A Formal Graph Model for RDF and Its Implementation
Vinh Nguyen Jyoti Leeka Olivier Bodenreider Kno.e.sis Center IIIT Delhi National Library of Medicine Wright State University India National Institute of Health Ohio, USA [email protected] Maryland, USA [email protected] [email protected] Amit Sheth Kno.e.sis Center Wright State University Ohio, USA [email protected]
ABSTRACT mapped to a directed labeled arc connecting the subject and Formalizing an RDF abstract graph model to be compati- the object nodes of that graph. Although these straightfor- ble with the RDF formal semantics has remained one of the ward mappings work for simple triples at the instance level, foundational problems in the Semantic Web. In this paper, challenges may arise where predicates also play the role of we propose a new formal graph model for RDF datasets. the subject in other triples as in Fig. 1. This model allows us to express the current model-theoretic Particularly, in the first triple, the predicate is mapped to semantics in the form of a graph. We also propose the con- a labeled arc connecting the two subject/object nodes of the cepts of resource path and triple path as well as an algorithm subgraph G1 representing the first triple. When the predi- for traversing the new graph. We demonstrate the feasibil- cate from the first triple becomes the subject or the object ity of this graph model through two implementations: one of the second triple, it is mapped to another node of the is a new graph engine called GraphKE, and the other is subgraph G2 representing the second triple. As a result, a extended from RDF-3X to show that existing systems can predicate is mapped to both a labeled arc in the subgraph also benefit from this model. In order to evaluate the em- G1 and a node within the subgraph G2. Despite the fact pirical aspect of our graph model, we choose the shortest that the predicate is shared between the two triples, the re- path algorithm and implement it in the GraphKE and the sulting graph is comprised of two disconnected subgraphs RDF-3X. Our experiments on both engines for finding the G1 and G2, making it impossible to traverse between the shortest paths in the YAGO2S-SP dataset give decent per- subgraphs. The Object 2 is unreachable from Subject 1. formance in terms of execution time. The empirical results In addition, it is unfortunate for the NLAN graph model show that our graph model with well-defined semantics can that such a scenario is common as RDF syntax allows us to be effectively implemented. make assertions about any resources, including predicates. Here we present two of such scenarios: schema triples and singleton property triples. We will take the singleton prop- 1. INTRODUCTION erty triples forward to motivate our work. As the adoption of Semantic Web grows, a large number
of datasets are generated and made publicly available, for Predicate 1 example, as in the Linked Open Data. Due to the graph Subject 1 Object 1 nature of these datasets, many graph-based algorithms for G1 RDF datasets have been developed such as query process- Predicate 2 ing [8, 16], graph matching [4, 6], semantic associations [2, Predicate 1 Object 2 arXiv:1606.00480v1 [cs.DB] 1 Jun 2016 3], path computing [10, 14], and centrality [5, 17]. These G2 existing graph algorithms employ the abstract graph model called Node-Labeled Arc-Node (NLAN), currently recom- mended by W3C [7]. Figure 1: Subgraphs G1 and G2 are disconnected in In this NLAN graph, the subject and object of a triple the Node-LabeledArc-Node diagram (NLAN). are mapped to two nodes of a graph, and the predicate is Schema triples. One common scenario where a predi- cate becomes the subject or object of another triple is through the RDF schema triples. A schema triple may describe a property using many other different properties such as Permission to make digital or hard copies of all or part of this work for rdfs:subPropertyOf, rdf:type, rdfs:domain, or rdfs:range. personal or classroom use is granted without fee provided that copies are For example, the predicates hasFamilyName and hasGivenName not made or distributed for profit or commercial advantage and that copies are asserted as sub properties of the predicate rdfs:label, bear this notice and the full citation on the first page. To copy otherwise, to and hence become the subjects in these triples: republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. hasFamilyName rdfs:subPropertyOf rdfs:label . Copyright 20XX ACM X-XXXXX-XX-X/XX/XX ...$15.00. hasGivenName rdfs:subPropertyOf rdfs:label . The Yago2S-SP dataset [12] contains 938 properties like these. We can also find schema triples in many other com- Table 1: Example triples using singleton properties mon datasets, such as DBPedia, etc. to represent the facts about the American politician Singleton Property triples. Another scenario comes Bill Clinton and his successors from the RDF datasets created by a recent work [12] where No Subject Predicate Object singleton properties are used to describe RDF statements. T1 BillClinton holdsPos#1 U.S.President A simple approach to make assertions about an RDF state- T2 holdsPos#1 singletonPropOf holdsPos ment is through reification. Reification creates an instance T3 holdsPos#1 hasSuccessor GeorgeWBush of class rdf:Statement to represent a statement and asserts T4 BillClinton holdsPos#2 ArkansasGovernor metadata about this statement through this instance. Al- T5 holdsPos#2 singletonPropOf holdsPos though this approach is intuitive, reifying one statement re- T6 holdsPos#2 hasSuccessor FrankWhite quires at least four triples, making it less attractive. Instead of reifying a statement, Nguyen et al. [12] propose a new ap- proach called the singleton property. The singleton property approach was shown to be more efficient than reification. Instead, we propose to use the singleton property ap- Each singleton property uniquely representing a statement proach to represent the facts as shown in Table 1. Indeed, can be annotated with different kinds of metadata describ- our example is motivated by the facts from the Yago2S-SP ing that statement such as provenance, time, and location. dataset, which will be used in the evaluation described in Therefore, the singleton properties also become subjects and Section 5. We create our example in the way that makes objects of other meta triples. Meta triples are the triples it intuitive to readers, and for readability, we eliminate all that describe other triples through the use of singleton prop- the URI prefixes. For each political position of Bill Clinton, erties. While singleton properties enrich RDF datasets with we create a singleton property to capture the information different kinds of metadata for the triples they represent, related to that context. Particularly, we create two single- the metadata subgraph is disconnected from the triple they ton properties holdsPos#1 and holdsPos#2. Then we attach describe. Traversing the NLAN graphs created by this ap- the predicate hasSuccessor to the two singleton properties proach also becomes a challenge for the RDF graph model. to represent the 3-ary relationship among the politician, the Being unable to traverse among triples in the scenarios de- position and the successor. scribed above due to the limited connectivity of the NLAN We will analyze how such a set of triples can be repre- graph is a critical issue, because it limits the capability of sented in the form of a graph by two existing approaches: answering reachability and shortest path queries on RDF the NLAN diagram [7] and the bipartite (BI) model [9]. datasets. A reachability query verifies if a path exists be- A triple may be represented as a NLAN diagram as ex- tween any two nodes in a graph. A shortest path query re- plained in the W3C RDF 1.1 Concepts and Syntax [7]. The turns the shortest distance in terms of the number of edges set of triples in Table 1 are represented as a NLAN diagram between any two nodes in a graph if the path exists. In in Fig. 2. We observe two problems with this modeling the NLAN graph model, both query types require travers- when dealing with such a set of triples. ing the graph from subject node to object node of a triple. Traversing from one node in subgraph G1 to another node U.S. holdsPos#1 Bill holdsPos#2 Arkansas in subgraph G2 in this way will not find any result because President Clinton Governor they do not share any node in common. The only resource these two triples share is the predicate of the first triple, holdsPos#1 holdsPos#2 which is also the subject of the second triple. Therefore, the singletonPropOf singletonPropOf ability to connect the triples represented in G1 with the cor- responding triples in G2 (schema triples or meta triples rep- hasSuccessor hasSuccessor resented through the singleton property) is desirable. Such holdsPos connectivity strengthens the robustness of the graph model. GeorgeW.Bush FrankWhite Next, we will take one sample set of RDF triples repre- sented by the singleton property approach as our motivat- Figure 2: Node-LabeledArc-Node diagram (NLAN) ing example. We will analyze the limitations of the existing for the example in Table 1. work through this motivating example and demonstrate our approach to address the challenges. First and more importantly, the NLAN resulting graph is not precisely a mathematical graph. In Fig. 2, the RDF 1.1 Motivating Example term holdsPos#1 is mapped to the predicate arc of triple T1, Consider the set of triples in Table 1, which will be used and to the subject node in T2, T3. Mapping the same RDF as a running example throughout the paper. In this exam- term holdsPos#1 to more than one different mathematical ple, we represent the facts that Bill Clinton is succeeded by object makes the object being referred to ambiguous. No George W. Bush as the president of the United States, and mapping function satisfies this because a function must map is succeeded by Frank White as the Governor of Arkansas. any source object to only one target. For this reason, this If we represent the facts as follows, these relationships are NLAN model cannot be compatible with the formal seman- unclear and incomplete because the graph does not repre- tics that is defined by several mapping functions. sent the political position context in which Bill Clinton is The second serious problem is that the disconnectedness succeeded. between subgraphs limits the possibility to traverse between BillClinton hasSuccessor GeorgeWBush . the triples that are connected through the predicates. In the BillClinton hasSuccessor FrankWhite . example at hand, the NLAN resulting graph is comprised of two disconnected subgraphs G1 (formed by two triples T1 GeorgeW.Bush hasSuccessor FrankWhite and T4) and G2 (formed by four triples T2, T3, T5 and T6). Although the two predicates holdsPos#1 and holdsPos#2 are shared between the triples from two subgraphs, no con- holdsPos#1 BillClinton holdsPos#2 nectivity between two subgraphs can be found in Fig. 2. As a consequence, although Bill Clinton is succeeded by George W. Bush, no path can be found from Bill Clinton to George U.S.President singletonPropOf Arkansas Governor W. Bush. From this perspective, the bipartite model pro- vides better graph connectivity than the NLAN model. The bipartite (BI) model proposed by Hayes [9] does not holdsPos encounter these problems of the NLAN model. It represents all subjects, predicates, and objects as nodes. It creates an auxiliary node and links this node to the subject, the Figure 4: Labeled Directed Multigraph with Triple predicate, and the object of this triple via three additional Nodes (LDM-3N) for the example in Table 1. arcs. However, the cost incurred in this approach with one extra node and three extra arcs for every triple is too high. model. We will formally define resource path and triple path 1.2 Our approach in Section 2.3. In this paper, we propose a new formal graph model which In terms of practical impact, we show that compared to represents RDF triples in a Labeled Directed Multigraph the NLAN model, our LDM-3N model does not introduce with Triple Nodes (LDM-3N). Similar to the bipartite model, cost to the system in terms of space even though it con- all subjects, predicates, and objects are mapped to nodes ceptually adds two edges to connect (subject-predicate) and of a LDM-3N graph. This model, however, differs from (predicate-object) in each triple. These initial and terminal the bipartite model in that it adds one pair of directed edges that allow us to disambiguate the nodes forming each edges (subject-predicate, predicate-object) to directly con- triple in the abstract model are unnecessary in the physical nect three nodes of the same triple. Furthermore, this ap- model. It is because the triples are already stored sepa- proach differs from the NLAN diagram in that the predicates rately in the indices. We use the same indices for traversing are mapped to nodes instead of labeled edges. in both models. In terms of execution time, traversing the As traversing the NLAN graph starts from a node and NLAN graph is supposed to be faster than traversing the explores its adjacent nodes, predicates as arcs never get ex- LDM-3N graph because exploration of the predicate nodes plored. In the LDM-3N model, predicates are mapped to is bypassed. However, that extra exploration in the LDM- nodes in the graph. As they are adjacent to the subjects, 3N graph allows for additional paths that are impossible to they are explored after the subject. The objects now be- find in the NLAN graph. We will report the extra time taken come adjacent to the predicates and get explored after the in traversing the LDM-3N graph through the shortest path predicates. Traversing the LDM-3N graph this way starting algoritm in Section 5. We summarize our contributions in this paper next: from the node Subject 1 will reach the node Object 2 after 3 hops, as shown in Fig. 3. • We develop a formal graph model (LDM-3N) for RDF. For traversing the new graph, we define two types of Subject 1 Predicate 1 Object 1 paths, resource path and triple path with a new traver- sal algorithm (in Section 2).
• We rewrite the model-theoretic formal semantics for Predicate 2 Object 2 RDF with three levels of interpretation (simple, RDF, and RDFS) based on the LDM-3N model (Section 3) and demonstrate the graph entailments using RDFS Figure 3: Subgraphs G1 and G2 from Fig. 1 are deduction rules (Section 3.4), which the NLAN and BI now connected in a Labeled Directed Multigraph models do not support. with Triple Nodes (LDM-3N). • We implement a new engine called GraphKE and ex- The LDM-3N graph representation of the example from tend the RDF-3X to support the LDM-3N graph model Table 1 is illustrated in Fig. 4. For the set of six triples in (Section 4). We implement and evaluate the empiri- Table 1, we map all subjects, predicates, and objects to the cal performance of the reachability and shortest path set of ten nodes. We link every three nodes representing the queries on both engines (Section 5) to demonstrate same triple by one pair of directed edges. For instance, the that our formal model is technically viable for both three nodes {BillClinton, holdsPos#1, U.S.President} are new and existing systems. I T connected by the pair of edges (e1, e1 ) to form the triple T1. We discuss related work and future work in Section 6 and Compared to the NLAN model, the LDM-3N graph model Section 7, respectively, and finally conclude the paper with provides better connectivity and makes it possible to find Section 8. the answers for the reachability and shortest path queries. While the node GeorgeWBush is unreachable from the node BillClinton in the NLAN model, it is not only reachable 2. RDF FORMAL GRAPH MODEL but also neatly presented as the resource path (BillClinton, We demonstrate how to represent any RDF triples in the holdsPos#1, hasSuccessor, GeorgeWBush) in the LDM-3N LDM-3N model. We start with the step-by-step modeling of simple facts in Section 2.1, then formalize the LDM-3N 2.2 Formalizing RDF Graph model in Section 2.2. Formally, any RDF dataset is a set of RDF triples. Let T = {t0, t1, ..., tn} be a set of triples on the vocabulary V, 2.1 Representing RDF Graph and let GRDF be the labeled directed multigraph with triple Referring back to the motivating example in Section 1.1, nodes (LDM-3N) of T . We will first create the graph GRDF the NLAN diagram represents the triple from T and then regenerate the set of original triples T from T3 : holdsPos#1 hasSuccessor GeorgeWBush GRDF . by mapping the predicate hasSuccessor to an arc which connects the two nodes holdsPos#1 and GeorgeWBush. We Proposition 2.1. (Forward transformation). Any set of argue that, just as subjects and objects are mapped to nodes, RDF triples can be transformed into a labeled directed multi- predicates should also be mapped to nodes since they are all graph with triple nodes GRDF . instances of the same class rdf:Resource. The next ques- Proof. Let V be the set of RDF terms in T . Let N and tion is “How do we link those three nodes to form a triple?” E be the set of nodes and the set of directed edges in the Existing models use either labeled edges (NLAN) or one graph GRDF , respectively. auxiliary node plus three labeled edges (BI). We will ap- The bijective function µ : V → N maps an RDF term in proach the question from another perspective. Although the V to a node in N. Let ti be a triple in T , ti = (si, pi, oi) ∈ T predicate is mapped to a node, this predicate node differs with 0 ≤ i ≤ n. Let Ni ⊂ N such that Ni = {nsi, npi, noi|nsi = from subject or object nodes because of its linking role. We Sn µ(si), npi = µ(pi), noi = µ(oi)}, then N = i=0 Ni. need to represent this link without causing a discrepancy The function : E → N × N is defined to map every edge like the one found in the NLAN model. I T I in E to an ordered-pair of nodes. Let ei , ei ∈ E : (ei ) = Here we have three nodes to form a triple. However, ev- T (nsi, npi) and (ei ) = (npi, noi). ery edge in the graph can only connect two nodes. Instead The bijective function τ : E → E maps an initial edge of adopting a complicated solution like using a hyperedge, I T to a terminal edge of the same triple. Then τ(ei ) = ei . how about using two regular edges? Obviously, two directed Let Ei ⊂ E be the set of two edges representing ti, Ei = edges can connect the three nodes into two pairs as illus- I T Sn {ei , ei }, and E = Ei. trated in Fig. 5 (a). However, if we add another triple, say i=0 Therefore, Gi = (Ni,Ei, , τ, µ) is the labeled directed T6: holdsPos#2 hasSuccessor FrankWhite, then an ambi- graph with triple nodes of the triple ti. guity arises in figuring out which two edges form the triple Sn Sn Finally, with N = Ni and E = Ei, the graph as in Fig. 5 (b). The next question is “How do we tie these i=0 i=0 GRDF = (N, E, , τ, µ) is a labeled directed multigraph with two edges together to represent that they are part of the triple nodes for all of the triples in T . same triple?” Proposition 2.2. (Backward transformation). Given the graph G (N, E, , τ, µ) transformed by Proposition 2.1, a holdsPos#1 hasSuccessor GeorgeW.Bush RDF set of RDF triples can be derived from GRDF . (a) I Proof. Let ei be any edge in E with 0 ≤ i ≤ n, the I T I holdsPos#1 GeorgeW.Bush corresponding terminal edge of ei in E is ei = τ(ei ). From I T this pair of edges ei and ei , we find the nodes connected by hasSuccessor the two edges by using the function. Let nsi, npi, noi ∈ N I T FrankWhite such that (nsi, npi) = (ei ), (npi, noi) = (ei ). holdsPos#2 (b) Let µ−1 be the reverse function of µ, then µ−1 : N → V returns the RDF term mapped to a graph node. Let −1 −1 si, pi, oi ∈ V such that si = µ (nsi), pi = µ (npi) and holdsPos#1 GeorgeW.Bush −1 oi = µ (noi). The three nodes form the original triple ti = (si, pi, oi). The set T of all RDF triples ti derived from hasSuccessor GRDF is as follows: T = {ti|∀i : ti = (si, pi, oi)}. holdsPos#2 FrankWhite (c) Proposition 2.3. The size of the graph GRDF on the set of triples T is 2|T |. Figure 5: A step by step construction of a LDM-3N graph for the two triples T3 and T6. Proof. The size of the graph GRDF is the number of edges needed to form all of the triples in T . For each triple Since we have two edges, we call the first edge between t = (s, p, o) ∈ T , we need 2 edges to form the triple. There- subject and predicate the initial edge, and we call the second fore, as the number of triples in T is |T |, we need 2|T | edge between predicate and object the terminal edge. We edges. create a mapping that maps every initial edge to its terminal edge. The pair of edges for a given triple i is denoted as 2.3 Traversing RDF Graph I T I T (ei , ei ), where ei is the initial edge and ei is the terminal A triple path is defined as a sequence of triple subgraphs I T I edge. It follows that (e3, e3 ) corresponds to T3 and (e6, where two adjacent triple subgraphs share one common node, T e6 ) to T6. The complete LDM-3N graph for T3 and T6 is that the subject node of the later triple subgraph is also the illustrated in Fig. 5 (c). object or predicate node of the previous triple subgraph. In Next, we will formalize the LDM-3N model for any set of the given LDM-3N graph GRDF , the triple path tp = (ti, RDF triples. ti+1, ..., tj ) where tk = (nsk, npk, nok) with i ≤ k ≤ j and tk+1 = (ns(k+1), np(k+1), no(k+1)) satisfy one of following • IL : V → IN, a labeling function, mapping literal conditions: (1) ns(k+1) = npk or (2) ns(k+1) = nok. values from V to IN. For example, (T , T ) from Table 1 is a triple path because 1 3 Let ·I be the interpretation function that maps all the URIs two triples share the node holdsPos#1. and literals in V to the set of nodes in IN. A ground triple A resource path is defined as a sequence of nodes such I (s p o .) is assigned true if all s, p, o ∈ V and ∃e1, e2 ∈ IE that (1) every two adjacent nodes are connected by an edge, I I I I (2) every three nodes connected by a pair of initial and ter- : IE (e1) = (s , p ), IE (e2) = (p , o ), and IT (e1) = e2. minal edges should form a triple. In the given LDM-3N The simple interpretation in [11] contains an extension function that maps a property to a set of resource pairs. graph GRDF , the resource path (n0, n1, ..., nk) satisfies the following conditions: In this simple interpretation, we define a set of new func- tions, IE and IT , for representing nodes and edges of the 1. (ni, ni+1) = (ei) ∈ E for all 0 ≤ i ≤ k − 1 underlying labeled directed multigraphs with triple nodes. To make this interpretation compatible with the existing T I 2. for every edge ei = (ni, ni+1) ∈ E and ∃ei ∈ E such ones, we incorporate the existing criteria for generic prop- I T that τ(ei ) = ei , erties and singleton properties into this interpretation. The I T I simple interpretation I also satisfies the following criteria: if ∃ei−1, ei−1 ∈ E such that ei−1 = (ni−1, ni) ∈ E and τ(eI ) = eT , then τ(eI ) = eT , or eT = eT . i−1 i−1 i−1 i i−1 i • IP, a set of generic property nodes, which is also a The second condition excludes the case where the two ini- subset of IN, IP ⊂ IN. tial and terminal edges from different triples are adjacent • IEXT , a function assigning to each property node a set to each other in a path. From the motivating example, the of pairs from IN. path (BillClinton, holdsPos#1, hasSuccessor, GeorgeW- IN ×IN IEXT : IP → 2 where IEXT (p) is the exten- Bush) satisfies these conditions and hence, it is a resource sion of generic property p. Particularly, IEXT (p) = path. We will describe the implementation of the shortest {(s, o)|∃e1, e2 ∈ IE : IE (e1) = (s, p), IE (e2) = (p, o), resource path in Section 4.1 and its evaluation in Section 5. and IT (e1) = e2}. 3. FORMAL SEMANTICS • IPs, called the set of singleton property nodes of I, as a subset of IN, This section explains how a labeled directed multigraph with triple nodes, or LDM-3N graph, can be exploited as • IS EXT (ps), a function mapping a singleton property an underlying model for RDF(S) formal semantics. Similar to a pair of resources. IS EXT : IPs → IN × IN . Par- to the model-theoretic semantics described in [11, 12], we ticularly, IS EXT (ps) = (s, o) such that ∃e1, e2 ∈ IE also represent three levels of interpretation: simple, RDF, : IE (e1) = (s, ps), IE (e2) = (ps, o), and IT (e1) = e2. and RDFS. However, we define new functions for modeling the underlying labeled directed multigraphs and use them 3.2 RDF interpretation to redefine class/property extension functions. The RDF interpretation of a vocabulary V is a simple interpretation I of the vocabulary V ∪ VRDF that satisfies 3.1 Simple interpretation the following criteria:
The simple interpretation I of the vocabulary V based I • p ∈ IP if ∃e1, e2 ∈ IE : IE (e1) = (p, rdf:type ), on the LDM model consists of: I I IE (e2) = (rdf:type , rdf:Property ), • IN, a non-empty set of nodes in I, and IT (e1) = e2. A generic property is an instance of the class rdf:Property. • IE, a set of directed edges in I, I • ps ∈ IPs if ∃e1, e2 ∈ IE : IE (e1) = (ps, rdf:type ), I I •IE : IE → IN × IN, mapping each edge to an ordered IE (e2) = (rdf:type , rdf:SingletonProperty ), pair of nodes. This function may be many-to-one be- and IT (e1) = e2. Every singleton property is an in- cause a multigraph allows multiple edges between two stance of the class rdf:SingletonProperty. nodes. • ps ∈ IPs if ∃e1, e2 ∈ IE, p ∈ IP : IE (e1) = (ps, I •IT : IE → IE, mapping an initial edge to a terminal rdf:singletonPropertyOf ), I edge of the same triple. This is a bijective function, IE (e2) = (rdf:singletonPropertyOf , p), and IT (e1) = −1 e2. A singleton property is connected to a generic and that the reverse function IT : IE → IE maps a terminal edge to its initial edge. property via the rdf:singletonPropertyOf.
• EL, a set of distinct labels to be assigned to the edges • if ps ∈ IPs then ∃!(e1, e2): IE (e1) = (s, ps), IE (e2) = in IE, (ps, o), IT (e1) = e2, with s, o ∈ IN and e1, e2 ∈ IE. This ensures only one occurrence of a singleton prop- • IEL : EL → IE, a labeling function, mapping labels erty as a predicate of a triple. from EL into the set IE of edges. All labeling functions • if “s”ˆˆrdf:XMLLiteral is in V and s is a well-typed are also bijective. XML Literal, then IL(“s”ˆˆrdf:XMLLiteral) ∈ LV and I • IS : V → IN, a labeling function, mapping URIs from ∃e1, e2 ∈ IE: IE (e1) = (s, rdf:type ), I I V into IN, IE (e2) = (rdf:type , rdf:XMLLiteral ), and IT (e1) = e2. All well-typed XML literals are in LV and they • LV , a set of literal values in IN, also are instances of class rdf:XMLLiteral. • if “s”ˆˆrdf:XMLLiteral is in V and s is an ill-typed • rdfs5: if (u, rdfs:subPropertyOf, v) and XML Literal, then (v, rdfs:subPropertyOf, x) IL(“s”ˆˆrdf:XMLLiteral) ∈/ LV and 6 ∃e1, e2 ∈ IE: then (u, rdfs:subPropertyOf, x). This rule states that I IE (e1) = (s, rdf:type ), the rdfs:subPropertyOf property is transitive. I I IE (e2) = (rdf:type , rdf:XMLLiteral ), and IT (e1) = if ∃e1, e2, e3, e4 ∈ IE in G : IT (e1) = e2, IT (e3) = e4, I e2. IE (e1) = (u, rdfs:subPropertyOf ), I LV does not contain ill-typed XML literal values. IE (e2) = (rdfs:subPropertyOf , v), I IE (e3) = (v, rdfs:subPropertyOf ), I 3.3 RDFS interpretation IE (e4) = (rdfs:subPropertyOf , x),
In the RDFS interpretation, we define the function ICEXT : then ∃e5, e6 ∈/ IE : IT (e5) = e6, IN I IN → 2 where ICEXT (y) is called the class extension of IE (e5) = (u, rdfs:subPropertyOf ), I y. Although some critiera can alternatively be expressed us- IE (e6) = (rdfs:subPropertyOf , x). The graph G en- 0 0 ing these two functions ICEXT and IEXT , here we explicitly tails G : IE = IE ∪ {e5, e6}. express these criteria as graph constraints to demonstrate their graph nature. We show only some of the more impor- • rdfs7: if (a, rdfs:subPropertyOf, b) and (u, a, y) then tant conditions here due to space constraints. (u, b, y). RDFS interpretation of a vocabulary V is an RDF in- All resources interlinked by a property are interlinked by its super property. terpretation I of the vocabulary V ∪ VRDF ∪ VRDF S that satisfies the following criteria: if ∃e1, e2, e3, e4 ∈ IE in G : IT (e1) = e2, I IE (e1) = (a, rdfs:subPropertyOf ), I IN IE (e2) = (rdfs:subPropertyOf , b), • ICEXT : IP → 2 , a function assigning to each class IE (e3) = (u, a), IE (e4) = (a, y) and IT (e3) = e4, then a set of nodes from IN. ICEXT (c) is called the class ∃e5, e6 ∈/ IE : IT (e5) = e6, IE (e5) = (u, b), IE (e6) = extension of class c. Particularly, ICEXT (c) = {s|s ∈ 0 0 I (b, y), and the graph G entails G : IE = IE∪{e5, e6}. IN, ∃e1, e2 ∈ IE : IE (e1) = (s, rdf:type ), IE (e2) = (rdf:typeI , c), and I (e ) = e }. T 1 2 • rdfs9: if (v, rdf:type, u) and (u, rdfs:subClassOf, x) then (v, rdf:type, x). • if ∃e , e , e , e ∈ IE: I (e ) = e , I (e ) = e , 1 2 3 4 T 1 2 T 3 4 A resource as a member of one class is also a member I (e ) = (x, rdfs:domainI ), I (e ) = (rdfs:domainI , y), E 1 E 2 of its superclass. I (e ) = (u, x), I (e ) = (x, v), E 3 E 4 if ∃e , e , e , e ∈ IE in G : I (e ) = e , I (e ) = e , then ∃e , e ∈ IE: I (e ) = e , 1 2 3 4 T 1 2 T 3 4 5 6 T 5 6 I (e ) = (u, rdfs:subClassOfI ), I (e ) = (u, rdf:typeI ), I (e ) = (rdf:typeI , y). E 1 E 5 E 6 I (e ) = (rdfs:subClassOfI , v), If one class is a domain of a property, then the class E 2 I (e ) = (v, rdfs:subClassOfI ), extension includes all subjects in the same triples with E 3 I (e ) = (rdfs:subClassOfI , x) the property. E 4 then ∃e5, e6 ∈/ IE: IT (e5) = e6, I IE (e5) = (u, rdfs:subClassOf ), • if ∃e1, e2, e3, e4 ∈ IE: IT (e1) = e2, IT (e3) = e4, I I I IE (e6) = (rdfs:subClassOf , x), IE (e1) = (x, rdfs:range ), IE (e2) = (rdfs:range , y), 0 0 and the graph G entails G : IE = IE ∪ {e5, e6}. IE (e3) = (u, x), IE (e4) = (x, v), then ∃e5, e6 ∈ IE: IT (e5) = e6, I I IE (e5) = (v, rdf:type ), IE (e6) = (rdf:type , y). 4. IMPLEMENTATION The class range of a property includes all objects in Here we describe the implementation of the two engines in the same triples with the property. order to demonstrate the feasibility of practical application of our graph model. Since we are proposing a new graph • if ∃e1, e2 ∈ IE: IT (e1) = e2, model with a new way of traversing the graph, our aim is I IE (e1) = (x, rdfs:subPropertyOf ), not to optimize the shortest path algorithm because there I and IE (e2) = (rdfs:subPropertyOf , y), then x, y ∈IP is a multitude of existing solutions for this purpose, such as and IEXT (x) ⊆ IEXT (y). The extension of a property [1]. Instead, our aim is to implement the two traversal algo- is a subset of the extension of its super property. rithms from two graph models on the same system so that we can fairly evaluate the two graph models. We also ex- • if ∃e1, e2 ∈ IE: IT (e1) = e2, plain how existing systems can adopt this model by making I IE (e1) = (x, rdfs:subClassOf ), changes to their traversal algorithm. I and IE (e2) = (rdfs:subClassOf , y), then ICEXT (x) ⊆ ICEXT (y). The extension of a class 4.1 Shortest Resource Path is a subset of its super class extension. We implemented Dijkstra’s algorithm on both NLAN and LDM-3N graph models to find the shortest resource path 3.4 RDFS Entailments as defined in Section 2.3. One difference between the two Here we present how the LDM-3N graph-based semantics algorithms is that traversing the LDM-3N graph visits the described in Section 3 can derive other graphs using RDFS predicates and explores their neighbor nodes while travers- rules. We demonstrate the entailments by using three rules: ing the NLAN does not. rdfs5, rdfs7, and rdfs9 from [11]. We choose these rules While traversing the graph, we use an in-memory priority since they are commonly used for reasoning with class and queue to store all the nodes to be visited and their distance property hierarchy. Other rules can also be applied in the to the source node. We also use a hash table visited to same way. Let G and G0 be the two LDM-3N graphs. store the set of visited nodes. For each visited node, we Algorithm 1 Dijkstra’s algorithm on the LDM-3N graph the Dijkstra’s algorithm 1 for the LDM-3N model in that 1: procedure LDM-3N Dijkstra(source, target) it does not visit the predicate as explained from lines 10-13 2: distance ← 0 of Algorithm 1. Instead, it only traverses from the curid 3: pqueue ← source to the obj and the distance dis + 1. We can observe that 4: while (curid ← pqueue pop(pqueue, dis)) 6= 0 do the distance associated with the obj in this model is shorter 5: if (curid = target) then than the one associated with the obj in the LDM-3N model. 6: return dis That is because the NLAN model traverses from sub to obj 7: else in one hop, while the LDM-3N model traverses from sub to 8: get all the pairs
GraphKE RDF-3X
Pairs 1T Avg1 5T Avg5 1T Avg1 HR 579 2688 4.64 1826 3.15 10627 18.35 SS 94 353 3.76 237 2.52 888 9.45 CS 164 770 4.695 519 3.17 2388 14.56 All 837 3811 4.55 2582 3.08 13903 16.61
lustrates the use of singleton properties in representing the Figure 6: Average time taken per shortest distance n-ary relationship between the politician, the position, and within CS. the successor. The politicians are connected to their po- sition and successor of that position by a set of singleton properties represented by the rounded squares.
Andrew Start Jacob End Card Lew
id_9eg59e_ 19y_n8hn0n id_1pz5ld6_x7 id_lc8pv5_x 7_1umh2ta 77_900yd3
Joshua id_900yd3_ White House id_1tyjwzl_1 William Bolten 19y_n8hn0n Chief of Staff 9y_n8hn0n M.Daley
id_1623d6a _x77_1tyjwz l id_lu06jg_1 id_1txrjdd_1 9y_n8hn0n id_1ofi5go_ 9y_n8hn0n x77_lu06jg
id_1huff3f_x Figure 7: Average time taken per shortest distance 77_1txrjdd within SS. Rahm Emanuel Pete Rouse
Figure 9: The shortest resource path and its respec- tive triple path from Andrew Card to Jacob Lew in Yago2S-SP. The resource path includes all the edges along the dashed line in the periphery of the figure, and the triple path includes all the triples where three nodes from a triple form a straight line. Traversing the resource path from start to end will find a sequence of five politicians (Andrew Card, Joshua Bolten, Rahm Emanuel, Pete Rouse, William M. Daley, Jacob Lew) in consecutive terms.
This graph results from running Dijkstra’s algorithm on Figure 8: Average time taken per shortest distance the LDM-3N graph model to find the shortest path from within HR. Andrew Card to Jacob Lew. The shortest distance is 15. Starting from Andrew Card, we follow the outer edges along the dashed line in 15 hops, we will find a sequence of politi- When we run the GraphKE with 5 threads, it finishes pro- cians Joshua Bolten, Rahm Emanuel, Pete Rouse, William cessing all input pairs in 2,582 seconds (44 minutes), com- M. Daley and then we will reach Jacob Lew. For every two pared to 3,811 seconds (64 minutes) in 1 thread. We observe politicians in the sequence, the former is succeeded by the that the time taken for each input pair increases in 5 threads, latter. but the overall time taken for all input pairs reduces 30% compared to 1 thread. 5.4 Discussion Findings. While investigating the results from our algo- We defined a new formal graph model for RDF that does rithm, we found many paths that are interesting to us. We not suffer from limitations of current alternative. To demon- draw one sample path in Figure 9. This LDM-3N graph il- strate viability for practical implementation and use, we showed two implementations - GraphKE as a ground up applied to these well-formalized graphs, while also leveraging implementation and adaptation of RDF-3X, for executing the RDF(S) semantics. We are also interested in studying reachability and shortest path queries. Based on the exper- the possibility of performing graph entailments directly on imental results in the Yago2S-SP dataset with 260 million the real graph structures. triples, we believe that the LDM-3N graph model can be effectively implemented in a practical system, either new or 8. CONCLUSION existing. We have presented a new formal graph model for RDF with examples clearly demonstrated. Our LDM-3N model 6. RELATED WORK allows us to represent any set of RDF triples in a formal Directly related to the formal graph model for RDF are graph. It also allows us to develop an underlying graph the NLAN diagram, which is currently used in the W3C model for the model-theoretic semantics of RDF(S) and the Recommendation documents, and the bipartite model. As RDFS entailments. To the best of our knowledge, this is the explained in prior sections, our approach differs from the first formal graph model that is compatible with the RDF NLAN approach in that predicates are mapped to nodes formal semantics. We have implemented the LDM-3N graph instead of arcs, and the nodes within the same triple are model in the new GraphKE engine and in the existing triple interconnected by a pair of initial and terminal edges. Our store RDF-3X. 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