Warded Datalog+/- has recently emerged as a powerful logic language for ontological reasoning on large knowledge graphs, ofering a very good trade-of between expressive power and data complexity. Achieving decidability and data tractability in practice, over complex recursive settings with existential quantification, requires reasoners to adopt specialized strategies that control the efects of recursion and ensure reasoning termination with small memory footprint. However, to enable these strategies, the Warded Datalog+/- settings must be in a “harmless” form, i.e., without joins on variables afected by existential quantification. We provide an algorithm to remove such “harmful” joins, supporting reasoning decidability and the full expressive power of the language while preserving the correctness of the task. The algorithm is integrated into the Vadalog system, a state-of-the-art Warded Datalog+/- -based reasoner. We employ it to solve the Strong Link problem, a relevant ifnancial task to find possible links between pairs of companies, based on the existence of a person who owns a significant share of both their stocks. Solving this problem allows to investigate possibly malevolent shareholdings and ownerships.
of a set Σ of Warded rules over a database instance is Recent years have witnessed a growing interest, among defined via the chase procedure [6]. Intuitively, it adds companies of distinct size and scope, towards building new facts to , possibly containing freshly generated and exploiting private corporate knowledge in the form symbols (technically, labelled nulls) [7] that act as placeof financial Knowledge Graphs (KG). This led to the ris- holders for the existentially quantified variables, until ing adoption of intelligent systems that allow to manage Σ is satisfied. Warded Datalog ± is implemented in the such extensional knowledge and enrich it with new infor- Vadalog system [3], a state-of-the-art reasoner that allows mation, inferred via eficient ontological reasoning mech- to perform ontological reasoning in complex scenarios. anisms and modeled by logical rules and ontologies in Critical Aspects in Reasoning. When reasoning on specific formalisms [ 1, 2]. Employing modern languages Datalog± settings, in the presence of recursion and exfor Knowledge Representation and Reasoning (KRR) [3] istential quantification, infinite labelled nulls could be enabled companies to solve relevant problems and tasks generated in the chase, causing the procedure not to terin distinct business domains, such as investment analysis, minate and inhibiting the decidability of the task [8]. In company ownership, shock propagation, fraud detection, this work, we tackle such issue in the context of Warded anti-money laundering, etc. Datalog± scenarios. Indeed, while the computational A Formalism for Ontological Reasoning. As main re- properties of the Warded fragment bode well for eficient quirements, KRR languages must exhibit full support for implementations, it is still required to apply specific techrecursion and joins as well as existential quantification, niques (namely, termination strategies) that properly conall aspects essential to guarantee the expressive power trol the interactions between recursion and existentials needed for ontological reasoning and KG traversal. At the in the chase. Consider the following example. same time, decidability and tractability of the reasoning must be sustained, basically limiting the data complexity Example 1. Scenario modeled with a Warded set Σ . to a polynomial degree [4]. Warded Datalog± [2] is a member (technically, a fragment) of the Datalog± fam- Bank(x) → ∃d Manager(x, d) ( ) ily [5] that recently rose among the logic languages for Acquires(x, y), Manager(x, d) → Manager(y, d) ( ) ontological reasoning on KGs. It covers these require- BankGroup(x, y) → ∃d Manager(x, d) ( ) ments, ofering a very good trade-of between expressive power and computational complexity and capturing Manager(x, d), Manager(y, d) → BankGroup(x, y) ( ) Σ represents a bank acquisition scenario. For each bank there exists a manager (rule ). If acquires a bank , also becomes manager of (rule ). If and have a common manager, they are in the same bank group (rule ) and vice versa (rule ).
Consider the database instance = {Bank(Unicredit), The applicability of the isomorphism termination stratBank(MPS), Acquires(Unicredit, MPS)} and the query : egy, to achieve reasoning decidability while preserving “what are all the BankGroups?” as ontological reasoning correctness of query answering, is essential in such comtask. It can be observed that the set of such bank groups plex domains. Therefore, reasoning settings that feature is finite, whereas the chase does not terminate. First, harmful joins and recursion require to be treated properly. we generate Manager(Unicredit, 0) and Manager(MPS, 1) Such is the case of the Strong Link problem [9], a highby activating from the facts Bank(Unicredit) and interest scenario in the context of company ownership. Bank(MPS), respectively. Then, we obtain Man- It consists in determining possible links between pairs of ager(MPS, 0) via , BankGroup(Unicredit,MPS) via the companies, based on the existence of a person who owns join on 0 in , Manager(Unicredit, 2), Manager(MPS, 2) a significant share of both their stocks. Solving this task by activating and so on. Indeed, the recursion involv- allows to investigate and monitor (possibly malevolent) ing , and causes the generation of an infinite set company shareholdings and ownerships. ⋃︀=3,...{Manager(Unicredit, ), Manager(MPS, )}. Motivated by this, we apply our HJE algorithm to the Warded Datalog± scenario of Strong Link, rewriting it into its Harmless Warded equivalent and enabling the Vadalog system to solve the task in an eficient fashion, while preserving correctness.
Enabling Reasoning Termination in Vadalog. To achieve reasoning termination in practice, while upholding the correctness of the task, the Vadalog system applies the isomorphism termination strategy in the chase: isomorphic copies of previously generated facts (i.e., same In detail, the main contributions of this paper are: name, same constants in same position and bijection between labelled nulls) are not explored, i.e., the chase steps • An integration of the theoretical bases for the starting from them are not performed and the derived Warded fragment. We introduce Harmless Warded facts are not generated. This strategy exploits the the- Datalog± and we discuss the role of harmful joins in oretical underpinning of Warded Datalog± known as Warded settings, with reference to expressive power reasoning boundedness, which states that facts derived and reasoning termination. We define and solve the from isomorphic origins would be in turn isomorphic, disarmament problem, which consists in rewriting a thus uninformative for query answering [9]. set of Warded rules with harmful joins into a Harmless
However, as a necessary condition for such exploita- Warded version, equivalent with respect to the chase. tion, the set of rules is required to be in a “harmless” • The Harmful Join Elimination algorithm, our rewritform, i.e., without a “harmful” type of joins between ing technique that exploits such bases to solve the variables afected by existential quantification (namely, disarmament problem and enable Vadalog’s terminain Harmless Warded Datalog± ). This is due to the fact tion strategy in practice on recursive reasoning setthat rules with these harmful joins could activate on tings with harmful joins. We illustrate the procedure labelled nulls, propagated from the existentials: there- and we apply it to the program in Example 1. fore, the suppression of isomorphic facts carrying such nulls could hamper their activation and, consequently, • A real-world application of the HJE algorithm on the undermine the correctness of the reasoning task. For Strong Link problem. We discuss the financial task and instance, in Example 1 Manager(MPS, 0) is isomorphic rewrite the corresponding setting with harmful joins to Manager(MPS, 1), yet its suppression would prevent into its Harmless Warded equivalent. We provide an the join in with Manager(Unicredit, 0) from generat- experimental evaluation of the now enabled reasoning ing BankGroup(Unicredit,MPS) and the query from being via the Vadalog system, extracting input data from the answered correctly. On the other hand, it is intuitive to open KG provided by DBpedia [12]. observe that precluding the use of harmful joins when modeling reasoning settings would afect the expressive power of the adopted KRR language.
In this work, we investigate the role of harmful joins in Warded Datalog± settings and we enable reasoning termination in their presence by applying the Harmful Join Elimination (HJE) algorithm [10], a technique to rewrite a set of Warded rules with harmful joins into an equivalent Harmless Warded form.
class of methodologies for Datalog rewriting. Among them, we mention its conversion into specific fragments, such as Guarded [13], Linear [14], and Disjunctive [15].
Similarly, by interpreting the rules as queries, several methods have been devised to rewrite Datalog± programs with existential rules [16, 17] and from distinct formalisms, such as Regular Path Queries [18] and DeA Financial Use Case. As a powerful Warded Datalog± - scription Logics [19, 20]. The importance of achieving debased reasoner, the Vadalog system is employed to ef- cidability of Datalog± reasoning settings, in the presence ifciently solve relevant tasks in real-world scenarios, of recursion and existential quantification, determined mainly related to the financial and business realm [ 11]. the research and the development of novel approaches to sustain termination of reasoning tasks [21, 5, 7]. Yet,
the HJE algorithm is, to the best of our knowledge, the generated from the application of the rules in Σ over ifrst technique to rewrite Warded Datalog ± rules into [6]. We denote chase graph (, Σ) as the directed an equivalent Harmless Warded version, that allows to graph with the facts from chase(, Σ) as nodes and an achieve reasoning termination and decidability. edge from a node to a node if is obtained from Overview. This paper is organized as follows. In Sec- (and possibly other facts) via a chase step, i.e., a rule tion 2, we recall relevant background notions. In Sec- in Σ [5]. Given a pair = (Σ , Ans), where Ans is an tion 3, we discuss the disarmament problem and we illus- n-ary predicate, we define the evaluation of over as trate the HJE algorithm in action. In Section 4, we provide the set of tuples (, Σ) = {¯ ∈ dom() | Ans(¯) ∈ the application of HJE on the Strong Link problem and chase(, Σ) }, where ¯ is a tuple of constants. We denote we present the experimental evaluation. We draw our reasoning task as the task of finding a database instance conclusions in Section 5. such that: (i) ¯ ∈ if Ans(¯) ∈ (, Σ) and (ii) for every other instance ′ such that ¯ ∈ ′ if ¯ ∈ (, Σ) , there is a homomorphism from to ′ [3].
In this section, we briefly recall some relevant notions to guide our discussion. Let , , and be disjoint countably infinite sets of constants, (labelled) nulls and (regular) In this section, we present our rewriting technique and variables, respectively. A (relational) schema S is a finite we apply it to Example 1. As the goal of this paper is to set of relation symbols (or predicates) with associated enable reasoning with the Vadalog system on a real-world arity. A term is a either a constant or variable. An atom scenario with harmful joins, we refer to the extended over S is an expression of the form (¯) , where ∈ S versions of the work [10, 22] for an in-depth discussion is of arity > 0 and ¯ is an -tuple of terms. A database of the algorithm and the theory behind. (instance) over S associates to each relation symbol in Without loss of generality (as more complex joins can S a relation of the respective arity over the domain of be broken into multiple steps [3]), we define a harmful constants and nulls. The members of the relations are join rule as a rule of the form: called tuples or facts [9].
(1, 1, ℎ), (2, 2, ℎ) → ∃ (, ) ( )
employs vadalog, a KRR language that implements where , and are atoms, [3] and [3] are afected Warded Datalog± as its logical core. A Warded Datalog± positions, 1, 2 ⊆ , 1, 2 ⊆ are disjoint tuples of program consists of a set of facts and rules. An existential harmless variables or constants, ℎ is a harmful variable. wruhleeirsea firs(tt-hoerdbeordsyen)taenndce ∀(t¯h∀e¯(h ea(d¯), a¯)re→co∃n¯jun(c¯t,io¯))ns, Harmless Warded Datalog± . As we have introduced, of atoms, over the respective predicates, with constants in order to exploit the reasoning boundedness property, and variables. For brevity, we may omit quantifiers and Warded programs with recursion and existentials must denote conjunction by comma. Let Σ be a set of rules not contain harmful joins, that is, they belong to the and [] a position (i.e., the -th term of a predicate with Harmless Warded fragment. A set of rules ∈ Harmless arity , where = 1, . . . , ). We define [] as afected if Warded Datalog± if the following conditions hold: (1) it is (i) appears in a rule in Σ with the -th term that con- Warded, i.e., all the dangerous variables in its rules appear tains an existentially quantified variable or, (ii) there is in a single body atom (the ward), which only shares harma rule of Σ such that a universally quantified variable less variables with the rest of the body; and (2) it does not is only in afected body positions and in position [] in contain harmful join rules. Indeed, while the chase prothe head of . A variable is harmful, with respect to a cedure remains potentially infinite in Harmless Warded rule, if appears only in afected positions, otherwise it settings due to the generation of infinite labelled nulls, is harmless. A rule that contains a harmful variable is a the absence of joins activating on such nulls renders their harmful rule. In Example 1, and are existential rules, identity irrelevant in the evaluation [9]. propagates the afected position Manager[2] and is a harmful join rule, i.e., a rule such that a harmful variable is involved in a join (namely, harmful join). If the harmful variable appears in the head of the rule, it is dangerous.
A rule with a dangerous variable is a dangerous rule.
Warded programs, the isomorphism termination strategy can thus be applied without afecting the correctness of the reasoning task: given two isomorphic facts and ′, only is explored in the chase, whereas the descending portions of the chase graph rooted in ′ are pruned. However, restricting vadalog to such fragment, while it preserves the computational qualities of Warded Datalog± , limits the expressiveness of the KRR language Chase and Reasoning. Let be a database and Σ a set of rules. The chase procedure is a fundamental algorithmic tool that enforces the satisfaction of Σ by expanding into a new instance chase(, Σ) with facts to joins between harmless variables. To avoid such re- ′ be unifiable with by substitution . The result of strictions, we define the disarmament problem for a set unfolding at with is the rule : (, →) . If Σ of Warded Datalog± rules as the task of finding a set the head of contains an existentially quantified variable Σ ′ of Harmless Warded rules that is equivalent to Σ . In ℎ, it replaces ℎ with a Skolem atom ℎ in , where is this context, two sets of rules are equivalent if they have an injective, deterministic and range disjoint function that the same meaning with respect to the chase [14], i.e., calculates the values for existentially quantified variables, chase(, Σ) = chase(, Σ ′) modulo fact isomorphism to control the identity of labelled nulls. for each database . It can be proved that the disarmament problem is always solvable, that is, for each Warded Moreover, to cover the activation of the harmful join in set Σ there is an equivalent Harmless Warded set Σ ′ [22]. on nulls that are propagated from causes involved in a recursion, the following folding operation [14] is applied.
with one or more harmful join rules. With the goal of Definition 3 ( Folding). Let be a rule , →, where enabling the isomorphism termination strategy, while is an atom and and are (conjunctions of) atoms, preserving vadalog’s expressive power, we propose a and let be a rule ′→, where is an atom and ′ technique to rewrite Σ into an equivalent Harmless form, is an atom or a conjunction of atoms. Let ′ be unifiable i.e., to solve the disarmament problem. with by substitution . The result of folding into is
We first provide some theoretical bases. By definition the rule : (, →) . of harmful variables, we observe that Σ always contains one or more rules that propagate the afectedness of such variables (i.e., the nulls in the chase), from the existentials to the harmful join. We define them as follows.
rule, we employ the harmful unfolding tree (hu-tree).
Apart from the more technical side [10], the hu-tree for ︀⟨ Σ , ⟩︀ can be defined as a rule-labelled tree-like structure Definition 1 ( Causes of Afectedness ). Let ∈ Σ be a where: (i) the root is labelled by ; (ii) for each , there harmful join rule and let ∈ {A,B} be an atom in the body exists a root-to-leaf path whose nodes are labelled by of . We define causes of afectedness as the sequences of the result of unfolding their parent nodes with the causes rules Γ = [ , . . . , 1] ( < |Σ |, ≥ 1) ∈ Σ , where: in (in order of appearance); (iii) for each ∈ (i) 1: 1(, ), 1→∃ℎ 2(, , ℎ) is a direct cause, i.e., involved in a recursion, there exists a node labelled by the an existential rule that causes a position to be afected; and result of folding its parent node into . By definition of (ii) : (, , ℎ), →+1(, , ℎ), 1 < ≤ , are unfolding and folding, it can be proved that the leaves indirect causes, i.e., rules that propagate the afectedness of are harmless rules that cover the generation of all from 1 to , such that +1 = . 1, . . . , are atoms, the facts derived in the chase from the activation of on 1, . . . , are (conjunctions of) atoms not containing ℎ labelled nulls [22]. With reference to Example 1, Figure 1 (as the rules are Warded). shows the path 21, built by unfolding with the causes in 21 (atoms are renamed for space reasons).
Man(x,d),Man(y,d)→BGroup(x,y) βГMan12 αГMan21
Acq(v1,v2),Man(v1,d), Man(y,d)→BGroup(v2,y)
αГMan12 Acq(v1,v2),Bank(v1),Man(y,d),
fdα(v1)→BGroup(v2,y) Acq(v1,v2),Bank(v1),Bank(v3), fdα(v1),fdα(v3)→BGroup(v2,v3) With reference to Example 1, the rules and are direct causes of afectedness, whereas is an indirect one. The sequences of causes for the atom 1 (resp., 2 in order of appearance in ) are: Γ 11 = [ ], Γ 12 = [, ], Γ 13 = [ ], Γ 14 = [, ]. Let = Γ ⌢Γ be the concatenation of the sequences Γ and Γ for the atoms , in : the causes in are labelled after the sequence they belong to. In our example 21 = [ Γ12 , Γ12 , Γ21 ]. Intuitively, our goal is to replace each with harmless rules that cover the generation of all the facts derived in the chase from the activation of . To learn how the propagated nulls that activate the harmful join afect the meaning, and to consequently build proper harmless rules, we compose along all . Such composition is performed via the unfolding operation [14], defined as follows.
Definition 2 ( Unfolding). Let be a rule , →, retical bases, we now present our rewriting algorithm, where and are atoms and is an atom or a con- which we named Harmful Join Elimination. Given a junction of atoms, and let be a rule →′, where ′ Warded set Σ with one or more harmful join rules , is an atom and an atom or a conjunction of atoms. Let HJE(Σ) = Σ ′ ∈ Harmless Warded Datalog± such that
In the grounding phase, HJE adds new harmless rules to cover the activation of on ground values, propagated from the database. It employs the Dom(ℎ) [9] atom, to ensure that the harmful join variables bind only to constants in the domain, and an artificial atom ′, where ∈ {, } of . With reference to Example 1, the following rules (namely, dom rules) are added to Σ ′.
HJE algorithm. We employ it to enable reasoning with the Vadalog system on the Strong Link problem, which we then solve and empirically evaluate.
The Strong Link Problem. Being able to determine and monitor the connections between companies and shareholders, as well as to investigate possible felonious activities in company ownership scenarios, is of highinterest in the financial and corporate realm. With such goal in mind, we consider the Strong Link problem over “significantly controlled companies”, that is, companies for which there exist “significant shareholders” who own more than 20% of their stocks [9]. Two distinct companies and are involved in a strong link if they share (at least) one significant shareholder. The Strong Link scenario can be modeled in Warded Datalog± as follows. Example 2. Strong Link modeled with Warded rules.
Company(x) → ∃p∃s Owns(p, s, x)
Owns(p, s, x) → Stock(x, s)
Owns(p, s, x) → PSC(x, p)
PSC(x, p), Controls(x, y) → ∃s Owns(p, s, y) PSC(x, p), PSC(y, p), x ̸= y → StrongLink(x, y) StrongLink(x, y) → ∃p∃s Owns(p, s, x) StrongLink(x, y) → ∃p∃s Owns(p, s, y)
Stock(x, s) → Company(x) (1) (2) (3) (4) (5) (6) (7) For each company there exists a person who owns a certain share (rule 1), which is part of the company stock (rule 2). If owns a share (rule 3), is a “person with significant control” (psc) for . For each company controlled by , owns a certain share of (rule 4). If two distinct companies and share a common psc, they are involved in a strong link (rule 5). Vice versa, if there is a strong link between and , there exists a person who owns a share of (rule 6) and (rule 7). If is a share of a stock for , then is a company (rule 8).
Indeed, reasoning on the Strong Link program does not terminate, as the existential quantification in rule 4, rule 6 and rule 7, and their recursion with rule 3 and rule 5, cause the generation of an infinite set of facts in the chase. Moreover, the isomorphism termination strategy cannot be applied, due to the presence of the harmful join rule 5 that hampers the reasoning boundedness of the program. Therefore, we apply HJE to enable reasoning with the Vadalog system. Note that rule 1, rule 6 and rule 7 are direct causes of afectedness for the atoms in rule 5, whereas rule 3 and rule 4 are indirect ones. Due to the recursion on such causes, the back-composition phase builds a hu-tree which consists of both unfolding and folding leaves. The harmless rules that replace rule 5 in Example 2 are provided below (atoms are renamed and some rules are merged for space reasons). As in Example 1, s are added via grounding, whereas s and s label hu-tree leaves deriving from unfolding and folding, respectively. Note that some rules were dropped during cleanup, as they would have never activated.
Dom(p), PSC(x, p) → PSC′(x, p) ( 1)
PSC′(x, p) → PSC(x, p) ( 2)
PSC′(x, p), PSC′(y, p), x ̸= y → SLink(x, y) ( 3) Comp(x), Contr(x, y) → SLink(x, y), SLink(y, x) ( 1,2) Comp(x), Contr(x, y), Contr(x, z) → SLink(y, z) ( 3) SLink(x, y), Contr(x, w), Contr(x, z) → SLink(w, z) ( 4) SLink(x, y), Contr(x, w), Contr(x, z) → SLink(z, w) ( 5) SLink(x, y), Contr(x, z) → SLink(x, z), SLink(z, x) ( 6,7) SLink(x, y), Contr(x, z) → SLink(y, z) ( 1)
SLink(x, y), Contr(x, z) → SLink(z, y) ( 2)
companies and ownership relations from the open KG provided by DBpedia [12], which publishes information for around 67K companies. We adopted datasets of increasing complexity, with 1K, 5K, 10K, 15K, 20K, 25K and 30K companies, respectively. The Vadalog system, with HJE integrated as part of its logic optimizer [3], was used as a “library” and invoked from specific Java test classes for end-to-end executions of the reasoning. We run the experiments on a local installation of the Vadalog system, with a MacBook Pro i7 with 2.5 GHz and 8 GB of RAM. We first applied HJE to the program, with average time of under 1 second. Then, we built two reasoning scenarios: • SpecStrongLinks (Spec), to obtain all the strong links of the company BBC, to vary the number of companies; • AllStrongLinks (All), to obtain all the possible pairs of strong links, to vary the number of companies.
for the reasoning to terminate and of number of strong links detected. Figure 3(a) illustrates the former perspective. The results prove the very good performance of the Vadalog system, even if tested on a local installation. Indeed, the Spec scenario requires more time than the All one for smaller datasets; when the number of companies increases, All tends to a steeper curve. On the other hand, Figure 3(b) shows the strong links founds between pairs of distinct companies in the All scenario. Finally, Figure 4 provides all the numerical results of the experiments.
Datalog± -based Vadalog system allows companies to solve relevant tasks in the financial and corporate realm.
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