A Direct Mapping of Relational Data to RDF

1 Introduction

Relational databases proliferate both because of their efficiency and their precise definitions, allowing for tools like SQL [SQLFN] to manipulate and examine the contents predictably and efficiently. Resource Description Framework (RDF) [RDF-concepts] is a data format based on a web-scalable architecture for identification and interpretation of terms. This document defines a mapping from relational representation to an RDF representation.

Strategies for mapping relational data to RDF abound. The direct mapping defines a simple transformation, providing a basis for defining and comparing more intricate transformations. This document includes an informal and a formal description of the transformation.

The Direct Mapping is intended to provide a default behavior for R2RML: RDB to RDF Mapping Language. It can be also used to materialize RDF graphs or define virtual graphs, which can be queried by SPARQL or traversed by an RDF graph API.

2 Direct Mapping Description (Informative)

The direct mapping defines an RDF Graph [RDF-concepts] representation of the data in any relational database. The direct mapping takes as input a relational database (data and schema), and generates an RDF graph that is called the direct graph. This graph is composed of relative IRIs that may be resolved against a base IRI per [RFC3987]. Foreign keys in relational databases establish a named reference from any row in a table to exactly one row in a (potentially different) table. The direct graph conveys these references, as well as each value in the rows.

2.1 Direct Mapping Example

The concepts in direct mapping can be introduced with an example RDF graph produced by a relational database. Following is SQL (DDL) to create a simple example with two tables with single-column primary keys and one foreign key reference between them:

CREATE TABLE Addresses (
	ID INT, 
	city CHAR(10), 
	state CHAR(2), 
	PRIMARY KEY(ID)
)

CREATE TABLE People (
	ID INT, 
	fname CHAR(10), 
	addr INT, PRIMARY KEY(ID), 
	FOREIGN KEY(addr) REFERENCES Addresses(ID)
)

INSERT INTO Addresses (ID, city, state) VALUES (18, "Cambridge", "MA")
INSERT INTO People (ID, fname, addr) VALUES (7, "Bob", 18)
INSERT INTO People (ID, fname, addr) VALUES (8, "Sue", NULL)

HTML tables will be used in this document to convey SQL tables. The primary key of these tables will be marked with the PK class to convey an SQL primary key such as ID in CREATE TABLE Addresses (ID INT, ... PRIMARY KEY(ID)). Foreign keys will be illustrated with a notation like "→ Address(ID)" to convey an SQL foreign key such as CREATE TABLE People (... addr INT, FOREIGN KEY(addr) REFERENCES Addresses(ID)).

People
PK		→ Address(ID)
ID	fname	addr
7	Bob	18
8	Sue	NULL

Addresses
PK
ID	city	state
18	Cambridge	MA

Given a base IRI http://foo.example/DB/, the direct mapping of this database produces a direct graph:

@base <http://foo.example/DB/>
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .


<People/ID=7> <rdf:type> <People> .
<People/ID=7> <People#ID> 7 .
<People/ID=7> <People#fname> "Bob" .
<People/ID=7> <People#addr> <Addresses/ID=18> .
<People/ID=8> <rdf:type> <People> .
<People/ID=8> <People#ID> 8 .
<People/ID=8> <People#fname> "Sue" .

<Addresses/ID=18> <rdf:type> <Addresses> .
<Addresses/ID=18> <Addresses#ID> 18 .
<Addresses/ID=18> <Addresses#city> "Cambridge" .
<Addresses/ID=18> <Addresses#state> "MA" .

In this expression, each row, e.g. (7, "Bob", 18), produces a set of triples with a common subject. The subject is an IRI formed from the concatenation of the base IRI, table name (People), primary key column name (ID) and primary key value (7). The predicate for each column is an IRI formed from the concatenation of the base IRI, table name and the column name. The values are either RDF literals formed from the lexical form of the column value, or, in the case of foreign keys, row identifiers (<Addresses/ID=18>). Note that these reference row identifiers must coincide with the subject used for the triples generated from the referenced row.

2.2 Preliminaries: Generating IRIs

In the process of translating relational data into RDF, the direct mapping must create IRIs for identifying tables, the columns in a table, and each row in a table. In this section, we assume that http://foo.example/DB is the the base IRI. All the examples in this section will contain relative IRIs which are to be understood as relative to this base IRI. The following are the IRIs that need to be generated:

Table IRI: The IRI that identifies a table is created by concatenating the base IRI with the table name. Specifically, if base_IRI is the base IRI and table_name is the table name, then base_IRI/table_name is the Table IRI for the table.
Column IRI:
- Single-column IRI: The IRI that identifies a column of a table is created by concatenating the base IRI with the table name and the column name. Specifically, if base_IRI is the base IRI, table_name is the table name and column_name is the column name, then base_IRI/table_name#column_name is the Column IRI for the column.
- Multi-column IRI: The IRI that identifies a sequence of two or more columns of a table is created by concatenating the base IRI with the table name and the column names. Specifically, if base_IRI is the base IRI, table_name is the table name and column_name_1, column_name_2, ..., column_name_k is a sequence of k columns (k > 1), then base_IRI/table_name#column_name_1,column_name_2,...,column_name_k is the Column IRI for the columns.
Row RDF Node:
- Row RDF Node for a row with a single-column primary key: The IRI that identifies a row is created by concatenating the base IRI with the table name, the column name of the primary key and the value of the row in that column. Specifically, if base_IRI is the base IRI, table_name is the table name, column_name is the column name of the primary key and value is the value of the row in that column, then base_IRI/table_name/column_name=value is the Row RDF Node (or Row IRI) for the row.
- Row RDF Node for a row with a multi-column primary key: The IRI that identifies a row is created by concatenating the base IRI with the table name, the names of the columns that constitute the primary key and the values of the row in those columns. Specifically, if base_IRI is the base IRI, table_name is the table name, column_name_1, column_name_2, ..., column_name_k is the sequence of k columns (k > 1) that constitute the primary key, and value_1, value_2, ..., value_k is the sequence of values of the columns that constitute the primary key of the row, then base_IRI/table_name/column_name_1=value_1,column_name_2=value_2,...,column_name_k=value_k is the Row RDF Node (or Row IRI) for the row.
- Row RDF Node for a row without a primary key: A fresh Blank Node is created, which is used as the Row RDF Node for the row.

Issue (hash-vs-slash):

The direct graph may be offered as Linked Open Data, raising the issue of distinguishing row identifiers from the information resources which describe them. This edition of this document presumes hash identifiers, allowing a GET on a row identifier to retrieve a small resource (i.e. not all rows from the same table) and distinguish between the retrieved resource People/ID=7 and the row People/ID=7. The "slash" alternative would offer a direct graph with identifiers like People/ID=7 but would demand the server respond to GET /People/ID=7 with a 303 redirect to some other resource.

Resolution:

None recorded.

2.2.1 IRIs generated for the initial example

Given the base IRI http://foo.example/DB/, the following are some of the IRIs that are used when translating into RDF the relational data given in the initial example:

For the table People, the following IRIs are considered in the translation process:

Table IRI:
```
<People> 
                     
```

Column IRIs:

<People#ID> 
<People#fname> 
<People#addr>

Row IRIs:

<People/ID=7> 
<People/ID=8>

For the table Addresses, the following IRIs are considered in the translation process:

Table IRI:
```
<Addresses> 
                     
```

Columns IRIs:

<Addresses#ID> 
<Addresses#city>  
<Addresses#state>

Row IRI:
```
<Addresses/ID=18>
                     
```

2.3 Mapping Rules

Each row in the database produces a set of RDF triples with a subject, predicate, and object composed as follows:

Shared Subject: A Row RDF Node, which may be an IRI or a Blank Node, is generated for each row.
Table Triples: The row generates a triple with the following:
- Predicate: the rdf:type property
- Object: the Table IRI for the table
Literal Triples: Each column with a non-null value, including the column(s) that constitute the primary key, and that either is not the only constituent of a foreign key or is the only constituent of a foreign key that references a candidate key, generates a triple with the following:
- Predicate: the Column IRI for the column
- Object: an RDF Literal with an XML Schema datatype corresponding to the SQL datatype of that value. String datatypes are expressed as an RDF plain literal
Reference Triples: Columns that constitute a foreign key and with non-null values in the row generate triples with the following:
- Predicate: the Column IRI for the columns that constitute the foreign key
- Object: the Row RDF Node for the corresponding referenced row (according to the foreign key)

Issue (primary-is-candidate-key):

Should the following exception be included in the definition of the direct mapping?

Primary-is-Candidate-Key Exception: If the primary key is also a candidate key K to table R:

The shared subject is the subject of the referenced row in R.
The foreign key K generates no reference triple.
Even if K is a single-column foreign key, it generates a literal triple.

Resolution:

None recorded.

2.3.1 Triples generated for the example in Section Direct Mapping Example

Next we show how the 11 triples in the example of Section Direct Mapping Example are classified into the above categories:

Triples generated from table People:

Table Triples:

<People/ID=7> <rdf:type> <People> .              
<People/ID=8> <rdf:type> <People> .

Literal Triples:

<People/ID=7> <People#ID> 7 .
<People/ID=7> <People#fname> "Bob" .
<People/ID=8> <People#ID> 8 .
<People/ID=8> <People#fname> "Sue" .

Reference Triple:

<People/ID=7> <People#addr> <Addresses/ID=18> .

Triples generated from table Addresses:

Table Triple:

<Addresses/ID=18> <rdf:type> <Addresses> .

Literal Triples:

<Addresses/ID=18> <Addresses#ID> 18 .
<Addresses/ID=18> <Addresses#city> "Cambridge" .
<Addresses/ID=18> <Addresses#state> "MA" .

2.4 Additional Examples and Corner Cases

2.4.1 Foreign keys referencing candidate keys

More complex schemas include compound and composite primary keys. In this example, the columns deptName and deptCity in the People table reference name and city in the Department table. The following is the schema of the augmented database:

CREATE TABLE Addresses (
	ID INT, 
	city CHAR(10), 
	state CHAR(2), 
	PRIMARY KEY(ID)
)

CREATE TABLE Deparment (
	ID INT, 
	name CHAR(10), 
	city CHAR(10), 
	manager INT, 
	PRIMARY KEY(ID), 
	UNIQUE (name, city), 
	FOREIGN KEY(manager) REFERENCES People(ID)
)

CREATE TABLE People (
	ID INT, 
	fname CHAR(10), 
	addr INT, 
	deptName CHAR(10), 
	deptCity CHAR(10), 
	PRIMARY KEY(ID), 
	FOREIGN KEY(addr) REFERENCES Addresses(ID), 
	FOREIGN KEY(deptName, deptCity) REFERENCES Department(name, city) 
)

The following is an instance of the augmented relational schema:

People
PK		→ Addresses(ID)	→ Department(name, city)
ID	fname	addr	deptName	deptCity
7	Bob	18	accounting	Cambridge
8	Sue	NULL	NULL	NULL

Addresses
PK
ID	city	state
18	Cambridge	MA

Department
PK	Unique Key		→ People(ID)
ID	name	city	manager
23	accounting	Cambridge	8

Per the People tables's compound foreign key to Department:

The row in People with deptName="accounting" and deptCity="Cambridge" references a row in Department with a primary key of ID=23.
The predicate for this key is formed from "deptName,deptCity", reflecting the order of the column names in the foreign key.
The referent identifier (object of the above predicate) is formed from the base IRI and "ID=23".

In this example, the direct mapping generates the following triples:

@base <http://foo.example/DB/>
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

<People/ID=7> <rdf:type> <People> .
<People/ID=7> <People#ID> 7 .
<People/ID=7> <People#fname> "Bob" .
<People/ID=7> <People#addr> <Addresses/ID=18> .
<People/ID=7> <People#deptName> "accounting" .
<People/ID=7> <People#deptCity> "Cambridge" .
<People/ID=7> <People#deptName,deptCity> <Department/ID=23> .
<People/ID=8> <rdf:type> <People> .
<People/ID=8> <People#ID> 8 .
<People/ID=8> <People#fname> "Sue" .

<Addresses/ID=18> <rdf:type> <Addresses> .
<Addresses/ID=18> <Addresses#ID> 18 .
<Addresses/ID=18> <Addresses#city> "Cambridge" .
<Addresses/ID=18> <Addresses#state> "MA" .

<Department/ID=23> <rdf:type> <Department> .
<Department/ID=23> <Department#ID> 23 .
<Department/ID=23> <Department#name> "accounting" .
<Department/ID=23> <Department#city> "Cambridge" .
<Department/ID=23> <Department#manager> <People#ID=8> .

The green triples above are generated by considering the new elements in the augmented database. It should be noticed that:

Although deptName is an attribute of table People that is part of a foreign key, the Literal Triple <People/ID=7> <People#deptName> "accounting" is generated by the direct mapping because deptName is not the sole column of a foreign key of table People.
The Reference Triple <People/ID=7> <People#deptName,deptCity> <Department/ID=23> is generated by considering a foreign key referencing a candidate key (instead of the primary key): (deptName, deptCity) is a multi-column foreign key in the table People which references the multi-column candidate key (name, city) in the table Department.

2.4.2 Multi-column keys

We note that primary keys may also be composite. For example, if the primary key for Department were (name, city) instead of ID in the example in Section Foreign keys referencing candidate keys, then the identifier for the only row in this table would be <Department/name=accounting,city=Cambridge>, and the following triples would have been generated by the direct mapping:

<Department/name=accounting,city=Cambridge> <rdf:type> <Department> . 
<Department/name=accounting,city=Cambridge> <Department#ID> 23 . 
<Department/name=accounting,city=Cambridge> <Department#name> "accounting" .
<Department/name=accounting,city=Cambridge> <Department#city> "Cambridge" .

2.4.3 Empty (non-existent) primary keys

Even if there is no primary key, rows generate a set of triples with a shared subject, but that subject is a blank node. For instance, assume that the following table is added to the schema of the example in Section Foreign keys referencing candidate keys (for keeping track of tweets in Twitter):

CREATE TABLE Tweets (
	tweeter INT,
	when TIMESTAMP,
	text CHAR(140),
	FOREIGN KEY(tweeter) REFERENCES People(ID)
)

The following is an instance of table Tweets:

Tweets
→ People(ID)
tweeter	when	text
7	2010-08-30T01:33	I really like lolcats.
7	2010-08-30T09:01	I take it back.

Given that table Tweets does not have a primary key, each row in this table is identified by a Blank Node. In fact, when translating the above table the direct mapping generates the following triples:

@base <http://foo.example/DB/>
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

_:a <rdf:type> <Tweets> .
_:a <Tweets#tweeter> <People/ID=7> .
_:a <Tweets#when> "2010-08-30T01:33"^^xsd:dateTime .
_:a <Tweets#text> "I really like lolcats." .

_:b <rdf:type> <Tweets> .
_:b <Tweets#tweeter> <People/ID=7> .
_:b <Tweets#when> "2010-08-30T09:01"^^xsd:dateTime .
_:b <Tweets#text> "I take it back." .

It is not possible to dereference blank nodes ("_:a" and "_:b" above). Queries or updates may be made to these nodes via SPARQL queries.

2.4.4 Referencing tables with empty primary keys

Rows in tables with no primary key may still be referenced by foreign keys. (Relational database theory tells us that these rows must be unique as foreign keys reference candidate keys and candidate keys are unique across all the rows in a table.) References to rows in tables with no primary key are expressed as RDF triples with blank nodes for objects, where that blank node is the same node used for the subject in the referenced row.

This example includes several foreign keys with mutual column names. For clarity; here is the DDL to clarify these keys:

CREATE TABLE Projects (
	lead INT,
        FOREIGN KEY (lead) REFERENCES People(ID),
        name VARCHAR(50), 
        UNIQUE (lead, name), 
        deptName VARCHAR(50), 
        deptCity VARCHAR(50),
        UNIQUE (name, deptName, deptCity),
        FOREIGN KEY (deptName, deptCity) REFERENCES Department(name, city)
)

CREATE TABLE TaskAssignments (
	worker INT,
        FOREIGN KEY (worker) REFERENCES People(ID),
        project VARCHAR(50), 
        PRIMARY KEY (worker, project), 
        deptName VARCHAR(50), 
        deptCity VARCHAR(50),
        FOREIGN KEY (worker) REFERENCES People(ID),
        FOREIGN KEY (project, deptName, deptCity) REFERENCES Projects(name, deptName, deptCity),
        FOREIGN KEY (deptName, deptCity) REFERENCES Department(name, city)
)

The following is an instance of the preceding schema:

Projects
Unique key
	Unique key
→ People(ID)		→ Department(name, city)
lead	name	deptName	deptCity
8	pencil survey	accounting	Cambridge
8	eraser survey	accounting	Cambridge

TaskAssignments
PK
	→ Projects(name, deptName, deptCity)
→ People(ID)		→ Departments(name, city)
worker	project	deptName	deptCity
7	pencil survey	accounting	Cambridge

In this case, the direct mapping generates the following triples from the preceding tables:

@base <http://foo.example/DB/>
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .
@prefix pencil: <http://foo.example/DB/TaskAssignment/worker=7,project=pencil+survey> .

_:c <rdf:type> <Projects> .
_:c <Projects#lead> <People/ID=8> .
_:c <Projects#name> "pencil survey" .
_:c <Projects#deptName> "accounting" .
_:c <Projects#deptCity> "Cambridge" .
_:c <Projects#deptName,deptCity> <Department/ID=23> .

_:d <rdf:type> <Projects> .
_:d <Projects#lead> <People/ID=8> .
_:d <Projects#name> "eraser survey" .
_:d <Projects#deptName> "accounting" .
_:d <Projects#deptCity> "Cambridge" .
_:d <Projects#deptName,deptCity> <Department/ID=23> .

pencil:_ <rdf:type> <TaskAssignments> .
pencil:_ <TaskAssignments#worker> <People/ID=7> .
pencil:_ <TaskAssignments#project> "pencil survey" .
pencil:_ <TaskAssignments#deptName> "accounting" .
pencil:_ <TaskAssignments#deptCity> "Cambridge" .
pencil:_ <TaskAssignments#deptName,deptCity> <Department/ID=23> .
pencil:_ <TaskAssignments#project,deptName,deptCity> _:c .

The absence of a primary key forces the generation of blank nodes, but does not change the structure of the direct graph or names of the predicates in that graph.

2.5 Hierarchical Tables

It is common to express specializations of some concept as multiple tables sharing a common primary key. In such cases, the primary keys of the inherited tables are in turn foreign keys to the table from which they derive.

Addresses
PK
ID	city	state
18	Cambridge	MA

Offices
PK
→ Addresses(ID)
ID	building	ofcNumber
18	32	G528

ExecutiveOffices
PK
→ Offices(ID)
ID	desk
18	oak

In this example, Offices are a specialization of Addresses and ExecutiveOffices are a specialization of Offices. The subjects for the triples implied by rows in Offices or ExecutiveOffices are the same as those for the corresponding row in Addresses.

@base <http://foo.example/DB/>
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

<Addresses/ID=18> <Addresses#ID> 18 .
<Addresses/ID=18> <Addresses#city> "Cambridge" .
<Addresses/ID=18> <Addresses#state> "MA" .

<Addresses/ID=18> <Offices#ID> 18 .
<Addresses/ID=18> <Offices#building> 32 .
<Addresses/ID=18> <Offices#ofcNumber> "G528" .

<Addresses/ID=18> <ExecutiveOffices#ID> 18 .
<Addresses/ID=18> <ExecutiveOffices#desk> "oak" .

The Primary-is-foreign Key Exception allows the generation of a triple with an RDF literal for the ID column in the Offices and ExecutiveOffices table (Offices.ID=18 and ExecutiveOffices.ID=18).

Issue (hier-table-at-risk):

This feature attempts to intricately model some existing modeling practice but adds significant complexity. This feature is at risk.

Resolution:

None recorded.

Issue (fk-pk-order):

What if fk is a rearrangement of the pk? E.g what if TaskAssignments, with a primary key (project, worker), had a foreign key (worker, project)?

Resolution:

None recorded.

Issue (many-to-many-as-repeated-properties):

The direct graph is arguably more faithful to the conceptual model if it reflects e.g. a person with multiple addresses (some many-to-many Person2Address table) as repeated properties. It is difficult to detect which tables with exactly two foreign keys and no other attributes are many-to-many. As a counter example, a Wedding table may have exactly two spouses but it's still not a many-to-many relation in most places.

Resolution:

None recorded.

Issue (formalism-model):

The RDB2RDF working group has not decided on a formalism for representing the direct mapping. We would appreciate feedback from the community in helping us choose between Section 5. Direct Mapping Definition and Section 6. Direct Mapping as Rules.

Resolution:

None recorded.

3 Direct Mapping Definition

3.1 Notations

The RDB and RDF data models make use of the commonly defined Abstract Data Types Set, List and MultiSet, used here as type constructors. For example, Set(A) denotes the type for the sets of elements of type A. We assume that they come with their common operations, such as the function size : Set → Int.

The definitions follow a type-as-specification approach, thus the models are based on dependent types. For example, { s:Set(A) | size(s) ≤ 1 } is a type denoting the sets for elements of type A, such that those sets have at most one element.

The denotational RDF semantics makes use of the set-builder notation for building the RDF sets.

The buttons below can be used to show or hide the available syntaxes.

3.2 Relational Data Model

3.2.1 RDB Abstract Data Type (Normative)

[1]	`Database`	::=	`Set(Table)`
	A relational database is a set of tables.
[2]	`Table`	::=	`(TableName, Header, List(CandidateKey), Set(ForeignKey), Body)`
	A relation has a name uniquely defining this table in the database; a header describing the domain of the data; a potentially empty list of candidate keys, possibly including a primary key; a potentially empty set of foreign keys; a body containing the rows of data.
[3]	`Header`	::=	`Set((ColumnName, Datatype))`
	A header is an associative array mapping each column to a SQL datatype.
[4]	`Body`	::=	`MultiSet(Row)`
	A body is a set of (potentially duplicate) rows.
[5]	`Row`	::=	`Set((ColumnName, CellValue))`
	A row is a associative array mapping each column in a row to a value.
[6]	`CellValue`	::=	`LexicalValue \| NULL`
	A cell value is either a lexical value or NULL, denoting the absence of value.
[7]	`ForeignKey`	::=	`(List(ColumnName), Table, CandidateKey)`
	A foreign key relies on a list of columns (their order matters) and points to a candidate key into another table.
[8]	`CandidateKey`	::=	`List(ColumnName)`
	A candidate key is made of a list of columns (their order matters).
[9]	`Datatype`	::=	`Int \| Float \| Date \| …`
	A datatype is a common SQL datatype.
[10]	`TableName`	::=	`String`
	A table name is a string.
[11]	`ColumnName`	::=	`String`
	A column name is a string.

3.2.2 RDB accessor functions (Normative)

[12]	`tablename`	:	`Table → TableName`
	Given a table, tablename returns its name.
[13]	`header`	:	`Table → Header`
	Given a table, header returns its header.
[14]	`candidateKeys`	:	`Table → List(CandidateKey)`
	Given a table, candidateKeys returns the list of candidate keys.
[15]	`primaryKey`	:	`Table → { s:Set(CandidateKey) \| size(s) ≤ 1 }`
	Given a table, primaryKey returns a set containing the primary key if it exists, otherwise it returns an empty set.
[16]	`foreignKeys`	:	`Table → Set(ForeignKey)`
	Given a table, foreignKeys returns the set of foreign keys.
[17]	`unary`	:	`ForeignKey → Boolean`
	Given a foreign key, unary tells if this is a unary foreign key, meaning it has exactly one column.
[18]	`lexicals`	:	`Table → Set({ c:ColumnName \| ! unary(c) })`
	Given a table, lexicals returns the set of columns that do not constitute a unary foreign key.
[19]	`body`	:	`Table → Body`
	Given a table, body returns its body.
[20]	`datatype`	:	`{ h:Header } → { c:ColumnName \| ∃ d, (c,d) ∈ h } → { d:Datatype \| (c,d) ∈ h }`
	Given a header and a column in this header, datatype returns the datatype associated with this column.
[21]	`table`	:	`{ r:Row } → { t:Table \| r ∈ t }`
	Given a row, table returns the table to which this row belongs.
[22]	`value`	:	`{ r:Row } → { a:ColumnName \| a ∈ r } → CellValue`
	Given a row and a column in this row, value returns the cell value (can be NULL) for this column.
[23]	`dereference`	:	{ r:Row } → { fk:ForeignKey \| fk ∈ foreignKeys(table(r)) } → { targetRow:Row \| let (columnNames, targetTable, ck) = fk in targetRow ∈ body(targetTable) and ∀ c_i^fk ∈ columnNames, ∀ c_j^ck ∈ ck, ∀ (c_k^r, v_k^r) ∈ r, ∀ (c_l^target, v_l^target) ∈ targetRow, i = j → c_i^fk = c_k^r → c_j^ck = c_l^target → v_k^r = v_l^target }
	Given a row and a foreign key from the table containing this row, dereference returns the row which is referenced by this foreign key. The values for the foreign key columns in `r` equal the values of the referenced columns in the returned row.

3.3 RDF Data Model (Non-normative)

Per RDF Concepts and Abstract Syntax, an RDF graph is a set of triples of a subject, predicate and object. The subject may be an IRI or a blank node, the predicate must be an IRI and the object may be an IRI, blank node, or an RDF literal.

This section recapitulates for convience the formal definition of RDF.

[24]	`Graph`	::=	`Set(Triple)`
	An RDF graph is a set of RDF triples.
[25]	`Triple`	::=	`(Subject, Predicate, Object)`
	An RDF triple is composed of a subject, predicate and object.
[26]	`Subject`	::=	`IRI \| BlankNode`
	A subject is either an IRI or a blank node.
[27]	`Predicate`	::=	`IRI`
	A predicate is always an IRI.
[28]	`Object`	::=	`IRI \| BlankNode \| Literal`
	An object is either an IRI, a blank node, or a literal.
[29]	`BlankNode`	::=	`RDF blank node`
	A blank node is an arbitrary term used only to establish graph connectivity.
[30]	`Literal`	::=	`PlainLiteral \| TypedLiteral`
	A literal is either a plain literal or a typed literal.
[31]	`PlainLiteral`	::=	`lexicalForm \| (lexicalForm, langageTag)`
	A plain literal has a lexical form and an optional language tag.
[32]	`TypedLiteral`	::=	`(lexicalForm, IRI)`
	An typed literal is composed of lexical form and a datatype IRI.
[33]	`IRI`	::=	`RDF URI-reference as subsequently restricted by SPARQL`
	An IRI is an RDF URI reference as subsequently restricted by SPARQL.
[34]	`lexicalForm`	::=	`a Unicode String`
	SQL string representing a value.

3.4 Denotational semantics (Normative)

In this model, Databases are inhabitants of RDB and they are denoted by mathematical objects living in the RDF domain. This denotational semantics is what we call the Direct Mapping.

Most of the functions defining the Direct Mapping are higher-order functions parameterized by a function φ : Row → Node. This function maps any row to a unique node IRI or Blank Node. φ is formally defined by the following axioms:

[35]	`row-iri`	:	`∀ db:Database, ∀ r:Row, r ∈ db → primaryKey(table(r)) ≠ ∅ → φ(r) is an IRI`
	If a row belongs to a table with a primary key then `φ` maps this row to a unique RDF IRI.
[36]	`row-blanknode`	:	`∀ db:Database, ∀ r:Row, r ∈ db → primaryKey(table(r)) = ∅ → φ(r) is a BlankNode`
	If a row belongs to a table with no primary key then `φ` maps this row to a unique BlankNode.

The Direct Mapping is defined by induction on the structure of RDB. Thus it is defined for any database. The entry point for the Direct Mapping is the function ⟦ ⟧^φ_database.

[37]	`⟦ ⟧^φ_database`	:	`Database → Graph`
	A mapping from a relational database to an RDF graph.
[38]	`⟦db⟧^φ_database`	=	`{ triple \| triple ∈ ⟦t⟧^φ_table \| t ∈ db }`
	The union of the triples expressing each table in the database.
[39]	`⟦ ⟧^φ_table`	:	`Table → Set(Triple)`
	A mapping from a table to a set of RDF triples.
[40]	`⟦t⟧^φ_table`	=	`{ triple \| triple ∈ ⟦r⟧^φ_row \| r ∈ body(t) }`
	The triples expressing each row in the table `t`
[41]	`⟦ ⟧^φ_row`	:	`Row → Set(Triple)`
	A mapping from a row to a set of RDF triples.
[42]	`⟦r⟧^φ_row`	=	`let s = φ(r) in { (s, p, o) \| (p, o) ∈ ⟦r, fk⟧^φ_ref \| fk ∈ foreignKeys(table(r)) } ⋃ { (s, p, o) \| (p, o) ∈ ⟦r, c⟧^φ_lex \| c ∈ lexicals(r) } ⋃ { (s, rdf:type, ue(tablename(table(r)))) }`
	The union of the triples coming from the foreign keys the lexical values (not contributing to a unary foreign key) the table name, which denotes an RDF type information
[43]	`⟦ , ⟧^φ_ref`	:	`(Row, ForeignKey) → (Predicate, Object)`
	A mapping from a foreign key in a row to an RDF predicate and an RDF object.
[44]	`⟦r, fk⟧^φ_ref`	=	`let p = ⟦table(r), fk⟧_col in let targetRow = dereference(r, fk) in let o = φ(targetRow) in (p, o)`
	The predicate based on the column name and the object refered by the foreign key `fk`.
[45]	`⟦ , ⟧_lex`	:	`(Row, Column) → { s:Set((Predicate, Object)) \| size(s) ≤ 1 }`
	A mapping from a column in a row to an optional pair of RDF predicate and object.
[46]	`⟦r, c⟧_lex`	=	`let p = ⟦table(r), fk⟧_col in let v = value(r, c) in let d = datatype(header(table(r))(c)) in if v is NULL then ∅ else if d is String then {(p, v)} else let datatype_iri = ⟦d⟧_datatype in {(p, (v, datatype_iri))}`
	If the cell value for this column is NULL: nothing; Otherwise: a predicate based on the column name and a typed literal made of the value in the cell plus the corresponding RDF datatype.
[47]	`⟦ , ⟧_col`	:	`(Row, List[Column]) → IRI`
	A mapping from a list of columns in a row to an IRI.
[48]	`⟦r, c*⟧_col`	=	`ue(tablename(table(r))) + '#' + ue(c₀) + ',' + ⋯ + ',' + ue(c_n-1)`
	A concatenation, with punctuation as separators, of the url-encoded table name and the url-encoded column names.
[49]	`⟦ ⟧_datatype`	:	`Datatype → IRI`
	A mapping from a SQL datatype to an XML Schema datatype IRI.
[50]	`⟦d⟧_datatype`	=	`if d is Int then XSD:integer else if d is Float then XSD:float else if d is Date then XSD:date ⋯`
	The XML Schema datatype for `d` as defined by IWD 9075 §9.5
[51]	`ue`	:	`String → String`
	An URL encoding per WSDL urlEncoded.

4 Direct Mapping as Rules (Normative)

In this section, we formally present the Direct Mapping as rules in Datalog syntax. The left hand side of each rule is the RDF Triple output. The right hand side of each rule consists of a sequence of predicates from the relational database and built-in predicates. The built-in predicates are divided into three groups. The first group contains some built-in predicates for dealing with repeated rows in a table without a primary key.

card(r, l, k): Given a table name r without a primary key and the list l of values [v₁, ..., v_n] for a row of table r, it returns in k the multiplicity of l in r (that is, k is the number of times row l appears in r)
n ≤ m: This is the usual order on positive integer values (given positive integers n and m, it holds if n is smaller than or equal to m)

The second group contains a predicate to deal with null values.

nonNull(v): Given a value v, it holds if v is not null

Finally, the third group of built-in predicates is used to generate IRIs for identifying tables and the columns in a table, and to generate IRIs or blank nodes for identifying each row in a table.

generateTableIRI(r, i): Given a table name r, it generates the Table IRI i of r
generateColumnIRI(r, l, i): Given a table name r and a non-empty list of columns l, it generates the Column IRI i for l
generateRowIRI(r, l₁, l₂, i): Given a table name r, a non-empty list l₁ of columns and a non-empty list l₂ of values (for the columns in l₁), it generates the Row RDF Node (or Row IRI) i for the given row
generateRowBlankNode(r, l, n, i): Given a table name r without a primary key, a list l of values for a row of table r and a positive integer n, it generates the Row RDF Node i for the n-th occurrence of row l in r (which is a Blank Node in this case). It is assumed that n is smaller than or equal to the multiplicity of l in r (that is, if card(r, l, k) holds, then 1 ≤ n ≤ k)

Consider again the example from Section Direct Mapping Example. It should be noticed that in the rules presented in this section, a formula of the form Addresses(X, Y, Z) indicates that the variables X, Y and Z are used to store the values of a row in the three columns of the table Addresses (according to the order specified in the schema of the table, that is, X, Y and Z store the values of ID, city and state, respectively). In particular, uppercase letters like X, Y, Z, S, P and O are used to denote variables. Moreover, double quotes are used in the rules to refer to the string with the name of a table or a column. For example, a formula of the form generateRowIRI("Addresses", ["ID"], [X], S) is used to generate the Row RDF Node (or Row IRI) for the row of table "Addresses" whose value in the primary key "ID" is the value stored in the variable X. The value of this Row IRI is stored in the variable S.

4.1 Generating Table Triples

4.1.1 Table has a primary key

Assume that r is a table with columns a₁, ..., a_m and such that [a₁, ..., a_n] is the primary key of r, where 1 ≤ n ≤ m. Then the following is the direct mapping rule to generate Table Triples from r:

Triple(S, "rdf:type", O) ← r(X₁, ..., X_m), generateRowIRI("r", ["a₁", ..., "a_n"], [X₁, ..., X_n], S), generateTableIRI("r", O)

For example, table Addresses in the Direct Mapping Example has columns ID, city and state, and it has column ID as its primary key. Then the following is the direct mapping rule to generate Table Triples from Addresses:

Triple(S, "rdf:type", O) ← Addresses(X₁, X₂, X₃), generateRowIRI("Addresses", ["ID"], [X₁], S), generateTableIRI("Addresses", O)

As a second example, consider table Department from the example in Section Foreign keys referencing candidate keys, which has columns ID, name, city and manager, and assume that (name, city) is the multi-column primary key of this table (instead of ID). Then the following is the direct mapping rule to generate Table Triples from Department:

Triple(S, "rdf:type", O) ← Department(X₃, X₁, X₂, X₄), generateRowIRI("Department", ["name","city"], [X₁, X₂], S), generateTableIRI("Department", O)

4.1.2 Table does not have a primary key

Assume that r is a table with columns a₁, ..., a_m and such that r does not have a primary key. Then the following is the direct mapping rule to generate Table Triples from r:

Triple(S, "rdf:type", O) ← r(X₁, ..., X_m), card("r", [X₁, ..., X_m], U), V ≤ U, generateRowBlankNode("r", [X₁, ..., X_m], V, S), 
                            generateTableIRI("r", O)

For example, table Tweets from Section Empty (non-existent) primary keys has columns tweeter, when and text, and it does not have a primary key. Then the following is the direct mapping rule to generate Table Triples from Tweets:

Triple(S, "rdf:type", O) ← Tweets(X₁, X₂, X₃), card("Tweets", [X₁, X₂, X₃], U), V ≤ U, generateRowBlankNode("Tweets", [X₁, X₂, X₃], V, S), 
                            generateTableIRI("Tweets", O)

4.2 Generating Literal Triples

4.2.1 Table has a primary key

Assume that r is a table with columns a₁, ..., a_m and such that [a₁, ..., a_n] is the primary key of r, where 1 ≤ n ≤ m. Then for every a_j (1 ≤ j ≤ m) that is not the only constituent of a foreign key of r or is the only constituent of a foreign key of r that references a candidate key, the direct mapping includes the following rule for r and a_j to generate Literal Triples:

Triple(S, P, X_j) ← r(X₁, ..., X_m), nonNull(X_j), generateRowIRI("r", ["a₁", ..., "a_n"], [X₁, ..., X_n], S), generateColumnIRI("r", ["a_j"], P)

For example, table Addresses in the Direct Mapping Example has columns ID, city and state, and it has column ID as its primary key. Then the following are the direct mapping rules to generate Literal Triples from Addresses:

Triple(S, P, X₁) ← Addresses(X₁, X₂, X₃), nonNull(X₁), generateRowIRI("Addresses", ["ID"], [X₁], S), generateColumnIRI("Addresses", ["ID"], P)
Triple(S, P, X₂) ← Addresses(X₁, X₂, X₃), nonNull(X₂), generateRowIRI("Addresses", ["ID"], [X₁], S), generateColumnIRI("Addresses", ["city"], P)
Triple(S, P, X₃) ← Addresses(X₁, X₂, X₃), nonNull(X₃), generateRowIRI("Addresses", ["ID"], [X₁], S), generateColumnIRI("Addresses", ["state"], P)

As a second example, consider again table Department from the example in Section Foreign keys referencing candidate keys, which has columns ID, name, city and manager, and assume that (name, city) is the multi-column primary key of this table (instead of ID). Then the following are the direct mapping rules to generate Literal Triples from Department:

Triple(S, P, X₁) ← Department(X₃, X₁, X₂, X₄), nonNull(X₁), generateRowIRI("Department", ["name", "city"], [X₁, X₂], S), 
                   generateColumnIRI("Department", ["name"], P)
Triple(S, P, X₂) ← Department(X₃, X₁, X₂, X₄), nonNull(X₂), generateRowIRI("Department", ["name", "city"], [X₁, X₂], S), 
                   generateColumnIRI("Department", ["city"], P)
Triple(S, P, X₃) ← Department(X₃, X₁, X₂, X₄), nonNull(X₃), generateRowIRI("Department", ["name", "city"], [X₁, X₂], S), 
                   generateColumnIRI("Department", ["ID"], P)

It is important to notice that no rule is generated from column manager, as this column is the only constituent of a foreign key that references a primary key: FOREIGN KEY(manager) REFERENCES People(ID).

4.2.2 Table does not have a primary key

Assume that r is a table with columns a₁, ..., a_m and such that r does not have a primary key. Then for every a_j (1 ≤ j ≤ m) that is not the only constituent of a foreign key of r or is the only constituent of a foreign key of r that references a candidate key, the direct mapping includes the following rule for r and a_j to generate Literal Triples:

Triple(S, P, X_j) ← r(X₁, ..., X_m), nonNull(X_j), card("r", [X₁, ..., X_m], U), V ≤ U, generateRowBlankNode("r", [X₁, ..., X_m], V, S), 
                   generateColumnIRI("r", ["a_j"], P)

For example, table Tweets from Section Empty (non-existent) primary keys has columns tweeter, when and text, and it does not have a primary key. Then the following are the direct mapping rules to generate Literal Triples from Tweets:

Triple(S, P, X₂) ← Tweets(X₁, X₂, X₃), nonNull(X₂), card("Tweets", [X₁, X₂, X₃], U), V ≤ U, generateRowBlankNode("Tweets", [X₁, X₂, X₃], V, S), 
                   generateColumnIRI("Tweets", ["when"], P)
Triple(S, P, X₃) ← Tweets(X₁, X₂, X₃), nonNull(X₃), card("Tweets", [X₁, X₂, X₃], U), V ≤ U, generateRowBlankNode("Tweets", [X₁, X₂, X₃], V, S), 
                   generateColumnIRI("Tweets", ["text"], P)

It is important to notice that no rule is generated from column tweeter, as this column is the only constituent of a foreign key that references a primary key: FOREIGN KEY(tweeter) REFERENCES People(ID).

4.3 Generating Reference Triples

For each foreign key from a table r₁ to a table r₂, one of the following four cases is applied.

4.3.1 Table r₁ has a primary key and table r₂ has a primary key

Assume that:

r₁ is a table with columns a₁, ..., a_i and such that [a₁, ..., a_j] is the primary key of r₁, where 1 ≤ j ≤ i
r₂ is a table with columns c₁, ..., c_k and such that [c₁, ..., c_m] is the primary key of r₂, where 1 ≤ m ≤ k
the foreign key indicates that the columns a_p₁, ..., a_{p_n} of r₁ reference the columns c_q₁, ..., c_{q_n} of r₂, where (1) 1 ≤ p₁, ..., p_n ≤ i, (2) 1 ≤ q₁, ..., q_n ≤ k, and (3) n ≥ 1

Then the direct mapping includes the following rule for r₁ and r₂ to generate Reference Triples:

Triple(S, P, O) ← r₁(X₁, ..., X_i), generateRowIRI("r₁", ["a₁", ..., "a_j"], [X₁, ..., X_j], S), 
                   r₂(U₁, ..., U_k), generateRowIRI("r₂", ["c₁", ..., "c_m"], [U₁, ..., U_m], O), 
                   nonNull(X_p₁), ..., nonNull(X_{p_n}), X_p₁ = U_q₁, ...,  X_{p_n} = U_{q_n},  generateColumnIRI("r₁", ["a_p₁", ..., "a_{p_n}"], P)

For example, ... to-do ...

 ... to-do ...

4.3.2 Table r₁ has a primary key and table r₂ does not have a primary key

Assume that:

r₁ is a table with columns a₁, ..., a_i and such that [a₁, ..., a_j] is the primary key of r₁, where 1 ≤ j ≤ i
r₂ is a table with columns c₁, ..., c_k, and it does not have a primary key
the foreign key indicates that the columns a_p₁, ..., a_{p_n} of r₁ reference the columns c_q₁, ..., c_{q_n} of r₂, where (1) 1 ≤ p₁, ..., p_n ≤ i, (2) 1 ≤ q₁, ..., q_n ≤ k, and (3) n ≥ 1

Then the direct mapping includes the following rule for r₁ and r₂ to generate Reference Triples:

Triple(S, P, O) ← r₁(X₁, ..., X_i), generateRowIRI("r₁", ["a₁", ..., "a_j"], [X₁, ..., X_j], S), 
                   r₂(U₁, ..., U_k), card("r₂", [U₁, ..., U_k], V₁), V₂ ≤ V₁, generateRowBlankNode("r₂", [U₁, ..., U_k], V₂, O), 
                   nonNull(X_p₁), ..., nonNull(X_{p_n}), X_p₁ = U_q₁, ...,  X_{p_n} = U_{q_n},  generateColumnIRI("r₁", ["a_p₁", ..., "a_{p_n}"], P)

For example, ... to-do ...

 ... to-do ...

4.3.3 Table r₁ does not have primary key and table r₂ has a primary key

Assume that:

r₁ is a table with columns a₁, ..., a_i, and it does not have a primary key
r₂ is a table with columns c₁, ..., c_k and such that [c₁, ..., c_m] is the primary key of r₂, where 1 ≤ m ≤ k
the foreign key indicates that the columns a_p₁, ..., a_{p_n} of r₁ reference the columns c_q₁, ..., c_{q_n} of r₂, where (1) 1 ≤ p₁, ..., p_n ≤ i, (2) 1 ≤ q₁, ..., q_n ≤ k, and (3) n ≥ 1

Then the direct mapping includes the following rule for r₁ and r₂ to generate Reference Triples:

Triple(S, P, O) ← r₁(X₁, ..., X_i), card("r₁", [X₁, ..., X_i], V₁), V₂ ≤ V₁, generateRowBlankNode("r₁", [X₁, ..., X_i], V₂, S), 
                   r₂(U₁, ..., U_k), generateRowIRI("r₂", ["c₁", ..., "c_m"], [U₁, ..., U_m], O), 
                   nonNull(X_p₁), ..., nonNull(X_{p_n}), X_p₁ = U_q₁, ...,  X_{p_n} = U_{q_n},  generateColumnIRI("r₁", ["a_p₁", ..., "a_{p_n}"], P)

For example, ... to-do ...

 ... to-do ...

4.3.4 Table r₁ does not have primary key and table r₂ does not have a primary key

Assume that:

r₁ is a table with columns a₁, ..., a_i, and it does not have a primary key
r₂ is a table with columns c₁, ..., c_k, and it does not have a primary key
the foreign key indicates that the columns a_p₁, ..., a_{p_n} of r₁ reference the columns c_q₁, ..., c_{q_n} of r₂, where (1) 1 ≤ p₁, ..., p_n ≤ i, (2) 1 ≤ q₁, ..., q_n ≤ k, and (3) n ≥ 1

Then the direct mapping includes the following rule for r₁ and r₂ to generate Reference Triples:

Triple(S, P, O) ← r₁(X₁, ..., X_i), card("r₁", [X₁, ..., X_i], V₁), V₂ ≤ V₁, generateRowBlankNode("r₁", [X₁, ..., X_i], V₂, S), 
                   r₂(U₁, ..., U_k), card("r₂", [U₁, ..., U_k], V₃), V₄ ≤ V₃, generateRowBlankNode("r₂", [U₁, ..., U_k], V₄, O), 
                   nonNull(X_p₁), ..., nonNull(X_{p_n}), X_p₁ = U_q₁, ...,  X_{p_n} = U_{q_n},  generateColumnIRI("r₁", ["a_p₁", ..., "a_{p_n}"], P)

For example, ... to-do ...

 ... to-do ...

A Direct Mapping of Relational Data to RDF

W3C Working Draft 24 March 2011

Abstract

Status of this Document

Table of Contents

Appendix

1 Introduction

2 Direct Mapping Description (Informative)

2.1 Direct Mapping Example

2.2 Preliminaries: Generating IRIs

2.2.1 IRIs generated for the initial example

2.3 Mapping Rules

2.3.1 Triples generated for the example in Section Direct Mapping Example

2.4 Additional Examples and Corner Cases

2.4.1 Foreign keys referencing candidate keys

2.4.2 Multi-column keys

2.4.3 Empty (non-existent) primary keys

2.4.4 Referencing tables with empty primary keys

2.5 Hierarchical Tables

3 Direct Mapping Definition

3.1 Notations

3.2 Relational Data Model

3.2.1 RDB Abstract Data Type (Normative)

3.2.2 RDB accessor functions (Normative)

3.3 RDF Data Model (Non-normative)

3.4 Denotational semantics (Normative)

4 Direct Mapping as Rules (Normative)

4.1 Generating Table Triples

4.1.1 Table has a primary key

4.1.2 Table does not have a primary key

4.2 Generating Literal Triples

4.2.1 Table has a primary key

4.2.2 Table does not have a primary key

4.3 Generating Reference Triples

4.3.1 Table r₁ has a primary key and table r₂ has a primary key

4.3.2 Table r₁ has a primary key and table r₂ does not have a primary key

4.3.3 Table r₁ does not have primary key and table r₂ has a primary key

4.3.4 Table r₁ does not have primary key and table r₂ does not have a primary key

5 References

A CVS History

A Direct Mapping of Relational Data to RDF

W3C Working Draft 24 March 2011

Abstract

Status of this Document

Table of Contents

Appendix

1 Introduction

2 Direct Mapping Description (Informative)

2.1 Direct Mapping Example

2.2 Preliminaries: Generating IRIs

2.2.1 IRIs generated for the initial example

2.3 Mapping Rules

2.3.1 Triples generated for the example in Section Direct Mapping Example

2.4 Additional Examples and Corner Cases

2.4.1 Foreign keys referencing candidate keys

2.4.2 Multi-column keys

2.4.3 Empty (non-existent) primary keys

2.4.4 Referencing tables with empty primary keys

2.5 Hierarchical Tables

3 Direct Mapping Definition

3.1 Notations

3.2 Relational Data Model

3.2.1 RDB Abstract Data Type (Normative)

3.2.2 RDB accessor functions (Normative)

3.3 RDF Data Model (Non-normative)

3.4 Denotational semantics (Normative)

4 Direct Mapping as Rules (Normative)

4.1 Generating Table Triples

4.1.1 Table has a primary key

4.1.2 Table does not have a primary key

4.2 Generating Literal Triples

4.2.1 Table has a primary key

4.2.2 Table does not have a primary key

4.3 Generating Reference Triples

4.3.1 Table r1 has a primary key and table r2 has a primary key

4.3.2 Table r1 has a primary key and table r2 does not have a primary key

4.3.3 Table r1 does not have primary key and table r2 has a primary key

4.3.4 Table r1 does not have primary key and table r2 does not have a primary key

5 References

A CVS History

4.3.1 Table r₁ has a primary key and table r₂ has a primary key

4.3.2 Table r₁ has a primary key and table r₂ does not have a primary key

4.3.3 Table r₁ does not have primary key and table r₂ has a primary key

4.3.4 Table r₁ does not have primary key and table r₂ does not have a primary key