Extending Shape Expressions for different types of knowledge graphs JoseEmilioLabra-Gayo WESO lab University of Oviedo Extending Shape Expressions for different types of knowledge graphs 1613-0073 881FEA8A36075F5FB04D0EA8524F9BDC GROBID - A machine learning software for extracting information from scholarly documents

Shape Expressions (ShEx) has been proposed as a concise and human-readable language to describe and validate RDF. Inspired by regular expressions, it offers an expressive formalism to describe graph structures based on regular bag expressions. Although plain RDF is one of the common data formats employed to represent knowledge graphs, there have been several proposals to either extend RDF with the so-called RDF-Star or RDF 1.2, or to employ other formalisms like property graphs. In this paper we present an overview and comparison of those approaches and propose three possible extensions of ShEx: ShEx-Star which can be used to validate RDF-Star, ShEx-N: that can be used when nodes also act as properties in RDF and PShEx, which can be used to describe property graphs. We present some examples and a semantics of each extension.
Introduction

Although Knowledge graphs have been successfully adopted by the industry, an important aspect of their practical application is the quality of the data that they contain. In order to increase their quality, it is necessary to have some mechanisms that can check the conformance of the data to some kind of schema. Knowledge graphs are usually considered schema-less, because there is no mandatory schema, however, in most cases the data curators have an implicit schema in mind. Having the possibility to materialize that implicit schema into a machine processable form that can be automatically verified can mitigate the risk of non-conformant data.

In practice, there are several types of technologies that can be used for knowledge graphs [1]: Directed edge-labeled graphs, whose main representative are RDF graphs, and Property graphs, which allow property-value pairs and labels to be associated with nodes and edges.

In the case of RDF, two main technologies have been proposed for validation: ShEx [2] and SHACL [3], which are based on the notion of a shape as a description of the topology of some specific kind of nodes. In this way, it is possible to define a schema as a set of shapes which describe the expected properties of some nodes, their expected cardinalities and the kind of nodes. These schemas, can be used to validate RDF data and check if it conforms to those shapes. We employ ShEx in the paper because it can be seen as a description language for RDF acting as grammar where a ShEx schema represents the set of all the RDF graphs that conform to it.

Envelope labra@uniovi.es (J. E. Labra-Gayo) GLOBE http://labra.weso.es (J. E. Labra-Gayo) Orcid 0000-0001-8907-5348 (J. E. Labra- Gayo) There has been a proposal for an extension of RDF called RDF-Star [4] and a Working Group is currently working taking that proposal as input to define what is currently called RDF 1.2 [5]. The proposal extends RDF with the possibility of having triples as subjects or objects in statements. The relationship between that extension and property graphs has already been studied [6].

Although ShEx was defined to describe and validate RDF, we consider that it can be extended to describe and validate RDF-Star (or RDF 1.2), so in this paper we explore a possible extension of the language in that direction. Apart from that, the shapes in ShEx are usually centered on describing nodes that act as either subjects or objects, but in RDF, it is also possible to add statements whose subjects or objects are also the predicates of other statements. We also explore a possible extension of ShEx where it is possible to define shapes about nodes that act as properties. Finally, although there have been several proposals to define schemas for property graphs, we consider that the grammar-based approach of ShEx can also be helpful, so we also explore what it would look like to extend ShEx for describing and validating property graphs.

The main contributions of this paper are to present three extensions of ShEx: ShEx-Star for RDF-Star (section 3), ShEx-N for describing nodes as properties (section 4) and PShEx for property graphs (section 5), with their abstract syntax and semantic definitions.

RDF and ShEx

Definition 1 (RDF triple and RDF Graph). Given a set of IRIs I , a set of blank nodes B and a set of literals Lit, an RDF triple is a tuple (𝑠, 𝑝, π‘œ) where 𝑠 ∈ I βˆͺ B is called the subject, 𝑝 ∈ I is called the predicate and π‘œ ∈ I βˆͺ B βˆͺ Lit is called the object. An RDF graph G is a set of RDF triples.

There are several syntaxes for RDF graphs like Turtle, N3, RDF/XML, etc. In this document, we will use Turtle.

Example 1 (Example of an RDF graph in Turtle). The following snippet contains a simple RDF graph with two nodes :a and :b. prefix : <http://example.org/> :a :name "Alice" ;

:knows :b . :b :firstname "Robert", "Julius" ;

:lastname "Smith" .

The neighbors of a node 𝑛 ∈ V in an RDF graph G are defined as π‘›π‘’π‘–π‘”β„Žπ‘ (𝑛,

G) = {(𝑛, 𝑝, 𝑦) | (𝑛, 𝑝, 𝑦) ∈ G} βˆͺ {(π‘₯, 𝑝, 𝑛) | (π‘₯, 𝑝, 𝑛) ∈ G} βˆͺ {(π‘₯, 𝑛, 𝑦) | (π‘₯, 𝑛, 𝑦) ∈ G}.

Shape Expressions (ShEx) were proposed as such a language in 2014 [2]. It was designed as a high-level and concise domain-specific language to describe RDF. The syntax of ShEx is inspired by Turtle and SPARQL, while the semantics was inspired by RelaxNG and XML Schema. In this section we describe a simplified abstract syntax of ShEx following [7] Intuitively, shape expressions define conditions about nodes while triple expressions define conditions about the neighborhood of nodes, and shapes qualify those neighborhoods by allowing (open shapes) or disallowing (closed shapes) triples with other predicates. We omit negation and disjunction operator to simplify the interactions between negation and recursion, which led to a stratified negation requirement in ShEx.

The restrictions imposed on shape expressions schemas in [8] also apply here. Namely, in a schema (L, 𝛿, S)

β€’ The shape label references used by the definition function 𝛿 are themselves defined, i.e. if @l appears in some shape definition, then l belongs to L; β€’ No definition 𝛿(l) uses a reference @l to itself, neither directly nor transitively, except while traversing a shape. For instance, 𝛿(l) = @l AND 𝑠𝑒 is forbidden, but 𝛿(l) = { 𝑝 βˆ’ β†’ @l} is allowed.

Example 2 (Example of ShEx schema). A ShEx schema that describes the RDF graph presented in example 1 can be defined as:

L = {π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘› } 𝛿(π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘›) = { ( π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘” | 𝑓 π‘–π‘ π‘‘π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘”*; π‘™π‘Žπ‘ π‘‘π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘”); π‘˜π‘›π‘œπ‘€π‘  βˆ’ βˆ’βˆ’βˆ’βˆ’ β†’ @π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘› * }

ShEx has several concrete syntaxes like a compact syntax (ShExC) and an RDF syntax defined based on JSON-LD (ShExJ) 2 . Example 3. An example of a ShEx schema using ShExC compact syntax. The semantics of ShEx schemas is based on a conformance relation parameterized by a shape assignment: we say that node 𝑛 in graph G conforms to shape expression 𝑠𝑒 with shape assignment 𝜏, and we write G, 𝑛, 𝜏 ⊨ 𝑠𝑒.

The following rules are defined similar to [9], where it is shown that there exists a unique maximal shape assignment 𝜏 max that allows us to define conformance independently of the shape assignment. The conformance relation is defined recursively on the structure of 𝑠𝑒 by the set of inference rules presented in table 1 where π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒) is the set of predicates that appear in a triple expression 𝑑𝑒 and can be defined as:

πΆπ‘œπ‘›π‘‘ π‘π‘œπ‘›π‘‘(𝑛) = π‘‘π‘Ÿπ‘’π‘’ G, 𝑛, 𝜏 ⊨ π‘π‘œπ‘›π‘‘ 𝐴𝑁 𝐷 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 1 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 2 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 1 AND 𝑠𝑒 2 π‘†β„Žπ‘Žπ‘π‘’π‘…π‘’π‘“ 𝛿(l) = 𝑠𝑒 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 G, 𝑛, 𝜏 ⊨ @𝑙 πΆπ‘™π‘œπ‘ π‘’π‘‘π‘†β„Žπ‘Žπ‘π‘’ π‘›π‘’π‘–π‘”β„Žπ‘ (𝑛, G) = 𝑑𝑠 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 G, 𝑛, 𝜏 ⊨ CLOSED {𝑑𝑒} π‘‚π‘π‘’π‘›π‘†β„Žπ‘Žπ‘π‘’ 𝑑𝑠 = {⟨π‘₯, 𝑝, π‘¦βŸ© ∈ π‘›π‘’π‘–π‘”β„Žπ‘ (𝑛, G) | 𝑝 ∈ π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒)} G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 G, 𝑛, 𝜏 ⊨ {𝑑𝑒}
Table 1

Inference rules for ShEx shape expressions

π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 ; 𝑑𝑒 2 ) = π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 ) βˆͺ π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 2 ) π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 | 𝑑𝑒 2 ) = π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 ) βˆͺ π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 2 ) π‘π‘Ÿπ‘’π‘‘π‘ ( 𝑝 βˆ’ β†’ 𝑑𝑒) = {𝑝} π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 * ) = π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒) π‘π‘Ÿπ‘’π‘‘π‘ (πœ–)

= βˆ… The rules for node constraints (πΆπ‘œπ‘›π‘‘) and conjunction are as expected. A node 𝑛 conforms to an open shape with triple expression 𝑑𝑒 if its neighborhood restricted to the triples with predicates from 𝑑𝑒 conform, meaning that triples whose predicates are not mentioned in 𝑑𝑒 are not constrained by the shape (rule π‘‚π‘π‘’π‘›π‘†β„Žπ‘Žπ‘π‘’). Conformance to a closed shape requires to consider the whole neighborhood of the node (rule πΆπ‘™π‘œπ‘ π‘’π‘‘π‘†β„Žπ‘Žπ‘π‘’).

Conformance to a triple expression uses a second relation defined on sets on neighborhood triples 𝑑𝑠 instead of nodes 𝑛. The set of neighborhood nodes 𝑑𝑠 of a graph G conforms to a triple expression 𝑑𝑒 with shape assignment 𝜏, written as G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒, as defined by the inference rules in table 2.

πΈπ‘Žπ‘β„Žπ‘‚π‘“ (𝑑𝑠 1 , 𝑑𝑠 2 ) ∈ π‘π‘Žπ‘Ÿπ‘‘(𝑑𝑠) G, 𝑑𝑠 1 , 𝜏 ⊩ 𝑑𝑒 1 G, 𝑑𝑠 2 , 𝜏 ⊩ 𝑑𝑒 2 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 ; 𝑑𝑒 2 𝑂𝑛𝑒𝑂𝑓 1 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 | 𝑑𝑒 2 𝑂𝑛𝑒𝑂𝑓 2 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 2 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 | 𝑑𝑒 2 𝑇 𝐢 1 𝑑𝑠 = {⟨π‘₯, 𝑝, π‘¦βŸ©} G, 𝑦, 𝜏 ⊨ 𝑠𝑒 G, 𝑑𝑠, 𝜏 ⊩ 𝑝 βˆ’ β†’ 𝑠𝑒 𝑇 𝐢 2 𝑑𝑠 = {βŸ¨π‘¦, 𝑝, π‘₯⟩} G, 𝑦, 𝜏 ⊨ 𝑠𝑒 G, 𝑑𝑠, 𝜏 ⊩ 𝑠𝑒 𝑝 βˆ’ β†’ π‘†π‘‘π‘Žπ‘Ÿ 2 (𝑑𝑠 1 , 𝑑𝑠 2 ) ∈ π‘π‘Žπ‘Ÿπ‘‘(𝑑𝑠) G, 𝑑𝑠 1 , 𝜏 ⊩ 𝑑𝑒 G, 𝑑𝑠 2 , 𝜏 ⊩ 𝑑𝑒 * G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 * π‘†π‘‘π‘Žπ‘Ÿ 1 G, βˆ…, 𝜏 ⊩ 𝑑𝑒 * Table 2

Inference rules for ShEx triple expressions

A shape assignment 𝜏 for graph G and S is called valid if for every node 𝑛 in G and every shape expression label l defined in S, if 𝑛@l ∈ 𝜏, then G, 𝑛, 𝜏 ⊨ @l.

According to Boneva et al [7], for every graph G, there exists a unique maximal valid shape assignment 𝜏 max such that if 𝜏 is a valid shape assignment for G and S, then 𝜏 βŠ† 𝜏 max .

RDF-Star and ShEx-Star𝛿(π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘›) = { ( π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘” | 𝑓 π‘–π‘ π‘‘π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘”*; π‘™π‘Žπ‘ π‘‘π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘”); β‰ͺ π‘˜π‘›π‘œπ‘€π‘  βˆ’ βˆ’βˆ’βˆ’βˆ’ β†’ @π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘› ≫{| π‘π‘’π‘Ÿπ‘‘π‘Žπ‘–π‘›π‘‘π‘¦ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ π·π‘’π‘π‘–π‘šπ‘Žπ‘™|}* }

The expression β‰ͺ 𝑝 βˆ’ β†’ 𝑠𝑒 ≫{|𝑑𝑒|} describes a triple term whose predicate is 𝑝 and whose object conforms to the shape expression 𝑠𝑒 and that can be the subject of triples conforming to triple expression 𝑑𝑒. The formal semantics can be described as:

𝑇 𝑇 𝐢 1 𝑑𝑠 = {⟨β‰ͺ 𝑑 ≫, 𝑝, π‘¦βŸ©} G, 𝑦, 𝜏 ⊨ 𝑠𝑒 π‘›π‘’π‘–π‘”β„Žπ‘ (β‰ͺ 𝑑 ≫, G) = 𝑑𝑠 β€² G, 𝑑𝑠 β€² , 𝜏 ⊩ 𝑑𝑒 G, 𝑑𝑠, 𝜏 ⊩β‰ͺ 𝑝 βˆ’ β†’ 𝑠𝑒 ≫ {|𝑑𝑒|} 𝑇 𝑇 𝐢 2 𝑑𝑠 = {⟨π‘₯, 𝑝, β‰ͺ 𝑑 β‰«βŸ©} G, π‘₯, 𝜏 ⊨ 𝑠𝑒 π‘›π‘’π‘–π‘”β„Žπ‘ (β‰ͺ 𝑑 ≫, G) = 𝑑𝑠 β€² G, 𝑑𝑠 β€² , 𝜏 ⊩ 𝑑𝑒 G, 𝑑𝑠, 𝜏 ⊩β‰ͺ 𝑠𝑒 𝑝 βˆ’ β†’ ≫ {|𝑑𝑒|}
Table 3

Inference rules for ShEx-Star new triple term expressions

ShEx-N: Describing nodes that act as properties

In the RDF data model, predicates can also act as subjects or objects or triples. This aspect is not taken into account in traditional ShEx, where the shapes describe the topology of nodes without considering their potential role as predicates. In order to capture these appearances in a single shape, it is possible to add a new kind of triple expression:

𝑑𝑒 ::= . . . Same definitions as in 2 | 𝑠𝑒 1 βˆ’ β†’ 𝑠𝑒 2 Triple constraint with focus node acting as predicate and subject conforming to 𝑠𝑒 1 and object conforming to 𝑠𝑒 2 The semantics of triple term constraints can be defined as:

𝑁 𝑃 1 𝑑𝑠 = {βŸ¨π‘ , π‘₯, π‘œβŸ©} G, 𝑠, 𝜏 ⊨ 𝑠𝑒 1 G, π‘œ, 𝜏 ⊨ 𝑠𝑒 2 G, 𝑑𝑠, 𝜏 ⊩ 𝑠𝑒 1 βˆ’ β†’ 𝑠𝑒 2
Table 4

Inference rules for ShEx-N Example 7 (Example of a ShEx-N schema). The following ShEx-N schema defines the shape 𝐹 π‘Ÿπ‘–π‘’π‘›π‘ β„Žπ‘–π‘π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦ which validates the node :knows in example 6:

𝛿(𝐹 π‘Ÿπ‘–π‘’π‘›π‘‘π‘†β„Žπ‘–π‘π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦) = { π‘ π‘˜π‘œπ‘ βˆΆπ‘Ÿπ‘’π‘™π‘Žπ‘‘π‘’π‘‘ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ [∢ 𝐹 π‘Ÿπ‘–π‘’π‘›π‘‘π‘ β„Žπ‘–π‘] ; @π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘› βˆ’ β†’ @π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘› } 𝛿(π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘›) = { βˆΆπ‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘” | βˆΆπ‘“ π‘–π‘Ÿπ‘ π‘‘π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘”* ; βˆΆπ‘™π‘Žπ‘ π‘‘π‘›π‘Žπ‘šπ‘’ βˆ’ βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ β†’ π‘†π‘‘π‘Ÿπ‘–π‘›π‘” ; π‘˜π‘›π‘œπ‘€π‘  βˆ’ βˆ’βˆ’βˆ’βˆ’ β†’ @π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘› * }
Property graphs and PShEx

Property graphs have become popular thanks to several commercial graph databases like Neo4j3 , JanusGraph4 or Sparksee 5 . A property graph has unique identifiers for each node/edge and allows to add property-value annotations to each node/edge in the arc as well as type annotations.

The following definition of a property graph follows [11].

Definition 4 (Property graph). Given a set of types T , a set of properties P, and a set of values V , a property graph G is a tuple ⟨N , E, 𝜌, πœ† 𝑛 , πœ† 𝑒 , 𝜎 ⟩ where

N ∩ E = βˆ…, 𝜌 ∢ E ↦ N Γ— N is a total function, πœ† 𝑛 ∢ N ↦ 𝐹 𝑖𝑛𝑆𝑒𝑑(T ), πœ† 𝑒 ∢ E ↦ T , and 𝜎 ∢ N βˆͺ E Γ— P ↦ 𝐹 𝑖𝑛𝑆𝑒𝑑(V ).

A property graph is formed by a set of node identifiers N and a set of edges E where 𝜌 associates a pair of nodes (𝑛 1 , 𝑛 2 ) to every 𝑒 ∈ E where 𝑛 1 is the subject and 𝑛 2 is the object, πœ† 𝑛 associates a set of types for node identifiers (notice that property graphs allow nodes to have more than one type), πœ† 𝑒 associates a types for each edge identifier, and 𝜎 associates a set of values to pairs (𝑖, 𝑝) such that 𝑖 ∈ N βˆͺ E is a node or edge and 𝑝 ∈ P is a property. Example 8. As an example, we will represent information that Alice knows Robert with a certainty of 0.5 T = {Person, knows} P = {name, certainty} V = {"Alice", "Robert", "Julius", "Smith", 0.5}

N = {𝑛 1 , 𝑛 2 } E = {π‘Ÿ 1 } 𝜌 = π‘Ÿ 1 ↦ (𝑛 1 , 𝑛 2 ) πœ† 𝑛 = 𝑛 1 ↦ {Person}, 𝑛 2 ↦ {Person} πœ† 𝑒 = π‘Ÿ 1 ↦ knows 𝜎 = (𝑛 1 , π‘›π‘Žπ‘šπ‘’) ↦ {"Alice"} (𝑛 2 , 𝑓 π‘–π‘Ÿπ‘ π‘‘π‘›π‘Žπ‘šπ‘’) ↦ {"Robert", "Julius"} (𝑛 2 , π‘™π‘Žπ‘ π‘‘π‘›π‘Žπ‘šπ‘’) ↦ {"Smith"} (π‘Ÿ 1 , π‘π‘’π‘Ÿπ‘‘π‘Žπ‘–π‘›π‘‘π‘¦) ↦ {0.5}

Figure 4 presents a possible visualization of a property graph. We define a ShEx extension called PShEx that can be used to describe and validate Property graphs. In property graphs, nodes and edges can have associated labels as well as a set of property/values. In this way, it is necessary to adapt the definition of ShEx to describe pairs or property/values. PShEx is composed of three main categories: shape expressions (𝑠𝑒) that describe the shape of nodes, triple expressions (𝑑𝑒) that describe the shape of edge relationships and property-value expressions (𝑝𝑣𝑠) that describe sets of property/values associated with node/edge identifiers.

Definition 5 (PShEx schema).

A PShEx Schema is a tuple ⟨L, π›ΏβŸ© where L set of shape labels, and 𝛿 ∢ L β†’ S is a total function from labels to shape expressions 𝑠𝑒 ∈ S defined using the abstract syntax:

𝑠𝑒

L = { π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘›} 𝛿(π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘›) = β„Žπ‘Žπ‘ π‘‡ 𝑦𝑝𝑒 𝐻 π‘’π‘šπ‘Žπ‘› AND βŒŠπ‘›π‘Žπ‘šπ‘’ ∢ π‘†π‘‘π‘Ÿπ‘–π‘›π‘” | 𝑓 π‘–π‘Ÿπ‘ π‘‘π‘›π‘Žπ‘šπ‘’ ∢ π‘†π‘‘π‘Ÿπ‘–π‘›π‘” * , π‘™π‘Žπ‘ π‘‘π‘›π‘Žπ‘šπ‘’ ∢ π‘†π‘‘π‘Ÿπ‘–π‘›π‘”βŒ‹ AND { π‘˜π‘›π‘œπ‘€π‘  βˆ’ βˆ’βˆ’βˆ’βˆ’ β†’ @π‘ƒπ‘’π‘Ÿπ‘ π‘œπ‘› βŒŠπ‘π‘’π‘Ÿπ‘‘π‘Žπ‘–π‘›π‘‘π‘¦ ∢ π·π‘’π‘π‘–π‘šπ‘Žπ‘™βŒ‹* }

In order to define the semantic specification of PShEx we will need to define the neighborhood of a node in a property graph.

The neighbors of a node 𝑛 ∈ N in a property graph G = ⟨N , E, 𝜌, πœ† 𝑛 , πœ† 𝑒 , 𝜎 ⟩ are defined as π‘›π‘’π‘–π‘”β„Žπ‘ (𝑛) = {(𝑛, 𝑝, 𝑦, 𝑣𝑠) | βˆƒπ‘£ ∈ E such that 𝜌(𝑣) = (𝑛, 𝑦) ∧ πœ† 𝑒 (𝑣) = 𝑝 ∧ 𝑣𝑠 = {(π‘˜, 𝑣) | 𝜎 (π‘˜, 𝑣) = 𝑀𝑠 ∧ 𝑣 ∈ 𝑀𝑠}}. For a node 𝑛, we define the property value set as π‘π‘Ÿπ‘œπ‘π‘£π‘ (𝑛) = {(π‘˜, 𝑣) | 𝜎 (π‘˜, 𝑣) = 𝑀𝑠 ∧ 𝑣 ∈ 𝑀𝑠}

The semantic specification of PShEx can be defined in a similar way to the ShEx one. Given a property graph G, and a shape assignment 𝜏, a node identifier 𝑛 ∈ N conforms with a shape expression 𝑠𝑒, which is represented as G, 𝑛, 𝜏 ⊨ 𝑠𝑒 and follows the rules presented in 5 where π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒) is the set of edge labels (or predicates) that appear in a triple expression 𝑑𝑒 and can be defined as:

πΆπ‘œπ‘›π‘‘ 𝑑𝑠 πœ† 𝑛 (𝑛) = 𝑣𝑠 π‘π‘œπ‘›π‘‘ 𝑑𝑠 (𝑣𝑠) = π‘‘π‘Ÿπ‘’π‘’ G, 𝑛, 𝜏 ⊨ π‘π‘œπ‘›π‘‘ 𝑑𝑠 𝐴𝑁 𝐷 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 1 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 2 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 1 AND 𝑠𝑒 2 πΆπ‘™π‘œπ‘ π‘’π‘‘π‘†β„Žπ‘Žπ‘π‘’ π‘›π‘’π‘–π‘”β„Žπ‘ (𝑛, G) = 𝑑𝑠 G, 𝑑𝑠, 𝜏 ⊩ 𝑠 β€² G, 𝑛, 𝜏 ⊨ CLOSED {𝑑𝑒} π‘‚π‘π‘’π‘›π‘†β„Žπ‘Žπ‘π‘’ 𝑑𝑠 = {⟨π‘₯, 𝑝, π‘¦βŸ© ∈ π‘›π‘’π‘–π‘”β„Žπ‘ (𝑛, G) | 𝑝 ∈ π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒)} G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 G, 𝑛, 𝜏 ⊨ {𝑑𝑒} π‘†β„Žπ‘Žπ‘π‘’π‘…π‘’π‘“ 𝛿(l) = 𝑠𝑒 G, 𝑛, 𝜏 ⊨ 𝑠𝑒 G, 𝑛, 𝜏 ⊨ @𝑙 𝑃𝑉 𝑆 π‘π‘Ÿπ‘œπ‘π‘£π‘ (𝑛) = 𝑠 G, 𝑠, 𝜏 ⊒ 𝑝𝑣𝑠 G, 𝑛, 𝜏 ⊨ 𝑝𝑣𝑠
Table 5

Rules for PShEx shape expressions

π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 ; 𝑑𝑒 2 ) = π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 ) βˆͺ π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 2 ) π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 | 𝑑𝑒 2 ) = π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 1 ) βˆͺ π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 2 ) π‘π‘Ÿπ‘’π‘‘π‘ ( 𝑝 βˆ’ β†’ 𝑑𝑒) = {𝑝} π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒 * ) = π‘π‘Ÿπ‘’π‘‘π‘ (𝑑𝑒) π‘π‘Ÿπ‘’π‘‘π‘ (πœ–)

= βˆ… As in the case of ShEx, the previous definition uses a second conformance relation defined on sets of triples 𝑑𝑠 instead of nodes 𝑛. The set of neighborhood nodes 𝑑𝑠 from a property graph G conforms to a triple expression 𝑑𝑒 with shape assignment 𝜏, written G, 𝑑𝑠, 𝜏 ⊩ 𝑠, as defined by the inference rules represented in table 6. In the case of PShEx we declare a new conformance relationship G, 𝑠, 𝜏 ⊒ π‘žπ‘  between a graph G a set 𝑠 ∈ 𝑃 Γ— 𝑉 of property-value elements, a shape assignment 𝜏 and a property-value specifier 𝑝𝑣𝑠 whose rules are defined in table 7 where π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑣𝑠) is the set of properties that appear in a property-value specifier 𝑝𝑠 and can be defined as:

π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠 1 , 𝑝𝑠 2 ) = π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠 1 ) βˆͺ π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠 2 ) π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠 1 | 𝑝𝑠 2 ) = π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠 1 ) βˆͺ π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠 2 ) π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠 * ) = π‘π‘Ÿπ‘’π‘‘π‘ (𝑝𝑠) π‘π‘Ÿπ‘œπ‘π‘ (𝑝 ∢ π‘π‘œπ‘›π‘‘ 𝑣 ) = {𝑝}

As in the case of ShEx, the semantics of ShEx schemas can be defined independently on shape assignments. A shape assignment 𝜏 for graph G and S is called valid if for every node 𝑛 in G and every shape expression label l defined in S, if 𝑛@l ∈ 𝜏, then G, 𝑛, 𝜏 ⊨ @l.

πΈπ‘Žπ‘β„Žπ‘‚π‘“ (𝑑𝑠 1 , 𝑑𝑠 2 ) ∈ π‘π‘Žπ‘Ÿπ‘‘(𝑑𝑠) G, 𝑑𝑠 1 , 𝜏 ⊩ 𝑑𝑒 1 G, 𝑑𝑠 2 , 𝜏 ⊩ 𝑑𝑒 2 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 ; 𝑑𝑒 2 𝑂𝑛𝑒𝑂𝑓 1 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 | 𝑑𝑒 2 𝑂𝑛𝑒𝑂𝑓 2 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 2 G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 1 | 𝑑𝑒 2 𝑇 π‘Ÿπ‘–π‘π‘™π‘’πΆπ‘œπ‘›π‘ π‘‘π‘Ÿπ‘Žπ‘–π‘›π‘‘ 𝑑𝑠 = {⟨π‘₯, 𝑝, 𝑦, π‘ βŸ©} G, 𝑦, 𝜏 ⊨ @l G, 𝑠, 𝜏 ⊒ π‘žπ‘  G, 𝑑𝑠, 𝜏 ⊩ 𝑝 βˆ’ β†’ @l π‘žπ‘  π‘†π‘‘π‘Žπ‘Ÿ 1 G, βˆ…, 𝜏 ⊩ 𝑑𝑒 * π‘†π‘‘π‘Žπ‘Ÿ 2 (𝑑𝑠 1 , 𝑑𝑠 2 ) ∈ π‘π‘Žπ‘Ÿπ‘‘(𝑑𝑠) G, 𝑑𝑠 1 , 𝜏 ⊩ 𝑑𝑒 G, 𝑑𝑠 2 , 𝜏 ⊩ 𝑑𝑒 * G, 𝑑𝑠, 𝜏 ⊩ 𝑑𝑒 * Table 6

Rules for PShEx triple expressions

𝑂𝑝𝑒𝑛𝑃𝑉 𝑠 𝑠 β€² = {(𝑝, 𝑣) ∈ 𝑠|𝑝 ∈ π‘π‘Ÿπ‘œπ‘π‘ (𝑝𝑠)} G, 𝑠 β€² , 𝜏 ⊒ 𝑝𝑠 G, 𝑠, 𝜏 ⊒ βŒŠπ‘π‘ βŒ‹ πΆπ‘™π‘œπ‘ π‘’π‘ƒπ‘‰ 𝑠 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 G, 𝑠, 𝜏 ⊒ βŒˆπ‘π‘ βŒ‰ πΈπ‘Žπ‘β„Žπ‘‚π‘“ 𝑃𝑠 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 1 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 2 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 1 , 𝑝𝑠 2 𝑂𝑛𝑒𝑂𝑓 𝑃𝑠 1 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 1 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 1 | 𝑝𝑠 2 𝑂𝑛𝑒𝑂𝑓 𝑃𝑠 2 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 2 G, 𝑠, 𝜏 ⊒ 𝑝𝑠 1 | 𝑝𝑠 2 π‘†π‘‘π‘Žπ‘Ÿπ‘ƒπ‘  1 G, βˆ…, 𝜏 ⊒ 𝑝𝑠 * π‘†π‘‘π‘Žπ‘Ÿπ‘ƒπ‘  2 (𝑠 1 , 𝑠 2 ) ∈ π‘π‘Žπ‘Ÿπ‘‘(𝑠) G, 𝑠 1 , 𝜏 ⊒ 𝑝𝑠 G, 𝑠 2 , 𝜏 ⊒ 𝑝𝑠 * G, 𝑠, 𝜏 ⊒ 𝑝𝑠 * π‘ƒπ‘Ÿπ‘œπ‘π‘’π‘Ÿπ‘‘π‘¦π‘‰ π‘Žπ‘™π‘’π‘’ 𝑠 = {(𝑝, 𝑀)} π‘π‘œπ‘›π‘£ 𝑣 (𝑀) = true G, 𝑠, 𝜏 ⊒ 𝑝 ∢ π‘π‘œπ‘›π‘‘ 𝑣
Table 7

Rules for PShEx property-value specifiers

Related work

ShEx was initially proposed in 2014 [2] as a concise and human readable language to describe and validate RDF. It was based on a variant of regular expressions called Regular Bag Expressions [12] which also supports recursive shapes. Combining negation with recursive shapes was later studied in [9] where a well founded formal semantics for ShEx for proposed based on stratification. Our definition of ShEx is based on that work, although we omitted negation, disjunction and EXTRA declarations of shape expressions in this paper. After ShEx was proposed, a W3C Data Shapes working group was chatered whose result was SHACL, proposed as a recommendation in 2017 [3]. The specification of SHACL didn't have an abstract syntax and left the semantics of recursive shapes as an implementation dependent feature enabling the appearance of several proposals that define an abstract syntax and add semantics for SHACL with negation and recursion [13,14,15,16]. A comparison between both ShEx and SHACL was provided in [17] while in [18], a simple language was defined that can be used as a common subset of both. We consider that some of our extensions to ShEx could also be applied to SHACL. The approach followed in this paper to extend ShEx was started in this paper [19] where we had already proposed an initial version of PShEx and WShEx [20], another extension of ShEx to support the Wikibase data model. WShEx is conceptually similar to PShEx although a distinct feature of the Wikidbase data model is that the values of properties can also be nodes in the graph, which can be considered as a generalized property graphs model. This model was called MARS (Multi-Attributed Relational Structures) in [21].

There are several proposals for property graphs schemas. GQL is an upcoming ISO standard (ISO39075) 6 which is currently being developed and addresses the property graph model. In order to provide support for GQL, PG-Schema [22] was proposed as a simple schema language for property graphs. PG-Schema does not support cardinality constraints on edges, which could be simulated using PG-Keys [23]. In [24], the authors propose a common framework for property graph schema languages based on first order logic rules which supports cardinality constraints. The closest proposal to PShEx would be the Property Graph Shapes Language (ProGS) [11] although that language is based on SHACL and some of the differences SHACL vs ShEx could also be applied to ProGS vs PShEx. For example, PShEx doesn't have property path expressions and ProGS doesn't have regular bag expressions. Another difference is that ProGS includes negation and recursion while in the version of PShEx included in this paper we omitted negation. Comparison between RDF-Star and property graphs at the data model have already been studied. In [25] proposes a common model for RDF, RDF-Star and property graphs that they call statement graph, which is inspired by the OneGraph [26] vision. The conversion between property graphs and RDF/RDF-Star is also studied in [6].

Conclusions and future work

We have presented three extensions to ShEx for different types of knowledge graphs: ShEx-* adds support for RDF-Star, ShEx-N adds a constraint to describe nodes that act as properties, and PShEx can be used to describe and validate property graphs. We presented an abstract syntax and formal semantics for each of the variants.

There are several lines of future work. On the theoretical level, it will be necessary to research the computational complexity implications of these features as well as the combination of these features with other features from ShEx like negation, disjunction, EXTRA declarations, and even the potential integration of all the features together in a single language. On the implementation level: it will be necessary to propose a concrete compact syntax for each variant that can be useful by data model engineers as well as devise algorithms that can be used to check the conformance of graphs to the corresponding schemas. Although we used ShEx in this paper, another line of future work would be to check which of the features proposed in this paper could also be applied to SHACL.

prefix : <http://example.org/> prefix xsd: <http://www.w3.org/2001/XMLSchema#> <Person> { (:name xsd:string | :firstname xsd:string * ; :lastname xsd:string ); :knows @<Person> * }
RDF-Star has been proposed as an extension of RDF where the subjects and objects can be triples. We present a formal definition of RDF-Star based on [10]: Definition 3 (RDF-Star). An RDF-Star triple is a tuple 𝑑 defined recursively as follows: Any RDF triple 𝑑 ∈ (I βˆͺ B) Γ— I Γ— (I βˆͺ B βˆͺ Lit) is an RDF-Star triple; and given RDF-Star triples 𝑑 and 𝑑 β€² and RDF terms 𝑠 ∈ (I βˆͺ B), 𝑝 ∈ I and π‘œ ∈ (I βˆͺ B βˆͺ Lit), the tuples (β‰ͺ 𝑑 ≫, 𝑝, π‘œ), (𝑠, 𝑝, β‰ͺ 𝑑 β€² ≫) and (β‰ͺ 𝑑 ≫, 𝑝, β‰ͺ 𝑑 β€² ≫) are RDF-Star triples. An RDF-Star graph is a set of RDF-Star triples. Example 4 (Example of an RDF-Star graph in Turtle-Star notation). The following snippet contains a simple RDF-Star graph with two nodes :a and :b.prefix : <http://example.org/> :a :name "Alice" . << :a :knows :b >> :certainty 0.5 . :b :firstname "Robert", "Julius" ; :lastname "Smith" .
Figure 2 :Figure 2: Basic RDF-Star example
Figure 3 :Figure 3: RDF example with a property as a node
Figure 4 :Figure 4: Example graph visualization of a property graph
1
Example 6 (Example of an RDF graph a node acting as a property). The following snippet contains a simple RDF graph where :knows is both a node and a property.
prefix : <http://example.org/>prefix skos: <http://www.w3.org/2004/02/skos/core#>:a :name "Alice" ;:knows :b .:b :firstname "Robert", "Julius" ;:lastname "Smith" .:knows skos:related :Friendship .:Friendshipskos:related:knows:a:b:name:firstname:firstname:lastname"Alice""Robert""Julius""Smith"
::= π‘π‘œπ‘›π‘‘ 𝑑 𝑠 Basic boolean condition on set of types 𝑑 𝑠 βŠ† T | 𝑠𝑒 1 AND 𝑠𝑒 2 As an example, we can define a PShEx schema that describes the property graph from example 8 where β„Žπ‘Žπ‘ π‘‡ 𝑦𝑝𝑒 𝑑 is a condition that is satisfied when the set of types of a node contains the type 𝑑, i.e. β„Žπ‘Žπ‘ π‘‡ 𝑦𝑝𝑒 𝑑 (𝑣𝑠) = true if 𝑑 ∈ 𝑣𝑠 and π‘†π‘‘π‘Ÿπ‘–π‘›π‘”, π·π‘’π‘π‘–π‘šπ‘Žπ‘™ are conditions on the values that are satisfied when the values have the corresponding type.Conjunction|@lShape label reference for l ∈ L|𝑝𝑣𝑠Property-value specifiers of a node|CLOSED {𝑑𝑒}Closed shape|{𝑑𝑒}Open shape𝑑𝑒 ::= 𝑑𝑒 1 ; 𝑑𝑒 2Each of 𝑑𝑒 1 and 𝑑𝑒 2|𝑑𝑒 1 | 𝑑𝑒 2Some of 𝑑𝑒 1 or 𝑑𝑒 2|𝑑𝑒 *Zero or more 𝑑𝑒|Open property-value specifiers 𝑝𝑠|βŒˆπ‘π‘ βŒ‰Closed property-value specifiers 𝑝𝑠𝑝𝑠 ::= 𝑝𝑠 1 , 𝑝𝑠 2Each of 𝑝𝑠 1 and 𝑝𝑠 2|𝑝𝑠 1 | 𝑝𝑠 2OneOf of 𝑝𝑠 1 or 𝑝𝑠 2|𝑝𝑠 *zero of more 𝑝𝑠|𝑝 ∢ π‘π‘œπ‘›π‘‘
𝑝 βˆ’ β†’ @l 𝑝𝑣𝑠 Triple constraint with property type 𝑝 whose nodes satisfy the shape l and property-values 𝑝𝑣𝑠 𝑝𝑣𝑠 ::= βŒŠπ‘π‘ βŒ‹ 𝑣 Property 𝑝 with value conforming to π‘π‘œπ‘›π‘‘ 𝑣 π‘π‘œπ‘›π‘‘ 𝑣 𝑠 is a boolean condition on sets of values 𝑣 𝑠 βŠ† V Example 9.
The full specification of ShEx is available at https://shex.io/shex-semantics/ See ShEx specification[8] for details. https://neo4j.com/ https://janusgraph.org/ https://www.sparsity-technologies.com/#sparksee https://www.iso.org/standard/76120.html
Acknowledgments

This work has been partially funded by the Project ANGLIRU: ANGLIRU: Applying kNowledge Graphs to research data interoperabiLIty and ReUsability, code: PID2020-117912RB. The discussions held during our participation in Dagstuhl seminar 24102 Shapes in graph data: theory and implementation helped us to write some of the ideas included in this paper.

AHogan EBlomqvist MCochez CAmato GDe Melo CGutiΓ©rrez SKirrane JELabra Gayo RNavigli SNeumaier A.-CNgonga Ngomo APolleres SMRashid ARula LSchmelzeisen JFSequeda SStaab AZimmermann 10.2200/S01125ED1V01Y202109DSK022 Knowledge Graphs, number 22 in Synthesis Lectures on Data, Semantics, and Knowledge Springer 2021 Shape Expressions: An RDF Validation and Transformation Language EPrud'hommeaux JELabra Gayo HSolbrig 10.1145/2660517.2660523 Proceedings of the 10th International Conference on Semantic Systems, SEMANTICS 2014 HSack AFilipowska JLehmann SHellmann the 10th International Conference on Semantic Systems, SEMANTICS 2014
Leipzig, Germany
 ACM Press September 4-5, 2014. 2014 HKnublauch DKontokostas Shapes Constraint Language (SHACL), W3C Recommendation 20 July 2017, W3C Recommendation, World Wide Web Consortium 2017 RCyganiak DWood MLanthaler RDF-star and SPARQL-star, Draft Community Group Report, World Wide Web Consortium 2021 RDF 1.2 Concepts and Abstract Syntax, W3C Working Draft OHartig P.-AChampin GKellogg ASeaborne World Wide Web Consortium 2024 Converting property graphs to rdf: a preliminary study of the practical impact of different mappings SKhayatbashi SFerrada OHartig 10.1145/3534540.3534695 Proceedings of the 5th ACM SIGMOD Joint International Workshop on Graph Data Management Experiences and Systems (GRADES) and Network Data Analytics (NDA), SIGMOD/PODS '22 the 5th ACM SIGMOD Joint International Workshop on Graph Data Management Experiences and Systems (GRADES) and Network Data Analytics (NDA), SIGMOD/PODS '22 ACM 2022 Semantics and Validation of Shapes Schemas for RDF IBoneva JELabra Gayo EGPrud'hommeaux The Semantic Web -ISWC 2017 -16th International Semantic Web Conference Lecture Notes in Computer Science CAmato MFernΓ‘ndez VA MTamma FLΓ©cuΓ© PCudrΓ©-Mauroux JFSequeda CLange JHeflin
Vienna, Austria
 Springer October 21-25, 2017. 2017 10587 Proceedings, Part I EPrud'hommeaux IBoneva JELabra Gayo GKellog Shape Expressions Language 2 2017 0 Semantics and validation of shapes schemas for rdf IBoneva JELabra Gayo EPrud'hommeaux International Semantic Web Conference 2017 Foundations of RDF* and SPARQL* -An Alternative Approach to Statement-Level Metadata in RDF OHartig Proceedings of the 11th Alberto Mendelzon International Workshop on Foundations of Data Management and the Web CEUR Workshop Proceedings JLReutter DSrivastava the 11th Alberto Mendelzon International Workshop on Foundations of Data Management and the Web
Montevideo, Uruguay; Sun SITE Central Europe
 June 7-9, 2017. 1912. 2017 ProGS: Property Graph Shapes Language PSeifer RLΓ€mmel SStaab 10.1007/978-3-030-88361-4_23 arXiv:2107.05566 International Semantic Web Conference Springer 2021 12922 Complexity and Expressiveness of ShEx for RDF SStaworko IBoneva JELabra Gayo SHym EGPrud'hommeaux HRSolbrig 18th International Conference on Database Theory, ICDT 2015 Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik 2015 31 Semantics and Validation of Recursive SHACL JCorman JLReutter OSavkoviΔ‡ The Semantic Web -ISWC 2018 -17th International Semantic Web Conference Lecture Notes in Computer Science DVrandečiΔ‡ KBontcheva MCSuΓ‘rez-Figueroa VPresutti ICelino MSabou LKaffee ESimperl
Monterey, CA, USA
 Springer October 8-12, 2018. 2018 11136 Proceedings, Part I Stable model semantics for recursive shacl MAndresel JCorman MOrtiz JLReutter OSavkovic MSimkus 10.1145/3366423.3380229 Proceedings of The Web Conference 2020 The Web Conference 2020 ACM 2020 WWW '20 Fixpoint semantics for recursive SHACL BBogaerts MJakubowski 10.4204/EPTCS.345.14 Proceedings 37th International Conference on Logic Programming (Technical Communications), ICLP Technical Communications 2021, Porto (virtual event) AFormisano YALiu BBogaerts ABrik VDahl CDodaro PFodor GLPozzato JVennekens NZhou 37th International Conference on Logic Programming (Technical Communications), ICLP Technical Communications 2021, Porto (virtual event)
EPTCS
 September 2021. 2021 345 Well-founded semantics for recursive shacl AChmurovic MΕ imkus CEUR Workshop Proceedings 3203. 2022 CEUR Workshop proceedings Validating RDF Data, volume 7 of Synthesis Lectures on the Semantic Web: Theory and Technology JELabra Gayo EPrud'hommeaux IBoneva DKontokostas 10.2200/s00786ed1v01y201707wbe016 2017 Morgan & Claypool Challenges in RDF Validation JELabra Gayo HGarcΓ­a-GonzΓ‘lez DFernΓ‘ndez-Alvarez EPrud'hommeaux 10.1007/978-3-030-06149-4_6 Current Trends in Semantic Web Technologies: Theory and Practice, Studies in Computational Intelligence GAlor-HernΓ‘ndez JLSΓ‘nchez-Cervantes ARodrΓ­guez-GonzΓ‘lez RValencia-GarcΓ­a Springer 2019 Creating knowledge graphs subsets using shape expressions JE LGayo arXiv:2110.11709 2021 Wshex: A language to describe and validate wikibase entities J.-ELabra-Gayo Proceedings of the 3rd Wikidata Workshop 2022 co-located with the 21st International Semantic Web Conference (ISWC2022), Virtual Event CEUR Workshop Proceedings LKaffee SRazniewski GAmaral KSAlghamdi the 3rd Wikidata Workshop 2022 co-located with the 21st International Semantic Web Conference (ISWC2022), Virtual Event
Hanghzou, China
 October 2022. 2022 3262 Logic on MARS: ontologies for generalised property graphs MMarx MKrâtzsch VThost 10.24963/ijcai.2017/165 Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI'17), International Joint Conferences on Artificial Intelligence CSierra the 26th International Joint Conference on Artificial Intelligence (IJCAI'17), International Joint Conferences on Artificial Intelligence 2017 Pg-schema: Schemas for property graphs RAngles ABonifati SDumbrava GFletcher AGreen JHidders BLi LLibkin VMarsault WMartens FMurlak SPlantikow OSavkovic MSchmidt JSequeda SStaworko DTomaszuk HVoigt DVrgoc MWu DZivkovic 10.1145/3589778 Proceedings of the ACM on Management of Data 1 2023 Pg-keys: Keys for property graphs RAngles ABonifati SDumbrava GFletcher KWHare JHidders VELee BLi LLibkin WMartens FMurlak JPerryman OSavković MSchmidt JSequeda SStaworko DTomaszuk 10.1145/3448016.3457561 Proceedings of the 2021 International Conference on Management of Data, SIGMOD/PODS '21 the 2021 International Conference on Management of Data, SIGMOD/PODS '21 ACM 2021 A formal design framework for practical property graph schema languages NBeeren GFletcher 10.48786/EDBT.2023.40 Proceedings 26th International Conference on Extending Database Technology, EDBT 2023 JStoyanovich JTeubner NMamoulis EPitoura JMühlig KHose SSBhowmick MLissandrini 26th International Conference on Extending Database Technology, EDBT 2023
Ioannina, Greece
 OpenProceedings March 28-31, 2023. 2023 EGelling GFletcher MSchmidt 10.48550/ARXIV.2304.13097 arXiv:2304.13097 Bridging graph data models: Rdf, rdf-star, and property graphs as directed acyclic graphs 2023 The onegraph vision: Challenges of breaking the graph model lock-in1 OLassila MSchmidt OHartig BBebee DBechberger WBroekema AKhandelwal KLawrence CMLopez Enriquez RSharda BThompson 10.3233/sw-223273 Semantic Web 14 2022