ ∩ Λ= ∅ , Γ ∩ W = ∅ , Λ ∩ W = ∅ . (23)

The following statements are equivalent:

1. For any γ ∈ Γ, λ ∈ Λ, we have that γ k λ | W , as in Deﬁnition 4, t

2. There exists a splitting WΓ, WΛ of W such that

W W WΓ ⊔ WΛ = W and Γ ∪ WΓ ∩ Λ ∪ WΛ = ∅ . (24)

9 Proof. • (Item 1 =⇒ Item 2). We consider two subsets Γ, Λ ⊂V such that (23) holds true, and we prove the existence of a a splitting WΓ,WΛ of W satisfying (23) in two steps.

′ W −W W W ′ W −W W First, we set WΓ = C B ∪ K Γ ⊂ W by deﬁnition (15f) of C and WΛ = C B ∪ K Λ ⊂ W , and we prove that W W ′ ′ Γ ∪ WΓ ∩ Λ ∪ WΛ = ∅ (25) We have that

W W W W ′ ′ W −W W −W W Γ ∪ WΓ ∩ Λ ∪ WΛ = Γ ∪C B ∪ K Γ ∩ Λ ∪ B ∪ K Λ W W = SW Γ ∩ SW Λ (by deﬁnition (18) of SW ) W W = SW γ ∩ SW λ γ[∈Γ λ[∈Λ by property (6) of the topological closure in an Alexandrov topology

W W = SW γ ∩ SW λ γ ,λ ∈Γ[∈Λ =∅ | {z } by (19) as γ k λ | W by assumption, with γ ∈ Γ ⊂ W c and λ ∈ Λ ⊂ W c by (23). t Thus, we have proven (25). ′ ′ Second, we are now going to prove that we can enlarge the subsets WΓ and WΛ to obtain a splitting of W satisfying Equation (24). ′ ′ W For this purpose, we set W = W \(WΓ ∪ WΛ) ⊂ W . The cousinhood relation C in (15f) is a partial equivalence relation on V, and it is easily seen to be an equivalence relation on W . This is f why we consider the partition W = ⊔i∈IWi of W , where, for each i ∈ I, the elements of the subset W belong to the same equivalence class of the equivalence relation CW ⊂ W 2 (understood as the i f f f restriction of the cousinhood relation CW to W ). We are now going to prove that f

W W W W ′ ′ ∀i ∈ I , either Wi ∩ Γ ∪ WΓ = ∅ or Wi ∩ Λ ∪ WΛ = ∅ . (26)

f f W W ′ The proof is by contradiction. Let i ∈ I be ﬁxed and suppose that both Wi ∩ Γ ∪ WΓ 6= W W W W W ′ ′ ∅ and Wi ∩ Λ ∪ WΛ 6= ∅. First, as Wi ∩ Γ ∪ WΓ 6= ∅, there wouldf exist λ ∈ Wi ∩ W W ′ (1) W ∗ (1) W ∗ Γ ∪ WΓf, hence there would exist wi ∈ Wfi such that λE wi (as Wi = E Wi by (10)),f and W W ∗ W −W W ′ W ∗ ′ there would exist γ ∈ Γ such that λE f∆ ∪C (B ∪ K ) γ (as Γf∪ WΓ =fE (Γ ∪ WΓ) = W ∗ W −W W ′ E ∆ ∪C (B ∪ K ) Γ by (10) and by deﬁnition of WΓ). Thus, we would have that W W (1) −W ∗ W ∗ W −W W ′ wi E E ∆ ∪C (B ∪ K ) γ. Second, as Wi ∩ Λ ∪ WΛ 6= ∅ and proceeding in the same (2) (2) −W ∗ W ∗ W −W W way, there would exist wi ∈ Wi and λ ∈ Λ suchf that wi E E ∆ ∪C (B ∪ K ) λ. Now, (1) (2) as wi and wi would be bothf in Wi they would be in the same equivalence class for the equivalence W (1) W (2) relation C , giving thus wi C wfi . Finally, we would obtain that

W W W −W ∗ W ∗ W −W ∗ W ∗ W −W W γ∆W c ∆ ∪ (B ∪ K )C E E C E E ∆ ∪C (B ∪ K ) ∆W c λ . (27) 10 Combining (27), Equation (35) and the deﬁnition (15g) of AW , we would obtain that γAW λ. Thus, we would arrive at a contradiction as we have, by assumption, γ k λ | W , hence ¬(γAW λ) by (21). t Thus, we have proven that the disjunction (26) holds true, from which we obtain a splitting I = IΓ ⊔ IΛ and a splitting W = WΓ ⊔ WΛ deﬁned by

W W f f f ′ IΓ = i ∈ I Wi ∩ Λ ∪ WΛ = ∅ and IΛ = I\IΓ , W W f WΓ = Wi and WΛ = Wi , i[∈I i[∈I f Γ f f Λ f which, by construction, satisﬁes

W W W W ′ ′ WΓ ∩ Λ ∪ WΛ = ∅ and WΛ ∩ Γ ∪ WΓ = ∅ . (28)

′ f ′ f Now, we deﬁne WΓ = WΓ ∪ WΓ and WΛ = WΛ ∪ WΛ, and we are going to prove that (24) holds true. f f To check the second part of (24), we calculate

W W W W ′ ′ Γ ∪ WΓ ∩ Λ ∪ WΛ = Γ ∪ WΓ ∪ WΓ ∩ Λ ∪ WΛ ∪ WΛ W W W W ′ ′ = Γ ∪ WΓ f∪ WΓ ∩ Λ ∪ WfΛ ∪ WΛ (by (6)) W W W W ′ f ′ ′ f = Γ ∪ WΓ ∩ Λ ∪ WΛ ∪ Γ ∪ WΓ ∩ WΛ =∅ by (25) =∅ by (28)f | W{z W} | W {z W } ′ ∪ WΓ ∩ Λ ∪ WΛ ∪ WΓ ∩ WΛ f =∅ by (28) f f | W {z W } = Wi ∩ Wj i[∈IΓ j[∈IΛ f f by (6) and by deﬁnition of WΓ and WΛ

f f W W = Wi ∩ Wj = ∅

i[∈IΓ j[∈IΛ f =∅f

| {z } W W as IΓ ∩ IΛ = ∅, using the postponed Lemma 8 which gives Wi ∩ Wj = ∅, for any i, j ∈ I with W W i 6= j. From the just proven equality Γ ∪ W ∩ Λ ∪ W = ∅, we readily get that W ∩ W = ∅. Γ Λ f f Γ Λ Therefore, to check the ﬁrst part of (24), it remains to calculate

W W ′ ′ ′ ′ ′ ′ WΓ ∪ WΛ = (WΓ ∪ WΛ) ∪ Wi ∪ Wi = (WΓ ∪ WΛ) ∪ W \(WΓ ∪ WΛ) = W . i[∈IΓ i[∈IΛ f f f ′ ′ =W =W \(WΓ∪WΛ) | {z }

11 • (Item 2 =⇒ Item 1). The proof is by contradiction. For this purpose, we suppose we suppose W W that there exists a splitting W = WΓ ⊔ WΛ such that Γ ∪ WΓ ∩ Λ ∪ WΛ = ∅ (Item 2) and that there exists γ ∈ Γ and λ ∈ Λ such that γAW λ (¬ Item 1). We show that we arrive at a contradiction. W W W W W Using the fact that γA λ and that {γ} ∩ {λ} ⊂ Γ ∪ WΓ ∩ Λ ∪ WΛ = ∅, we would obtain, by the postponed Lemma 7, that there would exist a nonempty subset Wγ,λ ⊂ W such that all the W elements of Wγ,λ would be in the same class for the C partial equivalence relation, and such that

W W W W {γ} ∩ Wγ,λ 6= ∅ and {λ} ∩ Wγ,λ 6= ∅ . (29)

W W W W W Now, using the fact that WΓ ∩ WΛ ⊂ Γ ∪ WΓ ∩ Λ ∪ WΛ = ∅, we would obtain that WΓ ∩ W WΛ = ∅, which, would imply that Wγ,λ would necessarily be included either in WΓ or in WΛ, by using the second part of Lemma 8. Therefore, Assuming that Wγ,λ ⊂ WΓ, we obtain a contradiction using Equation (29), as we would have that

W W W W W W ∅= 6 {λ} ∩ Wγ,λ ⊂ {λ} ∩ WΓ ⊂ Λ ∪ WΛ ∩ Γ ∪ WΓ = ∅ .

Proceeding in a similar way in the case Wγ,λ ⊂ WΛ leads to a similar contradiction.

We end with Lemma 7 and Lemma 8 which are instrumental in the proof of Proposition 6.

Lemma 7 Suppose that the assumptions of Proposition 6 are satisﬁed. Let γ ∈ Γ and λ ∈ Λ W W W be given such that γA λ and {γ} ∩ {λ} = ∅. Then, there exists wγ, wλ ∈ W such that W wγ and wλ are in the same equivalence class of the partial equivalence relation C (that is, W W W W W wγC wλ) and such that {γ} ∩ {wγ} =6 ∅ and {λ} ∩ {wλ} =6 ∅.

Proof. As a preliminary result, we prove that γ ∆ ∪BW ∪B−W ∪ KW λ contradicts the assumptions of Lemma 7. First, γ∆λ contradicts the assumption Γ ∩ Λ = ∅ in (23). Second, γBW λ W W contradicts the assumption {γ} ∩ {λ} = ∅. Indeed, using the deﬁnition (15b) of the conditional ascendent relation BW , and using the fact that γ ∈ Γ ⊂ W c, we have that γBW λ = γEE W ∗λ = W W ∗ W ∗ W W ∗ γ∆W c EE λ = γE λ. Thus, γB λ implies that γ ∈ E λ = {λ} and thus a contradiction as W W W W ∅ = {γ} ∩ {λ} ⊃ γ ∩ {λ} . Third, following the same lines, γB−W λ implies that λ ∈ {γ} W W and contradicts {γ} ∩ {λ} = ∅. Fourth, γKW λ = γE −W +E W +λ and thus γKW λ implies that W W E W +γ ∩E W +λ 6= ∅, hence that E W ∗γ ∩E W ∗λ 6= ∅, that is, {γ} ∩ {λ} 6= ∅ by (17). This again leads to a contradiction. Therefore, we get that ¬ γ ∆ ∪BW ∪B−W ∪ KW λ and γAW λ imply that γ BW ∪ KW CW B−W ∪ KW λ, W using the deﬁnition (15g) of the conditional active relatio n A . Thus, there exist wγ, wλ ∈ W such that W W −W W W γ B ∪ K wγ and wλ B ∪ K λ and wγ C wλ .

W W W W Now, we prove that γ B ∪ K wγ implies that we have {γ} ∩ {wγ } 6= ∅. Indeed, as W W W c γ B ∪ K wγ, we have two possibilities. First, suppose that γB wγ . Then, as γ ∈ W , this W W W ∗ W ∗ implies thatγ∆W c B wγ = γE wγ which implies that γ ∈ E wγ = {wγ } by (17). Therefore, W W W we get that {γ} ∩ {wγ } 6= ∅. Second, if γK wγ , then, as already seen at the beginning of the W W proof, we obtain that {γ} ∩ {wγ } 6= ∅.

12 W −W W Then, following similar arguments, we prove that wλ B ∪ K λ implies that we have {λ} ∩ W {wλ} 6= ∅. W W W W Finally, we have obtained that {γ} ∩ {wγ } 6= ∅ and {λ} ∩ {wλ} 6= ∅. Moreover wγ and W W wλ are in the same equivalence class of the partial equivalence relation C as wγC wλ. Thus, we have found two elements wγ , wλ ∈ W satisfying the conclusion of Lemma 7. This concludes the proof.

Lemma 8 Let W ′ and W ′′ be two subsets of W which are included in two distinct equivalence W W classes of the partial equivalence relation CW . Then, we have that W ′ ∩ W ′′ = ∅. W W Conversely, assume given a splitting W = W ′ ⊔ W ′′ such that W ′ ∩ W ′′ = ∅. Then, there does not exists w′ ∈ W ′ and w′′ ∈ W ′′ such that w′ and w′′ are in the same equivalence classes of CW

Proof. For the first assertion, we make a proof by contradiction. For this purpose, we consider W ′ and W ′′, two subsets of W which are included in two distinct equivalence classes of the W W partial equivalence relation CW , and we suppose that W ′ ∩ W ′′ 6= ∅. Then, by (10), there exists w′ ∈ W ′ and w′′ ∈ W ′′ and γ ∈ V such that γ ∈ E W ∗w′ and γ ∈ E W ∗w′′. Therefore we have ′ −W ∗ W ∗ ′′ ′ W −W W ′′ that w E E w , from which we deduce that w (∆ ∪ ∆W c B ∪B ∆W c ∪ K )w , using Equa- tion (34a). Moreover, as by assumption, W ′ and W ′′ are two subsets of W which are included in two distinct equivalence classes of CW , we have that w′ ∈ W , w′′ ∈ W and w′ 6= w′′. Hence, among the four possible cases corresponding to the union of four terms, only the last one is possible: we ′ W ′′ ′ W ′′ W must necessarily have that w ∆W K ∆W w , which implies that w C w by definition (15f) of C . Thus, w′ and w′′ belong to the same equivalence class of CW , but this contradicts that w′ ∈ W and w′′ ∈ W where W ′ and W ′′ are included in two distinct equivalence classes of CW . Now, we prove the converse assertion again by contradiction. For this purpose, consider a W W splitting W = W ′ ⊔ W ′′ such that W ′ ∩ W ′′ = ∅, and suppose that there exists w′ ∈ W ′ and w′′ ∈ W ′′ such that w′ and w′′ are in the same equivalence classes of CW or otherwise said, such that w′CW w′′. Using Equation (15f) and the fact that w′ 6= w′′, as W ′ ∩ W ′′ = ∅, we deduce that ′ W + ′′ N w ∆W K ∆W w . Hence, there exists k ∈ , k ≥ 1, and a sequence {wi}i∈J1,kK in W such that, W ′ W W ′′ ′ ′′ for all i ∈ J1, k− 1K, we have wiK wi+1 and w K w1 and wkK w . Setting w0 = w and wk+1 = w W W and using the property that γKW λ =⇒ {γ} ∩ {λ} 6= ∅ (shown at the beginning of the proof W W of Lemma 7), we get that {wi} ∩ {wi+1} 6= ∅ for i ∈ J0, kK. Now, the sequence {wi}i∈J0,k+1K is in W , with the first element, w′, in W ′ and the last one, w′′, in W ′′. As W = W ′ ⊔ W ′′, we can find two consecutive elements in the sequence such that one is in W ′ and the other one is in W ′′ and which are such that the intersection of their topological closure is not empty. Thus, we have W W obtained that W ′ ∩ W ′′ 6= ∅, which gives a contradiction. This ends the proof.

4 Conclusion

Together with its two companion papers [2, 5], this paper is a contribution to providing another perspective on conditional independence and do-calculus. In this paper, we consider

13 directed graphs (DGs), not necessarily acyclic, and we introduce a suitable topology on the set of vertices and the new notion of topological conditional separation on DGs. Then, we prove its equivalence with an extension of Pearl’s d-separation on DGs. What is more, we put forward a practical characterization of t-separation between subsets of vertices. The proofs partially rely on results proven in [2]. Checking topological separation is a two steps process. The ﬁrst one, which is combinatorial, consists in exploring the possible splitting of the conditioning set W and the second one consists in checking that the two closures induced by the splitting do not intersect. It should be noted that, once given the splitting, the second step is computationaly easy and thus the splitting appears as a “certiﬁcate” of conditional independence. By contrast, checking d-separation is a one step combinatorial process as it requires to check that all the paths that connect two variables are blocked.

A Additional material on Alexandrov topology

We use the material introduced in §2.2, and we provide additional results on Alexandrov topologies.

Proposition 9 Let E1, E2 be two binary relations on the set V. We have that

E1 ⊂E2 =⇒ TE2 ⊂ TE1 , (30)

TE1 ∪ E2 = TE1 ∩ TE2 . (31)

2 The product topology TE1 ⊗TE2 on the product set V coincides with the topology TE1×E2 in (8), 2 2 2 where E1 ×E2 ⊂V ×V is the product binary relation on the product set V :

TE1 ⊗ TE2 = TE1×E2 . (32)

2 The topological closure of a subset R⊂V w.r.t. the topology TE1 ⊗ TE2 = TE1×E2 is given by

E1×E2 ∗ −∗ R = E1 RE2 . (33)

Proof. The proof of (32) relies on the following identity between open rectangles:

Γ1 × Γ2 ∈TE1 ⊗TE2 ⇐⇒ Γ1 ∈TE1 and Γ2 ∈TE2

(by deﬁnition of the product topology TE1 ⊗TE2 ) ⇐⇒ E1Γ1 ⊂ Γ1 and E2Γ2 ⊂ Γ2 (by (9))

⇐⇒ (E1 ×E2)(Γ1 × Γ2) ⊂ Γ1 × Γ2 .

Regarding (33), by property (6) of an Alexandrov topology, we have that

E ×E E ×E R 1 2 = {(γ′, λ′)} 1 2 (γ′,λ[′)∈R

14 so that, for any γ, λ ∈V, we have

E ×E E ×E (γ, λ) ∈ R 1 2 ⇐⇒ ∃(γ′, λ′) ∈ R , (γ, λ) ∈ {(γ′, λ′)} 1 2 E E ⇐⇒ ∃(γ′, λ′) ∈ R , (γ, λ) ∈ {γ′} 1 × {λ′} 2 by property of the topological closure of rectangles in the product topology

′ ′ ∗ ′ ∗ ′ ⇐⇒ ∃(γ , λ ) ∈ R , γ ∈E1 γ , λ ∈E2 λ (by (10)) ∗ −∗ ⇐⇒ γE1 RE2 λ

This ends the proof.

Proposition 10 Let (V, E) be a graph, and V ⊂ V a subset. The following statements are equivalent:

1. The subset V is a connected component of the topological space (V, TE ),

′ −1 2. The subset V is a connected component of the topological space (V, TE )=(V, TE ),

3. The subset V is a connected component of the topological space (V, TE ∪ E−1 )=(V, TE ∩ TE−1 )= ′ (V, TE ∩ TE ), 4. The subset V is an equivalence class of the equivalence relation (E∪E −1)∗.

Proof. For any v ∈ V, we denote by vˆ ⊂ V the TE -connected component of the topological space (V, TE ) that contains v, that is, the union of all TE -connected subsets of V that contain v, and we prove that vˆ = (E∪E−1)∗v. For this purpose, we notice that, by (10), we have that −1 ∗ E ∪ E−1 (E∪E ) v = v , which is also the smallest TE -clopen set containing v by (31). On the one E ∪ E−1 hand, it is known, and can readily be shown, that vˆ ⊂ v . Indeed, if F is any TE -clopen set c containing v, then vˆ∩F and vˆ∩F are two disjoint TE -open sets whose union equal vˆ. As this latter set is TE -connected, and as v ∈ vˆ ∩ F , we deduce that vˆ ∩ F =v ˆ, hence that vˆ ⊂ F . On the other hand, it is well-known that the TE -connected component vˆ is closed. In an Alexandrov topology, c vˆ is also TE -open since the connected components form a partition of V, so that vˆ is a union of E ∪ E−1 TE -closed sets, hence is closed. Therefore, vˆ is a TE -clopen set, and we deduce that v ⊂ vˆ. − Therefore, we have obtained that vˆ = vE ∪ E 1 = (E∪E−1)∗v. Then, we easily deduce the equivalence between the four assertions. This ends the proof.

15 B Technical lemmas

Here below, the relations SW and S−W have been introduced in (18) and (22). The following lemma is proved in [2].

Lemma 11 ([2]) We have that

−W ∗ W ∗ W −W W E E = ∆ ∪ ∆W c B ∪ B ∆W c ∪K , (34a) W −W ∗ W ∗ W −W W C E E = C ∆ ∪ B ∆W c ∪K , (34b) CW E −W ∗E W ∗CW = CW , (34c) W −W ∗ W ∗ W −W W C E E ∆W c = C B ∪K ∆W c , (34d) −W ∗ W ∗ W W W W ∆W c E E C = ∆Wc B ∪K C . (34e) Lemma 12 We have that

W W W −W ∗ W ∗ W −W ∗ W ∗ W −W W ∆W c ∆ ∪ (B ∪K )C E E C E E ∆ ∪ C (B ∪K ) ∆W c W W W −W W (35) = ∆W c (B ∪K )C (B ∪K )∆W c

Proof. We have that

W W W −W ∗ W ∗ W −W ∗ W ∗ W −W W ∆W c ∆ ∪ (B ∪ K )C E E C E E ∆ ∪C (B ∪ K ) ∆W c −W ∗ W ∗ W −W ∗ W ∗ =∆W c E E C E E ∆W c (by developing) −W ∗ W ∗ W −W ∗ W ∗ W −W W ∪ ∆W c E E C E E C (B ∪ K ) ∆W c W W W −W ∗ W ∗ W −W ∗ W ∗ ∪ ∆W c (B ∪ K )C E E C E E ∆ W c W W W −W ∗ W ∗ W −W ∗ W ∗ W −W W ∪ ∆W c (B ∪ K )C E E C E E C (B ∪ K ) ∆W c −W ∗ W ∗ W −W ∗ W ∗ =∆W c E E C E E ∆W c −W ∗ W ∗ W −W W W −W ∗ W ∗ W W ∪ ∆W c E E C (B ∪ K )∆W c ( as C E E C = C by (34c)) W W W −W ∗ W ∗ ∪ ∆W c (B ∪ K )C E E ∆W c (also by (34c)) W W W −W W ∪ ∆W c (B ∪ K )C (B ∪ K )∆W c (also by (34c) applied twice) W W W −W W =∆W c B ∪ K C B ∪ K ∆W c (by (34d) and (34e)) W W −W W ∪ ∆W c B ∪ K (B ∪ K )∆ W c (by (34e)) W W W −W W ∪ ∆W c (B ∪ K )C B ∪ K ∆W c (by (34d)) W W W −W W ∪ ∆W c (B ∪ K )C (B ∪ K )∆ W c W W W −W W =∆W c (B ∪ K )C (B ∪ K )∆W c .

This ends the proof.

Lemma 13 We have that

−W −W ∗ W ∗ W W ∆W c S E E S ∆W c = ∆W c A ∆W c . (36)

16 Proof. First, we write

S−W E −W ∗E W ∗SW = ∆ ∪ BW ∪ KW CW E −W ∗E W ∗ ∆ ∪CW B−W ∪ KW S−W by (22) SW by (22) | {z } | {z } = (E −W ∗E W ∗) ∪ BW ∪ KW CW E −W ∗E W ∗ ∪ E −W ∗E W ∗CW B−W ∪ KW ∪ BW ∪ KW CW E −W ∗E W ∗CW B−W ∪ KW (by developing) =CW by (34c) | {z } = (E −W ∗E W ∗) ∪ BW ∪ KW CW E −W ∗E W ∗ ∪ E −W ∗E W ∗CW B−W ∪ KW ∪ BW ∪ KW CW B−W ∪ KW . Second, we obtain that

−W −W ∗ W ∗ W ∆W c S E E S ∆W c

−W ∗ W ∗ W W W −W ∗ W ∗ = ∆W c (E E )∆W c ∪ ∆W c B ∪ K C E E ∆W c W W W =C (B− ∪K )∆W c by (34d)

−W ∗ W ∗ W −W W | {z } ∪ ∆W c E E C B ∪ K ∆W c W W W =∆W c (B ∪K )C by (34e)

| W {z W W} −W W ∪ ∆W c B ∪ K C B ∪ K ∆W c −W ∗ W ∗ W W W −W W = ∆W c (E E )∆W c ∪ ∆W c B ∪ K C B ∪ K ∆W c because the three last terms in the union are all equal

W −W W = ∆W c (∆ ∪ ∆W c B ∪B ∆W c ∪ K )∆W c W W W −W W ∪ ∆W c B ∪ K C B ∪ K ∆W c (by (34a)) W −W W W W W −W W =∆W c ∆ ∪B ∪B ∪ K ∪ B ∪ K C B ∪ K ∆W c W W =∆W c A ∆W c . (by deﬁnition of A in (15g))

This ends the proof.

In Lemma 13, it was proved that the two relations S−W E −W ∗E W ∗SW and AW coincide when restricted to the subset W c. More generally, we give in this last lemma the relationship between these two relations.

Lemma 14 We have that

−W −W ∗ W ∗ W −W −W W W W −W W S E E S ∪ S B ∆W ∪ ∆W B S = A ∪ S ∪ S . (37) 17 Proof. We use the notation ΘW = BW ∪ KW and Θ−W = B−W ∪ KW = (ΘW )−1 to simplify the reading of the proof, so that

S−W E −W ∗E W ∗SW = (∆ ∪ BW ∪ KW CW )E −W ∗E W ∗(∆ ∪CW B−W ∪ KW ) (by deﬁnition of the relations SW and S−W in (18) and (22)) = (∆ ∪ ΘW CW )E −W ∗E W ∗(∆ ∪CW Θ−W ) (using the just deﬁned ΘW = BW ∪ KW and Θ−W = B−W ∪ KW ) = (E −W ∗E W ∗) ∪ (ΘW CW E −W ∗E W ∗) ∪ (ΘW CW Θ−W ) ∪ (E −W ∗E W ∗CW Θ−W ) (by developing and by (34c) giving CW E −W ∗E W ∗CW = CW ) −W ∗ W ∗ W W −W W W W −W W W W −W = (E E ) ∪ Θ C ∆ ∪B ∆W c ∪ K ∪ (Θ C Θ ) ∪ ∆ ∪ ∆W c B ∪ K C Θ W W −W ∗ W ∗ W W −W W as by (34b) Θ C E E = Θ C ∆ ∪B ∆W c ∪ K and by symmetry for the last term. W W −W W W −W Thus, using the last equality and performing a union with Θ C B ∆W ∪∆W B C Θ on both sides of the equality, we obtain

−W −W ∗ W ∗ W W W −W W W −W S E E S ∪ Θ C B ∆W ∪ ∆W B C Θ = (E −W ∗E W ∗) ∪ ΘW CW ∆ ∪B−W ∪ KW ∪ (ΘW CW Θ−W ) ∪ ∆ ∪BW ∪ KW CW Θ−W = (E −W ∗E W ∗) ∪ ΘW CW ∆ ∪ Θ−W ∪ (ΘW CW Θ−W ) ∪ ∆ ∪ Θ W CW Θ−W = (E −W ∗E W ∗) ∪ ΘW CW ∪ (ΘW CW Θ −W ) ∪CW Θ−W W −W W W W W W −W W −W = (∆ ∪ ∆W c B ∪B ∆W c ∪ K ) ∪ Θ C ∪ (Θ C Θ ) ∪C Θ (by (34a)) W −W W W W W W −W W −W = (∆ ∪ ∆W c B ∪B ∆W c ∪ K ) ∪ (∆ ∪ Θ C ) ∪(Θ C Θ ) ∪ (∆ ∪C Θ )

=S−W =SW W −W W W W −W −W W =∆ ∪ ∆W c B ∪B ∆W c ∪ K ∪ (Θ| C {zΘ ) ∪} S ∪ S . | {z }

W −W Finaly, using the last equality and performing a union with ∆W B ∪B ∆W on both sides of the equality, we obtain

−W −W ∗ W ∗ W W W −W W W −W W −W S E E S ∪ Θ C B ∆W ∪ ∆W B C Θ ∪ ∆W B ∪B ∆W =∆ ∪BW ∪B −W ∪ KW ∪ (ΘW CW Θ−W ) ∪ S−W ∪ SW =∆ ∪BW ∪B−W ∪ KW ∪ (ΘW CW Θ−W ) ∪ S−W ∪ SW = AW ∪ S−W ∪ SW . (by deﬁnition of AW in (15g))

This ends the proof.

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