Original: Relational Causal Models with Cycles: Representation and Reasoning
Revised: Relational Causal Models with Cycles: Representation and Reasoning

Original: Author names withheld

Original: Editor: Under Review for CLeaR

Original: Abstract
Revised: Abstract

Original: Causal reasoning in relational domains is fundamental to studying real-world social phenomena in which individual units can inﬂuence each other’s traits and behavior.
Revised: Causal reasoning in relational domains is fundamental to studying real-world social phenomena in which individual units can inﬂuence each other’s traits and behavior.

Original: Dynamics between interconnected units can be represented as an instantiation of a relational causal model; however, causal reasoning over such instantiation requires additional templating assumptions that capture feedback loops of inﬂuence.
Revised: Dynamics between interconnected units can be represented as an instantiation of a relational causal model; however, causal reasoning over such instantiation requires additional templating assumptions that capture feedback loops of inﬂuence.

Original: Previous research has developed lifted representations to address the relational nature of such dynamics but has strictly required that the representation has no cycles.
Revised: Previous research has developed lifted representations to address the relational nature of such dynamics but has strictly required that the representation has no cycles.

Original: To facilitate cycles in relational representation and learning, we introduce relational σ -separation, a new criterion for understanding relational systems with feedback loops.
Revised: To facilitate cycles in relational representation and learning, we introduce relational σ -separation, a new criterion for understanding relational systems with feedback loops.

Original: We also introduce a new lifted representation, σ - abstract ground graph which helps with abstracting statistical independence relations in all possible instantiations of the cyclic relational model.
Revised: We also introduce a new lifted representation, σ - abstract ground graph which helps with abstracting statistical independence relations in all possible instantiations of the cyclic relational model.

Original: We show the necessary and sufﬁcient conditions for the completeness of σ -AGG and that relational σ -separation is sound and complete in the presence of one or more cycles with arbitrary length.
Revised: We show the necessary and sufﬁcient conditions for the completeness of σ -AGG and that relational σ -separation is sound and complete in the presence of one or more cycles with arbitrary length.

Original: To the best of our knowledge, this is the ﬁrst work on representation of and reasoning with  relational causal models with cycles.
Revised: To the best of our knowledge, this is the ﬁrst work on representation of and reasoning with cyclic relational causal models.

Original: 1. Introduction
Revised: 1. Introduction

Original: Causal reasoning over relational systems is central to understanding real-world social phenomena, such as social inﬂuence and information diffusion.
Revised: Causal inference methods from observational data often assume that causal relationships can be represented in an acyclic graphical model. However, many real-world phenomena involve feedback loops or cycles that violate the acyclicity assumption. For example, supply and demand affect price and vice versa, hormone levels in the body affect each other and friends can impact each other’s choices. Common to these scenarios is that these interactions occur over time but any individual directed relationship may not be observed.

Revised: Instead what is observed is a set of values that are result of long term individual interactions.

Revised: This current state can be represented through a cycle.

Revised: The existing works on cyclic causal models primarily focus independently and identically distributed (i.i.d) data instances (Richardson, 1996, 1997; Strobl, 2019;

Revised: Rantanen et al., 2020).

Revised: However, in many real-world systems units are often interconnected in a complex network.

Revised: Causal reasoning over such relational systems is central to understanding real-world social phenomena, such as social inﬂuence and information diffusion.

Original: For example, a well-known study by Christakis and Fowler (2007) investigates whether obesity is contagious in a population where each person’s eating habits can affect their friends and family and vice-versa.
Revised: For example, a well-known study by Christakis and Fowler (2007) investigates whether obesity is contagious in a population where each person’s eating habits can affect their friends and family and vice-versa.

Original: Similarly, several real-world problems in epidemiology and computational social science encounter such mutual inﬂuence and interaction  among human subjects.
Revised: Similarly, several real-world problems in epidemiology and computational social science encounter such mutual inﬂuence and interaction [Figure] among human subjects.

Original: For example, effect of vaccination in mother and child (Shpitser, 2015; VanderWeele et al., 2012),time spent online by users in social network (Sherman andShpitser, 2018), inﬂuence of diet in family members (Ogburn et al., 2014), justices inﬂuencing each other in supreme court decisions (Ogburn et al., 2020) are all examples of mutual inﬂuence.
Revised: For example, effect of vaccination in mother and child (Shpitser, 2015; VanderWeele et al., 2012), and justices inﬂuencing each other in supreme court decisions (Ogburn et al., 2020) are all examples of mutual inﬂuence.

Original: Studying such phenomena requires causal reasoning over complex interactions between interconnected entities.
Revised: Studying such phenomena requires causal reasoning over complex interactions between interconnected entities.

Original: However, causal questions of mutual inﬂuence  can be hard to answer due to the lack of appropriate causal representations and reasoning mechanisms that can capture the cycles and feedback loops introduced by mutual inﬂuence.
Revised: However, causal questions of mutual inﬂuence in a relational system can be hard to answer due to the lack of appropriate causal representations and reasoning mechanisms. One way to represent interference is through SCMs which capture pairwise relationships (Ogburn et al., 2014; Shalizi and Thomas, 2011), where one individual’s inﬂuence on another is represented independently. Such representation is easy to reason about but it assumes that pairs of nodes are independent of other pairs of nodes.

Revised: Unfortunately, this does not reﬂect the properties of real-world social networks in which nodes are arbitrarily interconnected with each other.

Original: The development of relational causal models , which generalize over  causal models, was an important step towards capturing interactions between non-IID instances (Maier et al., 2013b,a; Lee and Honavar, 2015; Bhattacharya et al., 2020).
Revised: The development of relational causal models , which generalize over structural causal models, is an important step towards capturing interactions between non-i.i.d instances (Maier et al., 2013b,a; Lee and Honavar, 2015; Bhattacharya et al., 2020).

Original: Relational models involve multiple types of interacting entities with probabilistic dependencies among their attributes.
Revised: Relational models involve multiple types of interacting entities with probabilistic dependencies among their attributes.

Original: Maier et al.
Revised: Maier et al.

Original: (2013b) developed a lifted causal representation named abstract ground graph (AGG) that abstracts over all instantiations of a relational model.
Revised: (2013b) developed a lifted causal representation named abstract ground graph (AGG) that abstracts over all instantiations of a relational model.

Original: AGG enables reasoning about causal queries in relational causal models 
Revised: AGG enables reasoning about causal queries in relational causal models and relational causal discovery.

Original: [Figure]

Original: and relational causal discovery. However, existing relational causal models assume acyclicity and cannot capture feedback loops.
Revised: However, existing relational causal models assume acyclicity and do not allow for reasoning about identiﬁcation in the presence of feedback loops.

Original: Figure 1 shows an example of a relational model where employees develop products and product quality depends on  the performance levels of employees.
Revised: Figure 1 shows an example of a relational model where users react to news articles or posts on a speciﬁc topic (e.g., vaccines) generated by media agencies. The preference of the media regarding a topic (e.g., pro- vs anti-vaccination) is inﬂuenced by the engagement or feedback (e.g., positive/negative comments) it receives on its existing posts. The sentiment of a user towards a given post directly impacts their engagement in the post. The relational model representation tempts one to conclude that the sentiment of users regarding vaccination is independent of the preference of media agencies given engagements of the posts the users react to.

Revised: However, as we show in section 3.5, this is not necessarily true.

Original: The fact that product quality can also affect employee performance introduces a feedback loop in the graphical representation that cannot be reasoned about with existing representations.
Revised: Moreover, users’ sentiment can also be impacted by the engagements in posts they interact with. The dashed arrow in Figure 1 represents such a dependency which makes the model a cyclic one. Unfortunately, even though the model seems simple and realistic, the abstract ground graph (AGG) representation and relational d -separation no longer apply due to the presence of a feedback loop (i.e., a cycle). In this work, we speciﬁcally study cyclic RCMs and show that they offer the necessary representation to reason about a lot of real-world causal problems where popular assumptions do not hold.

Revised: We do not assume the Stable Unit Treatment Value Assumption (SUTVA), of causal inference, according to which the outcome of each unit depends only on the unit itself (Rubin, 1980).

Revised: This Assumption is violated in relational systems where the treatment and outcome of one unit can impact other units’ outcomes.

Original: In this work, we deﬁne a new abstract representation, σ - abstract ground graph ( σ -AGG) which generalizes over cyclic relational models.
Revised: To the best of our knowledge, this is the ﬁrst work that addresses representation of and reasoning about cyclic relational causal models. We deﬁne a new abstract representation, σ - abstract ground graph ( σ -AGG) which generalizes over cyclic relational models.

Original: In order to reason about relational queries in σ -AGG, we introduce relational σ -separation and provide proofs for its soundness and completeness for all instantiations of a relational model.
Revised: In order to reason about relational queries in σ -AGG, we introduce relational σ -separation and provide proofs for its soundness and completeness for all instantiations of a relational model.

Original: The implications of this new representation are twofold.
Revised: The implications of this new representation are twofold.

Original: First, it can lead the way for reasoning about causal effects like  contagionrepresented in relational models.
Revised: First, it can lead the way for reasoning about causal effects like interference and contagion in relational systems, which are of wide interest in the social sciences.

Original: Second, the abstract representation and its identiﬁed properties will lay a foundation for causal structure learning from relational models with cycles.
Revised: Second, the abstract representation and its identiﬁed properties will lay a foundation for causal structure learning from relational models with cycles.

Original: We show thenecessary and sufﬁcient conditions for the completeness of σ -AGG by ﬁrst resolving an open problem on the completeness of AGG.
Revised: We show the sufﬁcient conditions for the completeness of σ -AGG by ﬁrst resolving an open problem on the completeness of AGG.

Original: Finally, we discuss the Markov condition of relational σ -separation and its implications.
Revised: Finally, we discuss the Markov condition of relational σ -separation and its implications.

Original: 2. Related Work
Revised: 2. Related Work

Original: Maier et al.
Revised: Maier et al.

Original: (2013b) proposed a sound and complete abstract representation for relational data with multiple entities and relationships.
Revised: (2013b) proposed a sound and complete abstract representation for relational data with multiple entities and relationships.

Original: They deﬁned an abstract representation of relational dependence called abstract ground graph based on the assumption that the underlying relational model is acyclic.
Revised: They deﬁned an abstract representation of relational dependence called abstract ground graph based on the assumption that the underlying relational model is acyclic.

Original: Moreover, they introduced relational d -separation, a new criterion to reason about statistical independence in all the instantiations of a relational model.
Revised: Moreover, they introduced relational d -separation, a new criterion to reason about statistical independence in all the instantiations of a relational model.

Original: They proved the soundness and completeness of relational d -separation.
Revised: They proved the soundness and completeness of relational d -separation.

Original: The AGG and associated relational d -separation criterion together allows a formal basis of causal analysis in relational systems.
Revised: The AGG and associated relational d -separation criterion together allows a formal basis of causal analysis in relational systems.

Original: The same authors proposed a constraint-based causal structural learning algorithm called RCD (Maier et al., 2013a) which is ﬁrst of its kind.
Revised: The same authors proposed a constraint-based causal structural learning algorithm called RCD (Maier et al., 2013a) which is ﬁrst of its kind.

Original: Lee and Honavar (2015) presented a critical view on the generalizability of AGG where they constructed a counterexample which shows how AGG can fail to abstract all d -separation relationships in the ground graphs.
Revised: Lee and Honavar (2015) presented a critical view on the generalizability of AGG where they constructed a counterexample which shows how AGG can fail to abstract all d -separation relationships in the ground graphs.

Original: They also identiﬁed a shortcoming of the necessary conditions for considering a intersection variable and provided a revised set of conditions.
Revised: They also identiﬁed a shortcoming of the necessary conditions for considering a intersection variable and provided a revised set of conditions.

Original: In a different work, Lee and
Revised: In a different work, Lee and

Original: Honavar (2020) proposed a new abstract representation of relational causal models called the unrolled graph which guarantees a weaker sense of completeness.
Revised: Honavar (2020) proposed a new abstract representation of relational causal models called the unrolled graph which guarantees a weaker sense of completeness.

Original: However, neither of the two groups of work moved away from the assumption of acyclicity.
Revised: However, neither of the two groups of work moved away from the assumption of acyclicity.

Original: As a result their developments are not directly applicable to relational models with cycles or feedback loops.
Revised: As a result their developments are not directly applicable to relational models with cycles or feedback loops.

Original: Alternatively, chain graphs have been used for representing systems with pairwise feedback loops (Lauritzen and Richardson, ; Dawid, 2010; Ogburn et al., 2020).
Revised: Alternatively, chain graphs have been used for representing systems with pairwise feedback loops (Lauritzen and Richardson, 2002; Dawid, 2010; Ogburn et al., 2020).

Original: There is no prior work that focuses on representation and reasoning of relational models with cycles or feedback loops of arbitrary length.
Revised: There is no prior work that focuses on representation and reasoning of relational models with cycles or feedback loops of arbitrary length.

Original: Existing research on representation and reasoning about systems with cycles is based on propositional variables, not relational ones.
Revised: Existing research on representation and reasoning about systems with cycles is based on propositional variables, not relational ones.

Original: Spirtes (1995) showed that, in general case, without any speciﬁc assumption regarding the nature of dependence (linear, polynomial etc.), the d -separation relations in a directed cyclic graph are not sufﬁcient to entail all the corresponding conditional independence relations.
Revised: Spirtes (1995) showed that, in general case, without any speciﬁc assumption regarding the nature of dependence (linear, polynomial etc.), the d -separation relations in a directed cyclic graph are not sufﬁcient to entail all the corresponding conditional independence relations.

Original: Forr´e and Mooij (2017) introduced σ -separation, an alternative formulation for which the corresponding Markov property is shown to hold in a very general setting.
Revised: Forr´e and Mooij (2017) introduced σ -separation, an alternative formulation for which the corresponding Markov property is shown to hold in a very general setting.

Original: According to them, the σ -separation Markov property seems appropriate for a wide class of cyclic structural causal models with non-linear functional relationships between non-discrete variables.
Revised: According to them, the σ -separation Markov property seems appropriate for a wide class of cyclic structural causal models with non-linear functional relationships between non-discrete variables.

Original: 3. Preliminaries
Revised: 3. Preliminaries

Original: In this section we introduce the necessary terminology to discuss representation and abstraction of cyclic relational models.
Revised: In this section we introduce the necessary terminology to discuss representation and abstraction of cyclic relational models.

Original: 3.1. Directed Cyclic Graphs
Revised: 3.1. Directed Cyclic Graphs

Original: A Directed Cyclic Graph (DCG) is a graph G =
Revised: A Directed Cyclic Graph (DCG) is a graph G =

Original: (cid:104)V , E(cid:105) with nodes V and edges E ⊆ { ( u, v ) : u, v ∈
Revised: (cid:104)V , E(cid:105) with nodes V and edges E ⊆ { ( u, v ) : u, v ∈

Original: V, u (cid:54) = v } where ( u, v ) is an ordered pair of nodes.
Revised: V, u (cid:54) = v } where ( u, v ) is an ordered pair of nodes.

Original: We will denote a directed edge ( u, v ) ∈ E as u → v or v ← u , and call u a parent of v .
Revised: We will denote a directed edge ( u, v ) ∈ E as u → v or v ← u , and call u a parent of v .

Original: In this work, we restrict ourselves to DCG as the causal graphical model.
Revised: In this work, we restrict ourselves to DCG as the causal graphical model.

Original: A walk between two nodes u, v ∈ V is a tuple (cid:104) v 0 , e 1 , v 1 , e 2 , v 2 , ..., e n , v n (cid:105) of alternating nodes and edges in G ( n ≥ 0) , such that v 0 , ..., v n ∈ V , and e 1 , ..., e n ∈ E , starting with node v 0 = u and ending with node v n = v where the edge e k connects the two nodes v k − 1 and v k ∈ G for all k = 1 , ..., n .
Revised: A walk between two nodes u, v ∈ V is a tuple (cid:104) v 0 , e 1 , v 1 , e 2 , v 2 , ..., e n , v n (cid:105) of alternating nodes and edges in G ( n ≥ 0) , such that v 0 , ..., v n ∈ V , and e 1 , ..., e n ∈ E , starting with node v 0 = u and ending with node v n = v where the edge e k connects the two nodes v k −and v k ∈ G for all k = 1 , ..., n .

Original: If the walk contains each node at most once, it is called a path .
Revised: If the walk contains each node at most once, it is called a path .

Original: A directed walk (path) from v i ∈ V to v j ∈ V is a walk (path) between v i and v j such that every edge e k on the walk (path) is of the form v k − 1 → v k , i.e., every edge is directed and points away from v i .
Revised: A directed walk (path) from v i ∈ V to v j ∈ V is a walk (path) between v i and v j such that every edge e k on the walk (path) is of the form v k − 1 → v k , i.e., every edge is directed and points away from v i .

Original: We get the ancestors of node v j by repeatedly following the path(s) through the parents: AN G ( v j ) :
Revised: We get the ancestors of node v j by repeatedly following the path(s) through the parents: AN G ( v j ) :

Original: = { v i ∈
Revised: = { v i ∈

Original: V : v i = v 0 → v 1 → ... → v n = v j ∈ G} .
Revised: V : v i = v 0 → v 1 → ... → v n = v j ∈ G} .

Original: Similarly, we deﬁne the descendants of v i : DE G ( v i ) :
Revised: Similarly, we deﬁne the descendants of v i : DE G ( v i ) :

Original: = v j ∈ V : v i = v 0 → v 1 → ... → v n = v j ∈ G .
Revised: = v j ∈ V : v i = v 0 → v 1 → ... → v n = v j ∈ G .

Original: Each node is an ancestor and descendant of itself.
Revised: Each node is an ancestor and descendant of itself.

Original: A directed cycle is a directed path from v i to v j such that in addition, v j → v i ∈ E .
Revised: A directed cycle is a directed path from v i to v j such that in addition, v j → v i ∈ E .

Original: All nodes on directed cycles passing through v i ∈ V together form the strongly connected component SC G ( v i ) :
Revised: All nodes on directed cycles passing through v i ∈ V together form the strongly connected component SC G ( v i ) :

Original: = AN G ( v i ) ∩ DE G ( v i ) of v i .
Revised: = AN G ( v i ) ∩ DE G ( v i ) of v i .

Original: 3.2. σ -separation
Revised: 3.2. σ -separation

Original: The idea of σ -separation follows from d -separation, a fundamental notion in DAGs which was ﬁrst introduced by Pearl (1988):
Revised: The idea of σ -separation follows from d -separation, a fundamental notion in DAGs which was ﬁrst introduced by Pearl (1988):

Original: Deﬁnition 1 ( d -separation)
Revised: Deﬁnition 1 ( d -separation)

Original: A walk (cid:104) v 0 ...v n (cid:105) in DCG G = (cid:104)V , E(cid:105) is σ -blocked by C ⊆ V if:
Revised: A walk (cid:104) v 0 ...v n (cid:105) in DCG G = (cid:104)V , E(cid:105) is d -separated by C ⊆ V if:

Original: its ﬁrst node v 0 ∈ C or its last node v n ∈ C , or 2.
Revised: its ﬁrst node v 0 ∈ C or its last node v n ∈ C , or 2.

Original: it contains a collider v k / ∈ AN G ( C ) , or 3.
Revised: it contains a collider v k / ∈ AN G ( C ) , or 3.

Original: it contains a non-collider v k ∈ C .
Revised: it contains a non-collider v k ∈ C .

Original: If all paths in G between any node in set A ⊆ V and any node in set B ⊆ V are d -blocked by a set
Revised: If all paths in G between any node in set A ⊆ V and any node in set B ⊆ V are d -blocked by a set

Original: C ⊆ V , we say that A is d -separated from B by C , and we write A d | =
Revised: C ⊆ V , we say that A is d -separated from B by C , and we write A d | =

Original: G
Revised: G

Original: B | C .
Revised: B | C .

Revised: d -separation exhibits the global Markov property in DAGs which states that if two variables X and Y are d -separated given another variable Z in a DAG representation then X and Y are conditionally independent given Z in the corresponding distribution of the variables.

Original:  Spirtes (1995); Neal (2000) show that, without any speciﬁc assumption regarding the nature of dependence (linear, polynomial etc.), the d -separation relations are not sufﬁcient to entail all the corresponding conditional independence relations in a directed cyclic graph.
Revised: However, Spirtes (1995); Neal (2000) show that without any speciﬁc assumption regarding the nature of dependence (i.e. linear, polynomial), the d -separation relations are not sufﬁcient to entail all the corresponding conditional independence relations in a DCG.

Original: In a recent work, an alternative formulation called σ -separation is introduced which holds for a very general graphical settings (Forr´e and Mooij, 2017).
Revised: In a recent work, an alternative formulation called σseparation is introduced which holds for a very general graphical settings (Forr´e and Mooij, 2017).

Original: Here we consider a simpliﬁed version of the formal deﬁnition of σ -separation:
Revised: Here, we consider a simpliﬁed version of the formal deﬁnition of σ -separation:

Original: Deﬁnition 2 ( σ -separation) (Forr´e and Mooij, 2017)
Revised: Deﬁnition 2 ( σ -separation) (Forr´e and Mooij, 2017)

Original: A walk (cid:104) v 0 ...v n (cid:105) in DCG G = (cid:104)V , E(cid:105) is σ -blocked by C ⊆ V if:
Revised: A walk (cid:104) v 0 ...v n (cid:105) in DCG G = (cid:104)V , E(cid:105) is σ -blocked by C ⊆ V if:

Original: its ﬁrst node v 0 ∈ C or its last node v n ∈ C , or 2. 
Revised: its ﬁrst node v 0 ∈ C or its last node v n ∈ C , or 2. it

Original: it contains a collider v k / ∈ AN G ( C ) , or 3. 
Revised: contains a collider v k / ∈ AN G ( C ) , or 3. it contains a non-collider v k ∈ C that points

Original: it contains a non-collider v k ∈ C that points to aneighboring node on the walk in another strongly connected component (i
Revised: to a node on the walk in another strongly connected component (i.

Original: e., v k − 1 → v k → v k +1 with v k +1 / ∈ SC G ( v k ) , v k − 1 ← v k ← v k +1 with v k − 1 / ∈ SC G ( v k ) or v k − 1 ← v k → v k +1 with v k − 1 / ∈SC G ( v k ) or v k +1 / ∈ SC G ( v k ) ).
Revised: e , v k − 1 → v k → v k +1 with v k +1 / ∈ SC G ( v k ) , v k − 1 ← v k ← v k + with v k − 1 / ∈ SC G ( v k ) or v k − 1 ← v k → v k +1 with v k − 1 / ∈

Revised: SC

Original:  If all paths in G between any node in set A ⊆ V and any node in set B ⊆ V are σ -blocked by a set
Revised: G ( v k ) or v k +1 / ∈ SC G ( v k ) ). If all paths in G between any node in set A ⊆ V and any node in set B ⊆ V are σ -blocked by a set

Original: C ⊆ V , we say that A is σ -separated from B by C , and we write A σ | =
Revised: C ⊆ V , we say that A is σ -separated from B by C , and we write A σ | =

Original: G
Revised: G

Original: B | C .
Revised: B | C .

Revised: Forr ´ e and Mooij (2017) show that the global directed Markov property 1 holds for σ -separation in directed cyclic graphs, unlike d -separation.

Revised: This enables σ -separation to perform a similar role to d -separation but for directed cyclic graphs.

Revised: σ -separation is a generalization of d -separation.

Original: 3.3. σ -faithfulness
Revised: 3.3. σ -faithfulness

Original: σ - faithfulness refers to the property which states that all statistical dependencies found in the distribution generated by a given causal structure model is entailed in the graphical model in theform of σ -separation.
Revised: σ - faithfulness refers to the property which states that all statistical dependencies found in the distribution generated by a given causal structure model is entailed by the σ -separation relationships.

Original: Deﬁnition 3 ( σ -faithfulness)
Revised: Deﬁnition 3 ( σ -faithfulness)

Original: Given X A , X B , X C as the distributions of variables A , B , C respectively in solution X of a causal model M , σ -faithfulness states that if X A and X B are conditionally independent given X C , then A and B are σ -separated by C in the corresponding possibly cyclic graphical model G of M .
Revised: Given X A , X B , X C as the distributions of variables A , B , C respectively in solution X of a causal model M , σ -faithfulness states that if X A and X B are conditionally independent given X C , then A and B are σ -separated by C in the corresponding possibly cyclic graphical model G of M .

Original: 3.4. Relational Causal Models (RCM)
Revised: 3.4. Relational Causal Models (RCM)

Original: We adopt the deﬁnition of relational causal model used by previous work on relational causal discovery (Maier et al., 2013a; Lee and Honavar, 2020).
Revised: We adopt the deﬁnition of relational causal model used by previous work on relational causal discovery (Maier et al., 2013a; Lee and Honavar, 2020).

Original: We denote random variables and their realizations with uppercase and lowercase letters respectively, and bold to denote sets.
Revised: We denote random variables and their realizations with uppercase and lowercase letters respectively, and bold to denote sets.

Original: We use a simpliﬁed
Revised: We use a simpliﬁed

Original: Entity-Relationship model to describe relational data following previous work (Heckerman et al., 2007).
Revised: Entity-Relationship model to describe relational data following previous work (Heckerman et al., 2007).

Original: A relational schema S = (cid:104) E , R , A , card (cid:105) represents a relational domain where E , R and A refers to the set of entity, relationship and attribute classes respectively.
Revised: A relational schema S = (cid:104) E , R , A , card (cid:105) represents a relational domain where E , R and A refer to the set of entity, relationship and attribute classes respectively.

Original: It includes a cardinality function that constrains the number of times an entity instance can participate in a relationship.
Revised: It includes a cardinality function that constrains the number of times an entity instance can participate in a relationship.

Original: Figure 2a shows an example relational model that describes a simpliﬁed production system.
Revised: Figureshows an example relational model that describes a simpliﬁed user-media engagement system.

Original: The model[Figure] consists of two entity classes (Employee and Product), and one relationship class (Develops).
Revised: The model consists of three entity classes (User, Post, and Media), and two relationship classes (Reacts and Creates).

Original: The entity classEmployee has two attributes, performance and salary. The entity class Product also has two attributes: quality and proﬁt.
Revised: Each entity class has a single attribute.

Original: The cardinality constraints are shown with crow’s feet notation— an employee can develop multiple products and a product can be developed by multiple employees.
Revised: The cardinality constraints are shown with crow’s feet notation— a user can react to multiple posts, multiple users can react to a post, a post can be created by only a single media entity.

Original: A relational skeleton s is an instantiation of the relational schema S , represented by an undirected graph of entities and relationships.
Revised: A relational skeleton s is an instantiation of a relational schema S , represented by an undirected graph of entities and relationships.

Original: We denote the set of all skeletons for schema
Revised: Figure 2a shows an example skeleton of the relational model from Figure 1. It shows that Alice and Bob both react to post P1. Alice also reacts to post P2. P

Revised: Sentiment

Revised: ALICE

Revised: Sentiment

Revised: BOB

Revised: Engagement

Revised: P

Revised: Engagement P

Revised: Preference

Revised: M

Revised: R E A C T S

Revised: R E A C T S

Revised: R E A C T S

Revised: CR E A TE S

Revised: CR E A TE S

Revised: [Figure]

Revised: Relational Skeleton

Revised: Alice.

Revised: Sentiment

Revised: Bob.

Revised: Sentiment

Revised: P1.Engagement

Revised: M1.Preference

Revised: P2.Engagement (b) Ground graph [USER,REACTS,POST,CREATES,MEDIA,CREATES,POST].

Revised: Engagement [USER].Sentiment [USER,REACTS,POST].

Original: S as (cid:80) S .
Revised: Engagement [USER,REACTS,POST,CREATES,MEDIA]. Preference [USER,REACTS,POST,REACTS,USER].Sentiment [USER,REACTS,POST,REACTS,USER,REACTS,POST]. Engagement [USER,REACTS,POST,CREATES,MEDIA,CREATES,POST]. Engagement [USER,REACTS,POST,REACTS,USER,REACTS,POST].

Revised: Engagement ∩

Revised: [Figure]

Revised: and P2 both are created by media M1.

Revised: There could be inﬁnitely many possible skeletons for a given set

Original: Given a relational schema, we can specify relational paths, which intuitively correspond to ways of traversing the schema.
Revised: Given a relational schema, we can specify relational paths, which intuitively correspond to ways of traversing the schema.

Original: For the schema shown in Figure 2a, possible paths include [ Employee, Develops, P roduct ] (the products an employee develops), as well as [ Employee, Develops, P roduct, Develops, Employee ] (other employees who develops the same product).
Revised: For the schema shown in Figure 1, possible paths include [ User, Reacts, P ost ] (the posts a user reacts to), as well as [ User, Reacts, P ost, Reacts, User ] (other users who react to the same post).

Original: Relational variables consist of a relational path and an attribute that can be reached through that path.
Revised: Relational variables consist of a relational path and an attribute.

Original: For example, the relational variable [ Employee, Develops, P roduct ] .Quality corresponds to the quality of the product that an employee develops.
Revised: For example, the relational variable [ User, Reacts, P ost ] .Engagement corresponds to the overall engagements of the post that a user reacts to.

Original: The ﬁrst item  in the relational path corresponds to the perspective of the relational variable.
Revised: The ﬁrst item (i.e. User ) in the relational path corresponds to the perspective of the relational variable.

Original: The relational variable [ Employee, Develops, P roduct ] .Quality is represented from the perspective of Employee .

Original: A terminal set, P | i k is the terminal item on the relational path P =
Revised: A terminal set, P | i k is the terminal item on the relational path P =

Original: [ I j , ..., I k ] consisting of instances of class I k ∈ E ∪ R .
Revised: [ I j , ..., I k ] consisting of instances of class I k ∈ E ∪ R .

Original: A relational causal model M = (cid:104)S , D(cid:105) , is a collection of relational dependencies deﬁned over schema S .
Revised: A relational causal model M = (cid:104)S , D(cid:105) , is a collection of relational dependencies deﬁned over schema S .

Original: Relational dependencies consist of two relational variables, cause and effect.
Revised: Relational dependencies consist of two relational variables, cause and effect.

Original: As an example, consider the following relational dependency [ P roduct, develops, Employee ] .P erformance
Revised: As an example, consider the following relational dependency [ P ost, Reacts, User ] .Sentiment

Original: → [ P roduct ] .Quality which states that the quality of a product is affected by the performance of employees developing that product.
Revised: → [ P ost ] .Engagement which states that the engagement of a post is affected by the actions of users who react on that post.

Original: In Figure 2a, the arrows represents relational dependencies.
Revised: In Figure 1, the arrows represents relational dependencies.

Original: Note that, all causal dependencies are deﬁned with respect to a speciﬁc perspective.
Revised: Note that, all causal dependencies are deﬁned with respect to a speciﬁc perspective.

Original: 3.5. Ground Graph and Abstract Ground Graph
Revised: 3.5. Ground Graph and Abstract Ground Graph

Original: A realization of a relational model M with a relational skeleton is referred  as the ground graph
Revised: A realization of a relational model M with a relational skeleton is referred to as the ground graph

Original: GG M .
Revised: GG M .

Original: It is a directed graph consisting attributes of entities in the skeleton as nodes and relational dependencies among them as edges.
Revised: It is a directed graph consisting attributes of entities in the skeleton as nodes and relational dependencies among them as edges.

Original: A single relational model is actually a template for a set of possible ground graphs based on the given schema.
Revised: A single relational model is actually a template for a set of possible ground graphs based on the given schema.

Original: A ground graph has the same semantic as a graphical model.
Revised: A ground graph has the same semantic as a graphical model.

Original: Given a relational model M and a relational skeleton s , we can construct a ground graph GG M s by applying the relational dependencies as speciﬁed in the model to the speciﬁc instances of the relational skeleton.
Revised: Given a relational model M and a relational skeleton s , we can construct a ground graph GG M s by applying the relational dependencies as speciﬁed in the model to the speciﬁc instances of the relational skeleton.

Original: Figure 2b shows the ground graph for the  model of Figure 2a. An abstract ground graph (AGG) is an abstract representation that captures the dependencies among relational variables as a directed graph by generalizing dependencies in any possible ground graph. 
Revised: Figure 2b shows the ground graph for the relational model from abstract ground graph (AGG) is an abstract representation that solves the problem of generalization by the consistent all them a graph. are Hop threshold refers to the maximum length of the relational paths allowed in a speciﬁc AGG.

Original: There are two types of nodesexist inan AGG, relational variables and intersection variables.
Revised: There are two types of nodes in AGG, relational variables and intersection variables.

Original: Intersection variables are constructed from pairs of relational variables with non-empty intersections (Maier et al., 2013b).
Revised: Intersection variables are constructed from pairs of relational variables with non-empty intersections (Maier et al., 2013b).

Revised: For example, [ User, Reacts, P ost ] refers to the set of posts a user reacts to whereas [ User, Reacts, P ost, Reacts, User, Reacts, P ost ] refers to the set of other posts reacted by other users who also reacted to the same post as the given user.

Revised: These two sets of posts can overlap which is reﬂected by the corresponding intersection variable.

Original: Edges between a pair of nodes of AGG exists if the instantiations of those constituting relational variables contain a dependent pair in all ground graphs.
Revised: Edges between a pair of nodes of AGG exist if the instantiations of those constituting relational variables contain a dependent pair in all ground graphs.

Original: We deﬁne W as the set of nodes augmented with their corresponding intersection nodes for the set of relational variables W : W = W ∪ (cid:83) W ∈ W { W ∩ W (cid:48) | W ∩ W (cid:48) is an intersection node in σ -AGG M s } .
Revised: We deﬁne W as the set of nodes augmented with their corresponding intersection nodes for the set of relational variables W : W = W ∪ (cid:83) W ∈ W { W ∩ W (cid:48) | W ∩ W (cid:48) is an intersection node in AGG M s } .

Revised: Figure 2c presents the AGG from the perspective of User and with h = 6 corresponding to the model from Figure 1.

Revised: The AGG shows that the sentiment of a user is no longer independent of media preference given just engagements of the corresponding posts the user reacts to.

Revised: We also need to condition on the sentiment of other users who reacted to the same post.

Original: 3.6. Relational d-separation
Revised: 3.6. Relational d-separation

Original: Relational model describes a template for many possible instantiations of a relational schema.
Revised: Relational model describes a template for many possible instantiations of a relational schema.

Original: In order to reason about conditional independence facts entailed in all instances of a given relational template, Maier et al.
Revised: In order to reason about conditional independence facts entailed in all instances of a given relational template, Maier et al.

Original: (2013b) develop a relational counterpart forgraphical d -separation criteria.
Revised: (2013b) develop a relational counterpart for d -separation criteria.

Original: Two sets of relational variables X and Y from a given perspective are said to be d -separated by another set Z if and only if the terminal sets of X and Y are d -separated by the terminal set of Z from the given perspective in all possible ground graphs of the given model.
Revised: Two sets of relational variables X and Y from a given perspective are said to be d -separated by another set Z if and only if the terminal sets of X and Y are d -separated by the terminal set of Z from the given perspective in all possible ground graphs of the given model.

Original: Maier et al.
Revised: Maier et al.

Original: (2013b) introduce AGG as a means to reason about relational d -separation queries from a given perspective.
Revised: (2013b) introduce AGG as a means to reason about relational d -separation queries from a given perspective.

Original: The soundness and completeness of relational d -separation for AGG relies on the following assumptions:
Revised: The soundness and completeness of relational d -separation for AGG relies on the following assumptions:

Original: A 1 The relational model is acyclic.
Revised: A 1 The relational model is acyclic.

Original: A 2 There are no unobserved confounders in the relational model.
Revised: A 2 There are no unobserved confounders in the relational model.

Original: Here, soundness refers to the fact that any d -separation relationship found in AGG implies corresponding d -separation relationship in all ground graphs it represents whereas completeness claims that the d -separation facts that hold across all ground graphs are also entailed by d -separation on the AGG.
Revised: Here, soundness refers to the fact that any d -separation relationship found in AGG implies corresponding d -separation relationship in all ground graphs it represents whereas completeness claims that the d -separation facts that hold across all ground graphs are also entailed by d -separation on the AGG.

Original: The soundness of relational d -separation under AGG is already proved by Maier et al.
Revised: The soundness of relational d -separation under AGG is already proved by Maier et al.

Original: (2013b).
Revised: (2013b).

Original: However, the conditions under which completeness holds have been an open question since
Revised: However, the conditions under which completeness holds have been an open question since

Original: Lee and Honavar (2015) show that the initial formulation of Maier et al.
Revised: Lee and Honavar (2015) show that the initial formulation of Maier et al.

Original: (2013b) is not complete.
Revised: (2013b) is not complete.

Original: We resolve this question in Section 4.4 and show that AGG is also complete under certain realistic assumptions. 
Revised: We resolve this question in Section 4.4 and show that AGG is also complete under certain realistic assumptions. [Figure]

Original: 4. Relational Systems With Cycles
Revised: 4. Relational Systems With Cycles

Original: In this section, we develop deﬁnitions for cyclic relational causal models (RCM) and an abstract representation that allows causal reasoning and discovery.
Revised: In this section, we develop deﬁnitions for cyclic relational causal models (RCM) and an abstract representation that allows causal reasoning and discovery.

Original: We introduce a criterion for abstracting cyclic RCMs and provide proofs for its correctness.
Revised: We introduce a criterion for abstracting cyclic RCMs and provide proofs for its correctness.

Original: 4.1. Cyclic RCM
Revised: 4.1. Cyclic RCM

Original: A RCM is cyclic when the set of relational dependencies form one or more arbitrary length cycles.
Revised: An RCM is cyclic when the set of relational dependencies form one or more arbitrary length cycles.

Original: Deﬁnition 3 (Cyclic RCM)
Revised: Deﬁnition 4 (Cyclic RCM)

Original: A relational model M = ( S , D ) is said to be cyclic if the set of relational dependencies D constructs one or more directed cycles of arbitrary length.
Revised: A relational model M = ( S , D ) is said to be cyclic if the set of relational dependencies D constructs one or more directed cycles of arbitrary length.

Original: Cycles in RCM imply that the ground graphs are no longer guaranteed to be DAGs and they can contain cycles as well.
Revised: Cycles in RCM represent equilibrium states among a set of nodes in the ground graphs. It implies that the ground graphs are no longer guaranteed to be DAGs.

Original: In order to facilitate this, we propose a revised deﬁnition of relational dependency provided by Maier et al.
Revised: In order to facilitate this, we propose a revised deﬁnition of relational dependency provided by Maier et al.

Original: (2013b)
Revised: (2013b)

Original: by relaxing the restriction of having different attribute classes for cause and effect.
Revised: by relaxing the restriction of having different attribute classes for cause and effect.

Original: [Figure]

Original: Deﬁnition 4 (Relational Dependency)
Revised: Deﬁnition 5 (Relational Dependency)

Original: A relational dependency [ I j , ..., I k ] .X (cid:48)
Revised: A relational dependency [ I j , ..., I k ] .X (cid:48)

Original: → [ I j ] .X is a directed probabilistic dependence from any attribute class X (cid:48) to X through the relational path [ I j , ..., I k ] such that I j , ...,
Revised: → [ I j ] .X is a directed probabilistic dependence from any attribute class X (cid:48) to X through the relational path [ I j , ..., I k ] such that I j , ...,

Original: I k ∈ E ∪ R , X, X (cid:48)
Revised: I k ∈ E ∪ R , X, X (cid:48)

Original: ∈ A .
Revised: ∈ A .

Original: Note that it is possible to have X = X (cid:48) .
Revised: Note that it is possible to have X = X (cid:48) .

Original: Figure 1 shows an example relational model with cyclic dependencies.
Revised: Figure 1 shows an example relational model with cyclic dependencies (i.e. dashed arrows).

Original: Here we see a pair of dependencies [ P roduct, Develops, Employee ] .P erformance
Revised: We see a pair of dependencies [ P ost, Reacts, User ] .Sentiment

Original: → [ P roduct ] .quality
Revised: → [ P ost ] .Engagement

Original: and [ P roduct ] .Quality
Revised: and [ P ost ] .Engagement

Original: → [ P roduct, Develops, Employee ] .P erformance which are inverse to each other and forms a feedback loop.
Revised: → [ P ost, Reacts, User ] .Sentiment which are inverse to each other and form a feedback loop.

Revised: However, this mere feedback loop prohibits the use of AGG to answer relational causal query that asks whether a user’s

Revised: Sentiment about a post they reacted to is independent of the preference of media given the posts.

Original: The work by Maier et al.
Revised: Unfortunately, the work by Maier et al.

Original: (2013b) is not sufﬁcient to reason about conditional independence relationships in the ground graphs of such relational models since it is deﬁned speciﬁcally for acyclic models.
Revised: (2013b) is not sufﬁcient to reason about conditional independence relationships in the ground graphs of such relational models since it contain cycles.

Original: This motivates us to introduce a new criterion that enables  abstraction of relational queries with cyclic dependencies over all ground graphs.
Revised: This motivates us to introduce a new criterion that enables the abstraction of relational queries with cyclic dependencies over all ground graphs.

Original: 4.2. Relational σ -separation
Revised: 4.2. Relational σ -separation

Original: Conditional independence facts are only useful when they hold across all ground graphs that are consistent with the model.
Revised: Conditional independence facts are only useful when they hold across all ground graphs that are consistent with the model.

Original: Maier et al.
Revised: Maier et al.

Original: (2013b) show that relational d -separation is sufﬁcient to achieve that for acyclic models.
Revised: (2013b) show that relational d -separation is sufﬁcient to achieve that for acyclic models.

Original: However, such abstraction is not possible for cyclic models since the correctness of d -separation is not guaranteed for cyclic graphical models (Spirtes, 1995; Neal,2000).
Revised: However, such abstraction is not possible for cyclic models since the correctness of d -separation is not guaranteed for cyclic graphical models (Spirtes, 1995; Neal,

Original: In this work, we propose the following deﬁnition of relational σ -separation speciﬁcally for(possibly) cyclic relational models:
Revised: In this work, we propose the following deﬁnition of relational σ -separation speciﬁcally for cyclic relational models:

Original: Deﬁnition 5 (Relational σ -separation)
Revised: Deﬁnition 6 (Relational σ -separation)

Original: Let X , Y , and Z be three distinct sets of relational variables with the same perspective B ∈ E ∪ R deﬁned over relational schema S .
Revised: Let X , Y , and Z be three distinct sets of relational variables with the same perspective B ∈ E ∪ R deﬁned over relational schema S .

Original: Then, for relational model structure M , X and Y are σ -separated by Z if and only if, for all skeletons s ∈ (cid:80) S , X | b and Y | b are σ -separated by Z | b in ground graph GG M s for all  b ∈ s ( B )  .
Revised: Then, for relational model structure M , X and Y are σ -separated by Z if and only if, for all skeletons s ∈ (cid:80) S , X | b and Y | b are σ -separated by Z | b in ground graph GG M s for all instances b ∈ s ( B ) where s ( B ) refers to the instances of B in skeleton s .

Original: The deﬁnition directly follows from the deﬁnition of relational d -separation.
Revised: The deﬁnition directly follows from the deﬁnition of relational d -separation.

Original: If there exists even one skeleton and faithful distribution represented by the relational model for which X (cid:54)⊥⊥ Y | Z , then X | b and Y | b are not σ -separated by Z | b for b ∈ s ( B ) .
Revised: If there exists even one skeleton and faithful distribution represented by the relational model for which X (cid:54)⊥⊥ Y | Z , then X | b and Y | b are not σ -separated by Z | b for b ∈ s ( B ) .

Original: Figure 3a shows an example ground graph with a pairwise feedback loop.

Original: According to d -separation the following independence holds:

Original: Alice.

Original: Salary d | = Bob.

Original: Salary | { Alice.

Original: P erformance, Laptop.

Original: Quality } .

Original: However, according to σ -separation it doesn’t hold, rather it enforces a stricter conditioning:

Original: Alice.

Original: Salary σ | = Bob.

Original: Salary | { Alice.

Original: P erformance, Laptop.

Original: Quality,

Original: Bob

Original: .P erformance }

Original: which makes more intuitive sense.

Original: 4.3. σ -Abstract Ground Graph
Revised: 4.3. σ -Abstract Ground Graph

Original: We refer to the lifted representation for cyclic relational models as σ - abstract ground graph or σAGG.
Revised: We refer to the lifted representation for cyclic RCMs as σ - abstract ground graph or σ -AGG.

Original: A σ -AGG is constructed using the same extend method used to construct AGG (Maier et al., 2013b).
Revised: A σ -AGG is constructed using the same extend method used to construct AGG (Maier et al., 2013b).

Original: Deﬁnition 6 ( σ -Abstract Ground Graph) An abstract ground graph σ -AGG M =
Revised: Deﬁnition 7 ( σ -Abstract Ground Graph) An abstract ground graph σ -AGG M =

Original: ( V, E ) for relational model structure M =
Revised: ( V, E ) for relational model structure M =

Original: ( S , D ) , perspective B ∈ E ∪ R , and hop threshold h ∈ N 0 is a directed graph that abstracts the dependencies D for all ground graphs GG M s , where s ∈ (cid:80) S .
Revised: ( S , D ) , perspective B ∈ E ∪ R , and hop threshold h ∈ N 0 is a directed graph that abstracts the dependencies D for all ground graphs GG M s , where s ∈ (cid:80) S .

Original: The σ -AGG M s is a directed cyclic graph with the following nodes and edges:
Revised: The σ -AGG M s is a directed cyclic graph with the following nodes and edges:

Original: V = RV ∪ IV , where (a) RV is the set of relational variables with a path of length at most h
Revised: V = RV ∪ IV , where (a) RV is the set of relational variables with a path of length at most h

Original: + 1.
Revised: + 1.

Original: (b) IV are intersection variables between pairs of relational variables that could intersect 2.
Revised: (b) IV are intersection variables between pairs of relational variables that could intersect 2.

Original: E =
Revised: E =

Original: RVE ∪ IVE , where (a) RVE ⊂ RV × RV are the relational variable edges (b) IVE ⊂
Revised: RVE ∪ IVE , where (a) RVE ⊂ RV × RV are the relational variable edges (b) IVE ⊂

Original: ( IV × RV ) ∪ ( RV × IV ) are the intersection variable edges.
Revised: ( IV × RV ) ∪ ( RV × IV ) are the intersection variable edges.

Original: This is the set of edges that intersection variables “inherit” from the relational variables that they were created from
Revised: This is the set of edges that intersection variables “inherit” from the relational variables that they were created from

Original: Since the construction of an AGG and a σ -AGG M s is identical, they share mostly identical properties as deﬁned by Maier et al.
Revised: Since the construction of an AGG and a σ -AGG M s is identical, they share mostly identical properties as deﬁned by Maier et al.

Original: (2013b) for AGG.
Revised: (2013b) for AGG.

Original: The main difference being the existence of cycles.
Revised: The main difference being the existence of cycles.

Original: Consequently, the goal of σ -AGG M s is to reason about relational σ -separation queries instead of relational d -separation.
Revised: Consequently, the goal of σ -AGG M s is to reason about relational σ -separation queries instead of relational d -separation.

Revised: Figure 3b shows the σ -AGG M s corresponding to the cyclic RCM in Figure 1 with a pairwise feedback loop.

Revised: It is similar to the AGG in Figure 2c but allows cycles without violating the conditional independence statements under σ -separation which are otherwise undeﬁned with d -separation.

Original: 4.4. Soundness and Completeness of Relational σ -separation
Revised: 4.4. Soundness and Completeness of Relational σ -separation

Original: In order to discuss soundness and completeness of relational σ -separation we ﬁrst address the open problem of necessary conditions for the completeness of relational d -separation.
Revised: In order to discuss soundness and completeness of relational σ -separation we ﬁrst address the open problem of necessary conditions for the completeness of relational d -separation.

Original: Previous work has shown that the original claim of completeness of relational d -separation by Maier et al.
Revised: Previous work has shown that the original claim of completeness of relational d -separation by Maier et al.

Original: (2013a) cannot be guaranteed for any relational model (Lee and Honavar, 2015).
Revised: (2013a) cannot be guaranteed for any relational model (Lee and Honavar, 2015).

Original: A counterexample has been developed as well.
Revised: A counterexample has been developed as well.

Original: In this work, we show that relational d -separation is complete under the following assumption:
Revised: In this work, we show that relational d -separation is complete under the following assumption:

Original: A 3 The degree of any entity in the relational skeleton is greater than 1.
Revised: A 3 The degree of any entity in the relational skeleton is greater than 1.

Revised: Note that, this assumption is about the topology of the ground graphs, i.e., the network which deﬁnes how entities are connected to each other.

Revised: It only allows entities that are connected to at least two other entities.

Revised: For example, if the entities are users in a social network, the framework would only consider users who have degree at least two, i.e., are connected to at least two other users.

Revised: While this restricts the space of graph topologies allowable under the results in this work, many networks observed in real world domains, such as social networks, have minimum degree greater than one.

Original: Weﬁrst introduce the following lemma that establishes necessary conditions for AGGs to be realizable in ground graphs.
Revised: We introduce the following lemma that establishes sufﬁcient conditions for AGGs to be realizable in ground graphs.

Original: This result may be of independent interest, since it corrects a known deﬁciency in the original presentation of relational d -separation  (Maier et al., 2013b; Lee and Honavar, 2015).
Revised: This result may be of independent interest, since it provides sufﬁcient conditions for soundness in the original presentation of relational d -separation under additional assumptions (Maier et al., 2013b; Lee and Honavar, 2015).

Original: Lemma 1 Under assumption 3, every abstract ground graph can be realized as a ground graph.
Revised: Lemma 1 Under assumption 3, every abstract ground graph can be realized as a ground graph.

Original: That is, for every acyclic relational model M and skeleton s ∈ (cid:80) S any relational variable in
Revised: That is, for every acyclic relational model M and skeleton s ∈ (cid:80) S any relational variable in

Original: AGG M s has non-empty terminal sets in some ground graph GG M s .
Revised: AGG M s has non-empty terminal sets in some ground graph GG M s .

Original: Proof
Revised: Proof

Original: We ﬁrst consider the conditions under which empty terminal sets can occur, resulting in an
Revised: We ﬁrst consider the conditions under which empty terminal sets can occur, resulting in an

Original: AGG that is unrealizable in the ground graphs.
Revised: AGG that is unrealizable in the ground graphs.

Original: There are two necessary and sufﬁcient conditions for empty terminal sets to appear in all ground graphs corresponding to an AGG.
Revised: There are two necessary and sufﬁcient conditions for empty terminal sets to appear in all ground graphs corresponding to an AGG.

Original: First, there must be at least one intersection variable present in the AGG.
Revised: First, there must be at least one intersection variable present in the AGG.

Original: If no intersection variable exist in the AGG, then the completeness proof of relational d -separation by Maier et al.
Revised: If no intersection variable exist in the AGG, then the completeness proof of relational d -separation by Maier et al.

Original: (2013b) holds.
Revised: (2013b) holds.

Original: The second condition is that the intersection must be on a path consisting of only one-to-one relationships.
Revised: The second condition is that the intersection must be on a path consisting of only one-to-one relationships.

Original: In order to understand this condition, lets look at an example with the following relational paths from a hypothetical relational model which is a generalization of the counterexample given by Lee and
Revised: In order to understand this condition, lets look at an example with the following relational paths from a hypothetical relational model which is a generalization of the counterexample given by Lee and

Original: Honavar (2015) 1 :
Revised: Honavar (2015) 2 :

Original: • P =
Revised: • P =

Original: [ E i , . . .
Revised: [ E i , . . .

Original: , R j , . . .
Revised: , R j , . . .

Original: , E k ] • Q
Revised: , E k ] • Q

Original: =
Revised: =

Original: [ E i , . .
Revised: [ E i , . .

Original: R j , . .
Revised: R j , . .

Original: , R m , E k , . . .
Revised: , R m , E k , . . .

Original: , E q ] • S =
Revised: , E q ] • S =

Original: [ E i , . . .
Revised: [ E i , . . .

Original: , R j , . .
Revised: , R j , . .

Original: , R m , . . .
Revised: , R m , . . .

Original: , E s ] • S (cid:48) =
Revised: , E s ] • S (cid:48) =

Original: P + [ R m , . .
Revised: P + [ R m , . .

Original: , E s ]
Revised: , E s ]

Original: where E i , E k , E q , E s are some entity classes, R j ,
Revised: where E i , E k , E q , E s are some entity classes, R j ,

Original: R m are relationship classes, “ . . . ”
Revised: R m are relationship classes, “ . . . ”

Original: are arbitrary valid sequences of entities and relationships, and + represents the concatenation of relational paths.
Revised: are arbitrary valid sequences of entities and relationships, and + represents the concatenation of relational paths.

Original: Let’s assume two relational dependencies exist in the given model, P.X → Q.Y and S.Z → Q.Y where X, Y, Z are attributes of corresponding entity classes.
Revised: Let’s assume two relational dependencies exist in the given model, P.X → Q.Y and S.Z → Q.Y where X, Y, Z are attributes of corresponding entity classes.

Original: By deﬁnition, the corresponding edges P.X → Q.Y , S.Z → Q.Y appear in the AGG.
Revised: By deﬁnition, the corresponding edges P.X → Q.Y , S.Z → Q.Y appear in the AGG.

Original: Since S and S (cid:48) are intersectable an additional edge S.Z ∩ S (cid:48) .Z
Revised: Since S and S (cid:48) are intersectable an additional edge S.Z ∩ S (cid:48) .Z

Original: → Q.Y also appears in the AGG.
Revised: → Q.Y also appears in the AGG.

Original: Such a model can be realized in many possible ground graphs.
Revised: Such a model can be realized in many possible ground graphs.

Original: However, if we restrict the relationships to be strictly one-to-one, then there is only one skeleton structure possible to satisfy the relational dependencies at the cost of S (cid:48) 
Revised: However, if we restrict the relationships to be strictly one-to-one, then there is only one skeleton structure possible to satisfy the relational dependencies at the cost of S (cid:48) .Z

Original: .Z having empty terminal sets since an intance of R m can connect to only one instance of E k .
Revised: having empty terminal sets since an intance of R m can connect to only one instance of E k .

Original: If we allow many-to-many relationships then we can always construct a skeleton where an instance of R m connects to two instances of E k to produce non-empty terminal sets for both Q and S (cid:48) .
Revised: If we allow many-tomany relationships then we can always construct a skeleton where an instance of R m connects to two instances of E k to produce non-empty terminal sets for both Q and S (cid:48) .

Original: Since assumption 3 prohibits the second condition, it essentially implies that any relational variable in AGG M s results into non-empty terminal sets in corresponding ground graphs for every acyclic relational model M and skeleton s ∈ (cid:80) S which completes the proof for Lemma 1.
Revised: Since assumption 3 prohibits the second condition, it essentially implies that any relational variable in AGG M s results into non-empty terminal sets in corresponding ground graphs for every acyclic relational model M and skeleton s ∈ (cid:80) S which completes the proof for Lemma 1.

Original: The following proposition establishes the completeness of AGG for relational d -separation.
Revised: The following proposition establishes the completeness of AGG for relational d -separation under the assumption of a minimum degree greater than 1.

Original: Proposition 1 AGG is sound and complete for relational d -separation under assumption 3.
Revised: Proposition 1 AGG is sound and complete for relational d -separation under assumption 3.

Original: Proof Following lemma 1, the original proof of soundness and completeness of relational dseparation by Maier et al.
Revised: Proof Following lemma 1, the original proof of soundness and completeness of relational dseparation by Maier et al.

Original: (2013b) directly applies which proves proposition 1.
Revised: (2013b) directly applies which proves proposition 1.

Original: The correctness of our approach to relational σ -separation relies on several facts which are similar to the case for AGG: (1) σ -separation is valid for directed cyclic graphs; (2) ground graphs are directed cyclic graphs; and (3) σ -AGGs are directed cyclic graphs that represent exactly the edges that could appear in all possible ground graphs.
Revised: The correctness of our approach to relational σ -separation relies on several facts which are similar to the case for AGG: (1) σ -separation is valid for directed cyclic graphs; (2) ground graphs are directed cyclic graphs; and (3) σ -AGGs are directed cyclic graphs that represent exactly the edges that could appear in all possible ground graphs.

Original: Note that we no longer need assumption 1, but assumptions 2 and 3 are adopted from relational d -separation.
Revised: Note that we no longer need assumption 1, but assumptions 2 and 3 are adopted from relational d -separation.

Original: Using the previous deﬁnitions and lemmas, the following additional assumptions and sequence of results proves the correctness of our approach to identifying independence in cyclic relational models.
Revised: Using the previous deﬁnitions and lemmas, the following additional assumptions and sequence of results proves the correctness of our approach to identifying independence in cyclic relational models.

Original: A 4 The given cyclic relational model structure is σ -faithful.
Revised: A 4 The given cyclic relational model structure is σ -faithful.

Original: Theorem 1 The rules of σ -separation are sound and complete for cyclic directed graphs.
Revised: Theorem 1 The rules of σ -separation are sound and complete for cyclic directed graphs.

Original: Proof Forr´e and Mooij (2017) show that for quite general structural equation models HEDGes 2 always follow a directed global Markov property based on σ -separation which completes the proof for soundness since directed cyclic graphs are subsets of HEDGes.
Revised: Proof Forr´e and Mooij (2017) show that for quite general structural equation models HEDGes 3 always follow a directed global Markov property based on σ -separation which completes the proof for soundness since directed cyclic graphs are subsets of HEDGes.

Original: The completeness claim is already covered by Assumption 4.
Revised: The completeness claim is already covered by Assumption 4.

Original: Theorem 2 For every cyclic relational model structure M = ( S , D ) and skeleton s ∈ (cid:80) S such that relational variables involved in D are non-empty, the ground graph GG M s is a cyclic directed graph.
Revised: Theorem 2 For every cyclic RCM M = ( S , D ) and skeleton s ∈ (cid:80) S such that relational variables involved in D are non-empty, the ground graph GG M s is a cyclic directed graph.

Original: Proof
Revised: Proof

Original: Let’s assume for contradiction that there exists an acyclic ground graph g which is a realization of a given cyclic relational model M = ( S , D ) and skeleton s ∈ (cid:80) S .
Revised: Let’s assume for contradiction that there exists an acyclic ground graph g which is a realization of a given cyclic RCM M = ( S , D ) and skeleton s ∈ (cid:80) S .

Original: According to the deﬁnition of ground graphs, the edges of ground graphs are directly constructed based on the relational dependencies of the model.
Revised: According to the deﬁnition of ground graphs, the edges of ground graphs are directly constructed based on the relational dependencies of the model.

Original: Deﬁnition 3 states that a cyclic RCM consists of one or more cycles formed by the relational dependencies.
Revised: Deﬁnition 4 states that a cyclic RCM consists of one or more cycles formed by the relational dependencies.

Original: Assume a cycle in cyclic RCM is formed by the pair of relational dependencies as follows: a) P.j → Q.k , and b) Q.k → P.j where P and Q are relational paths from some perspective b and i, j refers to two attribute classes.
Revised: Assume a cycle in cyclic RCM is formed by the pair of relational dependencies as follows: a) P.j → Q.k , and b) Q.k → P.j where P and Q are relational paths from some perspective b and i, j refers to two attribute classes.

Original: By construction of g there must be two nodes a, b in g corresponding to P.j and Q.k respectively.
Revised: By construction of g there must be two nodes a, b in g corresponding to P.j and Q.k respectively.

Original: Moreover, the deﬁnition of ground graph requires two edges a → b and b → a to be present in the ground graph.
Revised: Moreover, the deﬁnition of g requires two edges a → b and b → a to be present in the ground graph.

Original: But sucha pair of edges constructs a cycle which is contradictory to the initial claim.
Revised: But such edges constructs a cycle which is contradictory to the initial claim.

Original: Thus, the ground graph g must be cyclic.
Revised: Thus, the ground graph g must be cyclic.

Original: Theorem 3 For every cyclic relational model structure M and perspective B ∈ E ∪ R , the σ -AGG M s is sound and complete for all ground graphs GG M s with skeleton s ∈ (cid:80) S .
Revised: Theorem 3 For every cyclic relational model structure M and perspective B ∈ E ∪ R , the σ -AGG M s is sound and complete for all ground graphs GG M s with skeleton s ∈ (cid:80) S .

Original: The proof follows from the proof of soundness and completeness of AGG (Maier et al., 2013b).
Revised: The proof follows from the proof of soundness and completeness of AGG (Maier et al., 2013b).

Original: The complete proof is provided in the Appendix.
Revised: The complete proof is provided in the Appendix.

Original: Theorem 4 The abstract ground graph σ -AGG M s is a cyclic directed graph if and only if the underlying relational model structure is cyclic.
Revised: Theorem 4 The abstract ground graph σ -AGG M s is a cyclic directed graph if and only if the underlying relational model structure is cyclic.

Original: Proof
Revised: Proof

Original: Let M be an arbitrary (possibly) cyclic relational model structure, and let B ∈ E ∪ R be an arbitrary perspective.
Revised: Let M be an arbitrary (possibly) cyclic relational model structure, and let B ∈ E ∪ R be an arbitrary perspective.

Original: It is clear by Deﬁnition 6 that every edge in the abstract ground graph σ -AGG M s is directed by construction.
Revised: It is clear by Deﬁnition 7 that every edge in the abstract ground graph σ -AGG M s is directed by construction.

Original: Assume for contradiction that no cycles exists in σ -AGG M s even if the relational dependencies form one or more cycles.
Revised: Assume for contradiction that no cycles exists in σ -AGG M s even if the relational dependencies form one or more cycles.

Original: Now assume the following two dependencies are part of the given relational model M : 1.
Revised: Now assume the following two dependencies are part of the given relational model M : 1.

Original: [ I j , ..., I k ] .X
Revised: [ I j , ..., I k ] .X

Original: → [ I j ] .Y ∈ D , 2.
Revised: → [ I j ] .Y ∈ D , 2.

Original: [ I j ] .Y
Revised: [ I j ] .Y

Original: →
Revised: →

Original: [ I j , ..., I k ] .X ∈ D where I j , ..., I k ∈ E ∪ R .
Revised: [ I j , ..., I k ] .X ∈ D where I j , ..., I k ∈ E ∪ R .

Original: By Deﬁnition 6, all edges inserted in σ -AGG M s are drawn from some dependency in M and edges in σ -AGG M s are constructed for all the dependencies in D .
Revised: By Deﬁnition 7, all edges inserted in σ -AGG M s are drawn from some dependency in M and edges in σ -AGG M s are constructed for all the dependencies in D .

Original: As a result there must be corresponding edges in the σ -AGG M s for both dependencies that form a cycle, which contradicts the assumption.
Revised: As a result there must be corresponding edges in the σ -AGG M s for both dependencies that form a cycle, which contradicts the assumption.

Original: Now, assume that a σ -AGG M s is acyclic even if the underlying relational model structure is cyclic.
Revised: Now, assume that a σ -AGG M s is acyclic even if the underlying RCM is cyclic.

Original: Using the same argument as above we can say that the edges in the σ -AGG M s constructed based on the dependencies in D .
Revised: Using the same argument as above we can say that the edges in the σ -AGG M s constructed based on the dependencies in D .

Original: If a cycle exists in the σ -AGG M s it directly implies the existence of a cycle in the relational model structure which leads to a contradiction.
Revised: If a cycle exists in the σ -AGG M s it directly implies the existence of a cycle in the RCM which leads to a contradiction.

Original: Thus the proof completes from both directions.
Revised: Thus the proof completes from both directions.

Original: Theorem 5 Relational σ -separation is sound and complete for σ -AGG.
Revised: Theorem 5 Relational σ -separation is sound and complete for σ -AGG.

Original: Let M be a (possibly) cyclic relational model structure, and let X , Y , and Z be three distinct sets of relational variables deﬁned over relational schema S .
Revised: Let M be a (possibly) cyclic relational model structure, and let X , Y , and Z be three distinct sets of relational variables deﬁned over relational schema S .

Original: Then, X and Y are σ -separated by Z on the abstract ground graph σ -AGG M s if and only if for all skeletons s ∈ (cid:80) S and for all perspectives
Revised: Then, X and Y are σ -separated by Z on the abstract ground graph σ -AGG M s if and only if for all skeletons s ∈ (cid:80) S and for all perspectives

Original: b ∈ s ( B ) , X | b and
Revised: b ∈ s ( B ) , X | b and

Original: Y | b are σ -separated by Z | b in ground graph GG M s .
Revised: Y | b are σ -separated by Z | b in ground graph GG M s .

Original: Proof
Revised: Proof

Original: We must show that σ -separation on an abstract ground graph implies σ -separation on all ground graphs it represents (soundness) and that σ -separation facts that hold across all ground graphs are also entailed by σ -separation on the abstract ground graph (completeness).
Revised: We must show that σ -separation on an abstract ground graph implies σ -separation on all ground graphs it represents (soundness) and that σ -separation facts that hold across all ground graphs are also entailed by σ -separation on the abstract ground graph (completeness).

Original: The proof follows from the proof of soundness and completeness of AGG (Maier et al., 2013b).
Revised: The proof follows from the proof of soundness and completeness of AGG (Maier et al., 2013b).

Original: Soundness:
Revised: Soundness:

Original: Assume that ¯ X and ¯ Y are σ -separated by ¯ Z on σ -AGG M s .
Revised: Assume that ¯ X and ¯ Y are σ -separated by ¯ Z on σ -AGG M s .

Original: Assume for contradiction that there exists an item instance b such that X | b and Y | b are not σ -separated by Z | b in the ground graph GG M s for some arbitrary skeleton s .
Revised: Assume for contradiction that there exists an item instance b such that X | b and Y | b are not σ -separated by Z | b in the ground graph GG M s for some arbitrary skeleton s .

Original: Then, there must exist a σ -connecting path p from some x ∈ ¯ X | b to some y ∈ ¯ Y | b given all z ∈ ¯ Z | b .
Revised: Then, there must exist a σ -connecting path p from some x ∈ ¯ X | b to some y ∈ ¯ Y | b given all z ∈ ¯ Z | b .

Original: By Theorem 3, σ -AGG M s is complete, so all edges in GG M s are captured by edges in σ -AGG M s .
Revised: By Theorem 3, σ -AGG M s is complete, so all edges in GG M s are captured by edges in σ -AGG M s .

Original: So, path p must be represented from some node in { N x | x ∈ N x | b } to some node in { N y | y ∈ N y | b } , where N x , N y are nodes in σ -AGG M s .
Revised: So, path p must be represented from some node in { N x | x ∈ N x | b } to some node in { N y | y ∈ N y | b } , where N x , N y are nodes in σ -AGG M s .

Original: If p is σ -connecting in GG M s , then it is σ -connecting in σ -AGG M s , implying that ¯ X and ¯ Y are not σ -separated by ¯ Z .
Revised: If p is σ -connecting in GG M s , then it is σ -connecting in σ -AGG M s , implying that ¯ X and ¯ Y are not σ -separated by ¯ Z .

Original: So, X | b and Y | b must be σ -separated by Z | b .
Revised: So, X | b and Y | b must be σ -separated by Z | b .

Original: Completeness
Revised: Completeness:

Original: Assume that X | b and Y | b are σ -separated by Z | b in the ground graph GG M s for all skeletons s for all b ∈ s ( B ) .
Revised: Assume that X | b and Y | b are σ -separated by Z | b in the ground graph GG M s for all skeletons s for all b ∈ s ( B ) .

Original: Assume for contradiction that ¯ X and ¯ Y are not σ -separated by ¯ Z on σ -AGG M s .
Revised: Assume for contradiction that ¯ X and ¯ Y are not σ -separated by ¯ Z on σ -AGG M s .

Original: Then, there must exist a σ -connecting path p for some relational variable X ∈ ¯ X to some Y ∈ ¯ Y given all Z ∈ ¯ Z .
Revised: Then, there must exist a σ -connecting path p for some relational variable X ∈ ¯ X to some Y ∈ ¯ Y given all Z ∈ ¯ Z .

Original: By Theorem 3, σ -AGG M s is sound, so every edge in σ -AGG M s must correspond to some pair of variables in some ground graph.
Revised: By Theorem 3, σ -AGG M s is sound, so every edge in σ -AGG M s must correspond to some pair of variables in some ground graph.

Original: So, if p is σ -connecting in σ -AGG M s , then there must exist some skeleton s such that p is σ -connecting in GG M s for some b ∈ s ( B ) , implying that σ -separation does not hold for that ground graph.
Revised: So, if p is σ -connecting in σ -AGG M s , then there must exist some skeleton s such that p is σ -connecting in GG M s for some b ∈ s ( B ) , implying that σ -separation does not hold for that ground graph.

Original: So, ¯ X and ¯ Y must be σ -separated by ¯ Z on σ -AGG M s .
Revised: So, ¯ X and ¯ Y must be σ -separated by ¯ Z on σ -AGG M s .

Original: Maier et al.
Revised: Maier et al.

Original: (2013b) show that relational d -separation is equivalent to the Markov condition on acyclic relational models.
Revised: (2013b) show that relational d -separation is equivalent to the Markov condition on acyclic relational models.

Original: However, it doesn’t hold for cyclic relational model.
Revised: However, it doesn’t hold for cyclic relational model.

Original: Here, we show how relational σ -separation is equivalent to the Markov condition on cyclic relational models.
Revised: Here, we show how relational σ -separation is equivalent to the Markov condition on cyclic relational models.

Original: Deﬁnition 7 (Relational σ -separation Markov Property) Let X, Y, Z be relational variables for perspective B ∈ E ∪ R deﬁned over relational schema S .
Revised: Deﬁnition 8 (Relational σ -separation Markov Condition) Let X, Y, Z be relational variables for perspective B ∈ E ∪ R deﬁned over relational schema S .

Original: For any solution ( X , (cid:15) ) of a relational model M which follows a simple SCM,
Revised: For any solution ( X , (cid:15) ) of a relational model M which follows a simple SCM,

Original: [Equation]
Revised: [Equation]

Original: if and only iffor ∀ x ∈ X | b , ∀ y ∈ Y | b , ∀ z ∈ Z [Equation] b
Revised: if and only if

Original: [Equation]
Revised: [Equation]

Original: in ground graph GG M s for all skeletons s ∈ (cid:80) S and for all b ∈ s ( B ) where ( X (cid:48) , (cid:15) (cid:48) ) refers to the solution of the SCM corresponding to the ground graphs.
Revised: in ground graph GG M s for all skeletons s ∈ (cid:80) S and for all b ∈ s ( B ) where ( X (cid:48) , (cid:15) (cid:48) ) refers to the solution of the SCM corresponding to the ground graphs.

Original: In other words, σ -separation of two relational variables X and Y given a third relational variable Z would imply X and Y are conditionally independent given Z if and only if, for all instances of X, Y, Z in all possible ground graphs, the same condition holds.
Revised: In other words, σ -separation of two relational variables X and Y given a third relational variable Z would imply X and Y are conditionally independent given Z if and only if, for all instances of X , Y , Z in all possible ground graphs, the same condition holds.

Original: For directed cyclic graphs, the global directed Markov property 3 holds for σ -separation (Forr´e and Mooij, 2017; Mooij et al., 2020).

Original: Since ground graphs of cyclic RCM are directed cyclic graphs and σ -separation on σ -AGG M s is sound and complete (by Theorem 5), we can conclude that relational σ -separation is equivalent to the relational Markov property.
Revised: Since ground graphs of cyclic RCM are directed cyclic graphs and σ -separation on σ -AGG M s is sound and complete (by Theorem 5), we can conclude that relational σ -separation is equivalent to the relational Markov property.

Original: 5. Conclusion
Revised: 5. Conclusion

Original: Cycles or feedback loops are common elements of many real-world system.
Revised: Cycles or feedback loops are common elements of many real-world system.

Original: Unfortunately, it is hardly studied in the ﬁeld of causal inference primarily because it breaks the nice properties of directed acyclic graphs.
Revised: Unfortunately, it is hardly studied in the ﬁeld of causal inference primarily because it breaks the nice properties of directed acyclic graphs.

Original: As a result, cycles and feedback loops are mostly avoided in the domain of relational causal model.
Revised: As a result, cycles and feedback loops are mostly avoided in the domain of relational causal model.

Original: In this study, we take a step forward to bridge this gap by developing an abstract representation and a criterion to reason about statistical relationships in relational models with or without cycles under a general framework.
Revised: In this study, we take a step forward to bridge this gap by developing an abstract representation and a criterion to reason about statistical relationships in relational models with or without cycles under a general framework.

Original: We show that the new criterion called σ -separation can consistently capture the statistical independence relationships of all possible instantiations of a relational causal model.
Revised: We show that the new criterion called σ -separation can consistently capture the statistical independence relationships of all possible instantiations of a relational causal model.

Original: We believe that this work will open the door for further development including but not limited to causal structure learning of relational models with cycles.
Revised: We believe that this work will open the door for further development including but not limited to causal structure learning of relational models with cycles.

Original: Appendix A. Preliminaries A.1. Directed Graphs with Hyperedges (HEDGes)
Revised: Appendix A. Preliminaries A.1. Directed Graphs with Hyperedges (HEDGes)

Original: A directed graph with hyperedges or hyperedged directed graph (HEDG) is a tuple G =
Revised: A directed graph with hyperedges or hyperedged directed graph (HEDG) is a tuple G =

Original: ( V , E , H ) , where ( V , E ) is a directed graph (with or without cycles) and H a simplicial complex over the set of vertices V of G .
Revised: ( V , E , H ) , where ( V , E ) is a directed graph (with or without cycles) and H a simplicial complex over the set of vertices V of G .

Original: A simplicial complex H over V is a set of subsets of V such that: 1) all single element sets { v } are in H for v ∈ V , and 2) if F ∈ H then also all subsets F (cid:48) ⊆
Revised: A simplicial complex H over V is a set of subsets of V such that: 1) all single element sets { v } are in H for v ∈ V , and 2) if F ∈ H then also all subsets F (cid:48) ⊆

Original: F are elements of
Revised: F are elements of

Original: V .
Revised: V .

Original: The general directed global Markov property (gdGMP) for the HEDGes is stated as follows:
Revised: The general directed global Markov property (gdGMP) for the HEDGes is stated as follows:

Original: Deﬁnition 8 (gdGMP) For all subsets X, Y, Z ⊆ V we have the implication:
Revised: Deﬁnition 9 (gdGMP) For all subsets X, Y, Z ⊆ V we have the implication:

Original: [Equation]
Revised: [Equation]

Original: A.2. Counterexample by Lee and Honavar (2015)
Revised: A.2. Counterexample by Lee and Honavar (2015)

Original: The following counterexample shows that AGG is not complete for relational d -separation.
Revised: The following counterexample shows that AGG is not complete for relational d -separation.

Original: Example.
Revised: Example.

Original: Let S
Revised: Let S

Original: =
Revised: =

Original: (cid:104)E , R , A , card (cid:105) be a relational schema such that: E = { E i } 

Revised: (cid:104)E , R , A , card (cid:105) be a relational schema such that: E = { E i }

Original: i
Revised: i

Original: =1 ; R = { R j } 3 j = with R 1 = (cid:104) E 1 , E 2 , E 4 (cid:105) , R 2 =
Revised: =1 ; R = { R j } 3 j = with R 1 = (cid:104) E 1 , E 2 , E 4 (cid:105) , R 2 =

Original: (cid:104) E 2 , E 3 (cid:105) , R 3 = (cid:104) E 3 , E 4 , E 5 (cid:105) ;
Revised: (cid:104) E 2 , E 3 (cid:105) , R 3 = (cid:104) E 3 , E 4 , E 5 (cid:105) ;

Original: A = { E 2 : { Y } , E 3 : { X } , E 5 : { Z }} ; and ∀ R ∈R ∀ E ∈E card ( R, E ) = one .
Revised: A = { E 2 : { Y } , E 3 : { X } , E 5 : { Z }} ; and ∀ R ∈R ∀ E ∈E card ( R, E ) = one .

Original: Let M =
Revised: Let M =

Original: (cid:104)S , D (cid:105) be a relational model with
Revised: (cid:104)S , D (cid:105) be a relational model with

Original: [Equation] →
Revised: D = { D 1 .X →

Original: [Equation] [Equation] Y [Equation]
Revised: [ I Y ] .Y, D 2 .Z

Original: Z → [Equation] Y ] .Y } such that D 1 =
Revised: → [ I Y ] .Y } such that D 1 =

Original: [ E 2 , R 2 , E 3 , R 3 , E 4 , R 1 , E 2 , R 2 , E 3 ] and D 2 =
Revised: [ E 2 , R 2 , E 3 , R 3 , E 4 , R 1 , E 2 , R 2 , E 3 ] and D 2 =

Original: [ E 2 , R 2 , E 3 , R 3 , E 5 ] .
Revised: [ E 2 , R 2 , E 3 , R 3 , E 5 ] .

Original: Let
Revised: Let

Original: P.X, Q.Y, S.Z, S (cid:48) .Z be four relational variables of the same perspective B = E 1 where their relational paths are distinct where
Revised: P.X, Q.Y, S.Z, S (cid:48) .Z be four relational variables of the same perspective B = E 1 where their relational paths are distinct where

Original: • P =
Revised: • P =

Original: [ E 1 , R 1 , E 2 , R 2 , E 3 ], • Q =
Revised: [ E 1 , R 1 , E 2 , R 2 , E 3 ] • Q =

Original: [ E 1 , R 1 , E 4 , R 3 , E 3 , R 2 , E 2 ], • S =
Revised: [ E 1 , R 1 , E 4 , R 3 , E 3 , R 2 , E 2 ] • S =

Original: [ E 1 , R 1 , E 4 , R 3 , E 5 ], • S (cid:48) =
Revised: [ E 1 , R 1 , E 4 , R 3 , E 5 ] • S (cid:48) =

Original: [ E 1 , R 1 , E 2 , R 2 , E 3 , R 3 , E 5 ]
Revised: [ E 1 , R 1 , E 2 , R 2 , E 3 , R 3 , E 5 ]

Original: Given the above example, Lee and Honavar (2015) make two claims:
Revised: Given the above example, Lee and Honavar (2015) make two claims:

Original: Claim 1. ( P.X (cid:54)⊥ S (cid:48) .Z | Q.Y )
Revised: Claim 1. ( P.X (cid:54)⊥ S (cid:48) .Z | Q.Y )

Original: AGG M
Revised: AGG M

Original: Claim 2.
Revised: Claim 2.

Original: There is no s ∈ (cid:80) S and b ∈ s ( B ) such that ( P.X | b (cid:54)⊥ S (cid:48) .Z
Revised: There is no s ∈ (cid:80) S and b ∈ s ( B ) such that ( P.X | b (cid:54)⊥ S (cid:48) .Z

Original: | b | Q.Y | b ) GG M s
Revised: | b | Q.Y | b ) GG M s

Original: Figure 4 shows the general pattern discussed in Section 4.4 regarding the construction of the counterexample by Lee and Honavar (2015).
Revised: Figure 4 shows the general pattern discussed in Section 4.4 regarding the construction of the counterexample by Lee and Honavar (2015).

Original: Appendix B. Proofs
Revised: Appendix B. Proofs

Original: The proof for Theorem 3 is given as follows:
Revised: The proof for Theorem 3 is given as follows:

Original: Proof
Revised: Proof

Original: Let M =
Revised: Let M =

Original: ( S , D ) be an arbitrary cyclic relational model structure and B ∈ E ∪ R an arbitrary perspective.
Revised: ( S , D ) be an arbitrary cyclic relational model structure and B ∈ E ∪ R an arbitrary perspective.

Original: Soundness: To prove that σ -AGG M s is sound, we must show that for every edge P k .X
Revised: Soundness: To prove that σ -AGG M s is sound, we must show that for every edge P k .X

Original: → P j .Y in σ -AGG M s , there exists a corresponding edge i k .X
Revised: → P j .Y in σ -AGG M s , there exists a corresponding edge i k .X

Original: → i j .Y in the ground graph GG M s for some skeleton s ∈ (cid:80) S , where i k ∈ P k | b
Revised: → i j .Y in the ground graph GG M s for some skeleton s ∈ (cid:80) S , where i k ∈ P k | b

Original: and i j ∈ P
Revised: and i j ∈ P

Original: j | b for some b ∈ s ( B ) .
Revised: j | b for some b ∈ s ( B ) .

Original: There are three subcases, one for each type of edge in an abstract ground graph: (a) Let [ B, ..., I k ] .X
Revised: There are three subcases, one for each type of edge in an abstract ground graph: (a) Let [ B, ..., I k ] .X

Original: → [ B, ..., I j ] .Y
Revised: → [ B, ..., I j ] .Y

Original: ∈
Revised: ∈

Original: RVE be an arbitrary edge in σ -AGG M s between a pair of relational variables.
Revised: RVE be an arbitrary edge in σ -AGG M s between a pair of relational variables.

Original: Assume for contradiction that there exists no edge i k .X
Revised: Assume for contradiction that there exists no edge i k .X

Original: → i j .Y in any ground graph:
Revised: → i j .Y in any ground graph:

Original: [Equation]
Revised: [Equation]

Original: By Deﬁnition 6 for σ -AGG M s , if [ B, ..., I k ] .X
Revised: By Deﬁnition 7 for σ -AGG M s , if [ B, ..., I k ] .X

Original: → [ B, ..., I j ] .Y ∈
Revised: → [ B, ..., I j ] .Y ∈

Original: RVE , then the model must have dependency [ I j , ..., I k ] .X
Revised: RVE , then the model must have dependency [ I j , ..., I k ] .X

Original: → [ I j ] .Y ∈
Revised: → [ I j ] .Y ∈

Original: D such that [ B, ..., I k ] ∈ extend ([ B, ..., I j ] , [ I j , ..., I k ]) .
Revised: D such that [ B, ..., I k ] ∈ extend ([ B, ..., I j ] , [ I j , ..., I k ]) .

Original: So, by the deﬁnition of ground graphs, there is an edge from every i k .X to every i j .Y , where i k is in the terminal set for i j along [ I j , ..., I k ] .
Revised: So, by the deﬁnition of ground graphs, there is an edge from every i k .X to every i j .Y , where i k is in the terminal set for i j along [ I j , ..., I k ] .

Original: Therefore, there exists a ground graph GG M s such that i k .X
Revised: Therefore, there exists a ground graph GG M s such that i k .X

Original: → i
Revised: → i

Original: j .Y ∈
Revised: j .Y ∈

Original: GG M s , which contradicts the assumption.
Revised: GG M s , which contradicts the assumption.

Original: (b) Let P 1 .X ∩ P 2 .X
Revised: (b) Let P 1 .X ∩ P 2 .X

Original: → [ B, ..., I j ] .Y
Revised: → [ B, ..., I j ] .Y

Original: ∈ IVE be an arbitrary edge in σ -AGG M s between an intersection variable and a relational variable, where P 1 =
Revised: ∈ IVE be an arbitrary edge in σ -AGG M s between an intersection variable and a relational variable, where P 1 =

Original: [ B, ..., I m , ..., I k ] and
Revised: [ B, ..., I m , ..., I k ] and

Original: P 2 =
Revised: P 2 =

Original: [ B, ..., I n , ..., I k ] with I m (cid:54) =
Revised: [ B, ..., I n , ..., I k ] with I m (cid:54) =

Original: I n .
Revised: I n .

Original: By Deﬁnition 6, if the σ -abstract ground graph has edge P 1 .X ∩ P 2 .X
Revised: By Deﬁnition 7, if the σ -abstract ground graph has edge P 1 .X ∩ P 2 .X

Original: →
Revised: →

Original: [ B, ..., I j ] .Y ∈ IVE , then either P 1 .X → [ B, ..., I j ] .Y ∈ RVE or
Revised: [ B, ..., I j ] .Y ∈ IVE , then either P 1 .X → [ B, ..., I j ] .Y ∈ RVE or

Original: P 2 .X
Revised: P 2 .X

Original: → [ B, ..., I j ] .Y ∈ RVE .
Revised: → [ B, ..., I j ] .Y ∈ RVE .

Original: Then, as shown in case (a), there exists an i j ∈ [ B, ..., I j ] | b such that i k .X
Revised: Then, as shown in case (a), there exists an i j ∈ [ B, ..., I j ] | b such that i k .X

Original: → i
Revised: → i

Original: j .Y ∈
Revised: j .Y ∈

Original: GG M s , which contradicts the assumption.
Revised: GG M s , which contradicts the assumption.

Original: (c) Let [ B, ..., I k ] .Y
Revised: (c) Let [ B, ..., I k ] .Y

Original: → P 1 .X ∩ P 2 .X
Revised: → P 1 .X ∩ P 2 .X

Original: ∈
Revised: ∈

Original: IVE be an arbitrary edge in σ -AGG M s between an intersection variable and a relational variable, where P 1 =
Revised: IVE be an arbitrary edge in σ -AGG M s between an intersection variable and a relational variable, where P 1 =

Original: [ B, ..., I m , ..., I j ] and P 2 =
Revised: [ B, ..., I m , ..., I j ] and P 2 =

Original: [ B, ..., I n , ..., I j ] with I m (cid:54) =
Revised: [ B, ..., I n , ..., I j ] with I m (cid:54) =

Original: I n .
Revised: I n .

Original: The proof follows case (b) to show that there exists a skeleton s ∈ (cid:80) S and b ∈ s ( B ) such that for all i k ∈
Revised: The proof follows case (b) to show that there exists a skeleton s ∈ (cid:80) S and b ∈ s ( B ) such that for all i k ∈

Original: [ B, ..., I k ] | b there exists an i j ∈ P 1 ∩ P 2 | b such that i k .X
Revised: [ B, ..., I k ] | b there exists an i j ∈ P 1 ∩ P 2 | b such that i k .X

Original: → i
Revised: → i

Original: j .Y ∈
Revised: j .Y ∈

Original: GG M s .
Revised: GG M s .

Original: Completeness: To prove that the σ -abstract ground graph σ -AGG M s is complete, we show that for every edge i k .X
Revised: Completeness: To prove that the σ -abstract ground graph σ -AGG M s is complete, we show that for every edge i k .X

Original: → i j .Y in every ground graph GG M s where s ∈ (cid:80) S , there is a set of corresponding edges in σ -AGG M s .
Revised: → i j .Y in every ground graph GG M s where s ∈ (cid:80) S , there is a set of corresponding edges in σ -AGG M s .

Original: Speciﬁcally, the edge i k .X
Revised: Speciﬁcally, the edge i k .X

Original: → i j .Y yields two sets of relational variables for some b ∈ s ( B ) , namely P k .X
Revised: → i j .Y yields two sets of relational variables for some b ∈ s ( B ) , namely P k .X

Original: =
Revised: =

Original: { P k .X | i
Revised: { P k .X | i

Original: k ∈ P k | b } and P j .Y =
Revised: k ∈ P k | b } and P j .Y =

Original: { P j .Y | i j ∈ P
Revised: { P j .Y | i j ∈ P

Original: j | b } .
Revised: j | b } .

Original: Note that all relational variables in both P k .X and P j .Y are nodes in σ -AGG M s , as are all pairwise intersection variables.
Revised: Note that all relational variables in both P k .X and P j .Y are nodes in σ -AGG M s , as are all pairwise intersection variables.

Original: We show that for all P k .X ∈ P k .X and for all
Revised: We show that for all P k .X ∈ P k .X and for all

Original: P j .Y ∈
Revised: P j .Y ∈

Original: P
Revised: P

Original: j .Y either (a)
Revised: j .Y either (a)

Original: P k .X
Revised: P k .X

Original: → P j .Y ∈
Revised: → P j .Y ∈

Original: σ -AGG M s (b)
Revised: σ -AGG M s (b)

Original: P k .X
Revised: P k .X

Original: ∩ P (cid:48) k .X
Revised: ∩ P (cid:48) k .X

Original: → P j .Y ∈
Revised: → P j .Y ∈

Original: σ -AGG M s where P (cid:48)
Revised: σ -AGG M s where P (cid:48)

Original: k .X
Revised: k .X

Original: ∈ P k .X , or (c) P k .X
Revised: ∈ P k .X , or (c) P k .X

Original: → P j .Y ∩ P (cid:48)
Revised: → P j .Y ∩ P (cid:48)

Original: j .Y 
Revised: j .Y ∈

Original: ∈ σ -AGG M s where P (cid:48)
Revised: σ -AGG M s where P (cid:48)

Original: j .Y ∈ P j .Y .
Revised: j .Y ∈ P j .Y .

Original: Let s ∈ (cid:80) S be an arbitrary skeleton, let i k .X
Revised: Let s ∈ (cid:80) S be an arbitrary skeleton, let i k .X

Original: → i
Revised: → i

Original: j .Y ∈
Revised: j .Y ∈

Original: GG M s be an arbitrary edge drawn from [ I j , ..., I k ] .X
Revised: GG M s be an arbitrary edge drawn from [ I j , ..., I k ] .X

Original: → [ I j ] .Y ∈ D , and let
Revised: → [ I j ] .Y ∈ D , and let

Original: P k .X ∈ P k .X, P j .Y ∈
Revised: P k .X ∈ P k .X, P j .Y ∈

Original: P j .Y be an arbitrary pair of relational variables.
Revised: P j .Y be an arbitrary pair of relational variables.

Original: (a) If P k ∈ extend ( P j , [ I j , ..., I k ]) , then P k .X
Revised: (a) If P k ∈ extend ( P j , [ I j , ..., I k ]) , then P k .X

Original: → P j .Y ∈
Revised: → P j .Y ∈

Original: σ -AGG M s by Deﬁnition 6.
Revised: σ -AGG M s by Deﬁnition 7.

Original: (b) If P k / ∈ extend ( P j , [ I j , ..., I k ]) , but ∃ P (cid:48)
Revised: (b) If P k / ∈ extend ( P j , [ I j , ..., I k ]) , but ∃ P (cid:48)

Original: k ∈ extend ( P j , [ I j , ..., I k ]) such that P (cid:48)
Revised: k ∈ extend ( P j , [ I j , ..., I k ]) such that P (cid:48)

Original: k .X
Revised: k .X

Original: ∈ P k .X , then P (cid:48)
Revised: ∈ P k .X , then P (cid:48)

Original: k .X
Revised: k .X

Original: → P j .Y ∈ σ -AGG M s , and P k .X
Revised: → P j .Y ∈ σ -AGG M s , and P k .X

Original: ∩ P (cid:48) k .X
Revised: ∩ P (cid:48) k .X

Original: → P j .Y ∈
Revised: → P j .Y ∈

Original: σ -AGG M s by Deﬁnition 6.
Revised: σ -AGG M s by Deﬁnition 7.

Original: (c) If ∀ P ∈ extend ( P j , [ I j , ..., I k ])( P.X / ∈ P k.X ) , then ∃ P (cid:48)
Revised: (c) If ∀ P ∈ extend ( P j , [ I j , ..., I k ])( P.X / ∈ P k.X ) , then ∃ P (cid:48)

Original: j such that i j ∈ P (cid:48)
Revised: j such that i j ∈ P (cid:48)

Original: j | b and P k ∈ extend ( P (cid:48)
Revised: j | b and P k ∈ extend ( P (cid:48)

Original: j , [ I j , ..., I k ]) .
Revised: j , [ I j , ..., I k ]) .

Original: Therefore, P (cid:48)
Revised: Therefore, P (cid:48)

Original: j .Y ∈ P
Revised: j .Y ∈ P

Original: j .Y,
Revised: j .Y,

Original: P k .X
Revised: P k .X

Original: → P (cid:48)
Revised: → P (cid:48)

Original: j .Y ∈
Revised: j .Y ∈

Original: σ -AGG M s , and P k .X → P (cid:48) j .Y ∩ P j .Y ∈ σ -AGG M s by Deﬁnition 6.
Revised: σ -