6. Tuple Tables

As explained in Section 4, a data store uses tuple tables as containers for facts – that is, triples and other kind of data that RDFox® should process. Each tuple table is identified by a name that is unique for a data store. Moreover, each tuple table has a minimal and maximal arity, which are numbers determining the smallest and the largest numbers of RDF resources in a fact stored in the tuple table. In most cases, the minimal and maximal arity are the same, in which case they are called just arity.

6.1. Types of Tuple Tables

RDFox supports three kinds of tuple tables.

• In-memory tuple tables are the most commonly used kind of tuple table, which, as the name suggests, store facts in RAM. The RDF dataset of a data store is represented using the in-memory tuple tables DefaultTriples and Quads. Tuple table DefaultTriples has arity three and contains the triples of the default graph, and tuple table Quads has arity four and contains the triples of every named graph. RDFox provides ways to add and delete facts in in-memory tuple tables. See Section 5.2 for more detail on how RDFox stores RDF datasets.

• Built-in tuple tables contain facts that can be useful in various applications of RDFox. Their content is determined by RDFox and cannot be modified by users. They are described in more detail in Section 6.5.

• Data source tuple tables provide a ‘virtual view’ over data in non-RDF data sources, such as CSV files, relational databases, or a full-text Apache Solr index. Such tuple tables must be created explicitly by the user, and doing so requires specifying how the external data is to be transformed into a format compatible with RDF. The facts in data source tuple tables are ‘virtual’ in the sense that they are constructed automatically by RDFox based on the data in the data source — that is, there is no way to add/delete such facts directly. Finally, data source tuple tables can be of arbitrary arity — that is, such tuple tables are not limited to containing just triples. Data source tuple tables and the process of importing external data are described in detail in Section 7.

6.2. Fact Domains

Each fact in a tuple is associated with one or more fact domains.

• The explicit fact domain contains facts that were imported explicitly by the user.

• The derived fact domain contains facts that were not imported explicitly by the user, but were derived by a rule.

• The all fact domain contains all facts — that is, all is the union of explicit and derived.

A fact can belong to more than one domain. For example, facts added to the store are stored into the explicit domain, and during reasoning they are transferred into the all domain.

Only the explicit fact domain can be directly affected by users. That is, all explicitly added facts are added to the explicit domain, and only those facts can be deleted. It is not possible to manually delete derived facts since the meaning of such deletions is unclear.

Many RDFox operations accept a fact domain as an argument. For example, SPARQL query evaluation takes a fact domain as an argument, which determines what subset of the facts the query should be evaluated over. Thus, if a query is evaluated with respect to the explicit domain, it will ‘see’ only the facts that were explicitly added to a data store, and it will ignore the facts that were derived by reasoning.

6.3. Managing and Using Tuple Tables

RDFox provide ways for creating and deleting tuple tables: this can be accomplished in the shell using the tupletable command (see Section 15.2.51) and the corresponding APIs described in Section 16.11. When creating a tuple table, one must specify a list of key-value parameters that determine what kind of tuple table is to be created. The parameters for data source tuple tables depend on the type of data source and are described in detail in Section 7. Moreover, the parameters for in-memory and built-in tuple tables are described in Section 6.4 and Section 6.5, respectively.

Facts in tuple tables can be accessed during querying and reasoning. In queries, tuple tables DefaultTriples and Quads can be accessed using standard SPARQL syntax for querying RDF datasets. Access to arbitrary tuple tables in the data store is established using the proprietary SPARQL operator TT and the reserved IRI rdfox:TT, both of which are described in Section 9.4. In rules, tuple tables DefaultTriples and Quads can be accessed using the dedicated syntax for default graph atoms (see Section 10.4.1.1) and named graph atoms (see Section 10.4.1.2), respectively. Access to arbitrary tuple tables is established using the general atom syntax described in Section 10.4.1.3.

RDFox provides different ways of updating the content of in-memory tuple tables. The content of tuple tables DefaultTriples and Quads can be updated by adding or removing RDF data using the shell command import (see Section 15.2.25) and the corresponding APIs described in Section 16.9. Similarly, one can update the content of arbitrary in-memory tuple tables by importing facts using the Datalog general atom syntax (see Section 10.4.1.3). Another way of updating the content of in-memory tuple tables is to use the SPARQL Update Language. One can use standard syntax to update the tuple tables DefaultTriples and Quads, and they can use the operator TT and the reserved IRI rdfox::TT (see Section 9.4) to update the content of arbitrary in-memory tuple tables. The final way of updating the content of in-memory tuple tables is via reasoning. For example, adding an OWL ontology to the default graph or to a named graph will derive facts in DefaultTriples and Quads, respectively. Similarly, adding a set of Datalog rules to the data store, will update the tuple tables referenced in their head atoms.

6.4. In-Memory Tuple Tables

RDFox uses in-memory tuple tables to store facts imported by the users. In-memory tuple tables DefaultTriples and Quads are created automatically when a fresh data store is created to represent the default RDF dataset of the data store. Moreover, in-memory tuple tables can be created and deleted using the tuple table management APIs. During table creation, the user can specify the following parameters.

Parameter	Default value	Description
type	—	Specifies the type of the in-memory tuple table. The available options are unary-table, binary-table, triple-table, quad-table-lg, and quad-table-sg.
max-tuple-capacity	(as in the data store)	Specifies the maximum number of tuples that the new tuple table will be able to hold. The main purpose of this parameter is to reduce the amount of address space that the tuple table will use. The default value is the value of the data store parameter with the same name.
init-tuple-capacity	(as in the data store)	Provides a hint as to how many facts the system should expect to store initially in the tuple table. When importing large data sets, setting this parameter to be roughly equal to the number of facts to be imported could improve the speed of importation.

6.5. Built-In Tuple Tables

Every data store contains a fix set of built-in tuple tables whose content is determined by RDFox. Built-in tables are different to other tuple tables in that they don’t necessarily store facts explicitly, they may have a variable arity, they may not be allowed in rules, and they may impose restrictions on what positions need to be fixed when accessed. As with other tuple tables, built-in tuple tables are referenced in queries using the proprietary TT operator or the reserved IRI rdfox:TT (see Section 9.4). Similarly, when allowed in rules, built-in tuple tables are referenced using general atoms (see Section 10.4.1.3).

Each built-in tuple table is identified by a fixed name, which cannot be changed. When a data store is created, all built-in tuple tables supported by RDFox will be created automatically. Like with other tuple tables, RDFox allows built-in tuple tables to be deleted. Once deleted, a built-in table can be recreated as outlined in Section 6.3 by specifying the tuple table name without any parameters. Note that names of built-in tuple tables cannot be used for the creation of other types of tuple tables.

6.5.1. SKOLEM

The SKOLEM tuple table can have arity from one onwards. Moreover, in each fact in this tuple table, the last resource of the fact is a blank node that is uniquely determined by all remaining arguments. This can be useful in queries and/or rules that need to create new objects. This is explained using the following example.

Example: Let us assume we are dealing with a dataset where each person is associated with zero or more companies using the :worksFor relationship. For example, our dataset could contain the following triples.

:Peter :worksFor :Company1 .
:Peter :worksFor :Company2 .
:Paul  :worksFor :Company1 .

Now assume that we wish to attach additional information to each individual employment. For example, we might want to say that the employment of :Peter in :Company1 started on a specific date. To be able to capture such data, we will ‘convert’ each :worksFor link to a separate instance of the :Employment class; then, we can attach arbitrary information to such instances. This presents us with a key challenge: for each combination of a person and company, we need to ‘invent’ a fresh object that is uniquely determined by the person and company.

This problem is solved using the SKOLEM built-in tuple table. In particular, we can restructure the data using the following rule.

:Employment[?E],
:employee[?E,?P],
:inCompany[?E,?C] :-
    :worksFor[?P,?C],
    SKOLEM("Employment",?P,?C,?E) .

The above rule can be understood as follows. Body atom :worksFor[?P,?C] selects all combinations of a person and a company that the person works for. Moreover, atom SKOLEM("Employment",?P,?C,?E) contains all facts where the value of ?E is uniquely determined by the fixed string "Employment", the value of ?P, and the value of ?C. Thus, for each combination of ?P and ?C, the built-in tuple table will produce a unique value of ?E, which is then used in the rule head to derive new triples.

How a value of ?E is computed from the other arguments is not under application control: each value is a blank node whose name is guaranteed to be unique. However, what matters is that the value of ?E is always the same whenever the values of all other arguments are the same. Thus, we can use the following rule to specify the start time of Peter’s employment in Company 1.

:startDate[?E,"2020-02-03"^^xsd:date] :- SKOLEM("Employment",:Peter,:Company1,?E) .

After evaluating these rules, the following triples will be added to the data store. We use blank node names such as _:new_1 for clarity: the actual names of new blank nodes will be much longer in practice.

_:new_1 rdf:type   :Employment            .
_:new_1 :employee  :Peter                 .
_:new_1 :inCompany :Company1              .
_:new_1 :startDate "2020-02-03"^^xsd:date .
_:new_2 rdf:type   :Employment            .
_:new_2 :employee  :Peter                 .
_:new_2 :inCompany :Company2              .
_:new_3 rdf:type   :Employment            .
_:new_3 :employee  :Paul                  .
_:new_3 :inCompany :Company1              .

When creating fresh objects using the SKOLEM built-in tuple table, it is good practice to incorporate object type into the argument. The above example achieved this by passing a fixed string "Employment" as the first argument of SKOLEM. This allows us to create another, distinct blank node for each combination of a person and a company by simply varying the first argument of SKOLEM.

Atoms involving the SKOLEM built-in tuple table must satisfy certain binding restrictions in rules and queries. Essentially, it must be possible to evaluate a query/rule so that, once an SKOLEM atom is reached, either the value of the last argument, or the values of all all but the last argument must be known. This is explained using the following example.

Example: The following query cannot be evaluated by RDFox — that is, the system will respond with a query planning error.

SELECT ?P ?C ?E WHERE {
    TT SKOLEM { "Employment" ?P ?C ?E }
}

This query essentially says “return all ?P, ?C, and ?E where the value of ?E is uniquely defined by "Employment", ?P, and ?C”. The problem with this is that the values of ?P and ?C have not been restricted in any way, so the query should, in principle, return infinitely many answers.

To evaluate the query, one must provide the values of ?P and ?C, or for ?E, either explicitly as arguments or implicitly by binding the arguments in other parts of the query. Thus, both of the following queries can be successfully evaluated.

SELECT ?E WHERE {
    TT SKOLEM { "Employment" :Paul :Company2 ?E }
}

SELECT ?T ?C ?P WHERE {
    BIND (_:new_1 as ?E)
    TT SKOLEM { ?T ?P ?C ?E }
}

The latter query aims to unpack _:new_1 into the values of ?T, ?C, and ?P for which _:new_1 is the uniquely generated fresh blank node. Note that such ?T, ?C, and ?P may or may not exist, depending on the algorithm RDFox uses to generate blank nodes. The following is a more realistic example of blank node ‘unpacking’.

SELECT ?T ?C ?P WHERE {
    ?E rdf:type :Employment .
    TT SKOLEM { ?T ?P ?C ?E }
}

6.5.2. SHACL

SHACL constraint validation in RDFox can be performed using the following RDFox tuple tables.

SHACL { DataGraph [FactDomain = rdfox:all] ShapesGraph S P O }
SHACL_NN { DataGraph [FactDomain = rdfox:all] ShapesGraph S P O }
SHACL_ND { DataGraph [FactDomain = rdfox:all] S P O }
SHACL_DN { [FactDomain = rdfox:all] ShapesGraph S P O }
SHACL_DD { [FactDomain = rdfox:all] S P O }

The tables differ in whether the validated graph (i.e. the SHACL data graph) and the graph storing the constraints (i.e. the SHACL shapes graph) are named graphs or the default graph. In particular, tuple tables SHACL, SHACL_NN and SHACL_ND validate the content of the named graph DataGraph, while tuple tables SHACL_DN and SHACL_DD validate the content of the default graph. Similarly, tuple tables SHACL, SHACL_NN and SHACL_DN perform validation using the SHACL shapes stored in the named graph ShapesGraph, while tuple tables SHACL_ND and SHACL_DD perform validation using the shapes stored in the default graph.

In all variants, the FactDomain argument specifies the domain of the facts in the data graph that will be validated. This argument is optional with default value rdfox:all and possible values rdfox:explicit, rdfox:derived, and rdfox:all, corresponding to the respective fact domain values described in Section 6.2. The last three arguments receive the subject, the predicate and the object of each triple in the validation report that results from validating the data graph with respect to the constraints in the shapes graph.

Basic SHACL Validation

Example: Assume that the following data graph about employees and their employers is imported into the named graph :data.

@prefix sh: <http://www.w3.org/ns/shacl#>.
@prefix : <https://rdfox.com/examples/shacl#>.

:John a :Employee;
    :worksFor :Company1.

:Jane a :Employee;
    :worksFor [ a :Employer ].

Furthermore, assume that the following shapes graph, which asserts that each value of the property :worksFor is of type :Employer, is imported into the named graph :shacl.

@prefix sh: <http://www.w3.org/ns/shacl#>.
@prefix : <https://rdfox.com/examples/shacl#>.

:ClassShape
    sh:targetClass :Employee ;
    sh:path :worksFor ;
    sh:class :Employer.

One can now query the SHACL tuple table to generate the validation report resulting from the validation of the data graph :data using the shapes graph :shacl as follows.

PREFIX : <https://rdfox.com/examples/shacl#>

SELECT ?s ?p ?o {
    TT SHACL { :data :shacl ?s ?p ?o }
}

The validation report should look as follows, modulo blank node names and prefix abbreviations:

_:anonymous1001 rdf:type sh:ValidationReport .
_:anonymous1001 sh:conforms false .
_:anonymous1001 sh:result _:anonymous1002 .
_:anonymous1002 rdf:type sh:ValidationResult .
_:anonymous1002 sh:focusNode :John .
_:anonymous1002 sh:sourceConstraintComponent sh:ClassConstraintComponent .
_:anonymous1002 sh:sourceShape :ClassShape .
_:anonymous1002 sh:resultPath :worksFor .
_:anonymous1002 sh:value :Company1 .
_:anonymous1002 sh:resultSeverity sh:Violation .
_:anonymous1002 sh:resultMessage "The current value node is not a member of the specified class <https://rdfox.com/examples/shacl#Employer>." .

Saving a Validation Report

A validation report can be saved into a named graph using the INSERT update of SPARQL. This is illustrated in the following example.

Example: The following update saves the validation report into the named graph :report:

PREFIX sh: <http://www.w3.org/ns/shacl#>
PREFIX : <https://rdfox.com/examples/shacl#>

INSERT {
    GRAPH :report { ?s ?p ?o }
}
WHERE {
    TT SHACL { :data :shacl ?s ?p ?o }
}

Rejection of Non-Conforming Updates

Certain use cases may require the content of a data store to be kept consistent with SHACL constraints at all times — that is, any updates that result in a violation of a SHACL constraint should be rejected. To achieve this behavior in RDFox, one can query the SHACL tuple table before committing a transaction as follows and, in case any violations are detected, adding an instance of the rdfox:ConstraintViolation class in the default graph; As discussed in Section 11.5, the latter will prevent a transaction from committing. This technique is demonstrated in the following example.

Example: Consider the data and shape graphs from the previous examples and assume the insertion of the data graph is performed using the following RDFox commands.

begin

import > :data data.ttl

INSERT { ?report a rdfox:ConstraintViolation } \
    WHERE { TT SHACL { :data :shacl ?report sh:conforms false } }

# the transaction fails
commit

The INSERT update checks whether the SHACL constraints are satisfied, and if not, adds the value of ?report as an instance of rdfox:ConstraintViolation. As discussed earlier, the constraints are not satisfied for the data in this example, so the WHERE part of the update will bind variable ?report to _:anonymous1001; thus, triple _:anonymous1001 a rdfox:ConstraintViolation will be added to the default graph, which will prevent the transaction from completing successfully.

In contrast, if we fix the data prior to committing the transaction as in the following example, the transaction will be successfully committed.

begin

import > :data data.ttl

# the following tuple makes the data in data.ttl consistent with the SHACL graph
import > :data ! :Company1 a :Employer.

INSERT { ?report a rdfox:ConstraintViolation } \
    WHERE { TT SHACL { :data :shacl ?report sh:conforms false } }

# the transaction succeeds
commit

If we now attempt to remove the triple :Company1 a :Employer using the same approach, the transaction in question will be rejected, since the remaining data would no longer conform with the constraints in the SHACL graph.

begin

# attempting to remove a tuple that would invalidate the remaining of the data
import > :data - ! :Company1 a :Employer.

INSERT { ?report a rdfox:ConstraintViolation } \
    WHERE { TT SHACL { :data :shacl ?report sh:conforms false } }

# the transaction fails
commit

If we want the error message to contain additional information about the constraint violation, we can insert other triples with the rdfox:ConstraintViolation instance in the subject postion into the default graph, for example:

begin

import > :data - ! :Company1 a :Employer.

INSERT { \
    ?s a rdfox:ConstraintViolation . \
    ?s ?p ?o \
} WHERE { \
    TT SHACL { :data :shacl ?s ?p ?o} . \
    FILTER(?p IN (sh:sourceShape, sh:resultMessage, sh:value)) \
}

commit

This should produce an error message like this:

An error occurred while executing the command:
    The transaction could not be committed because it would have introduced the following constraint violation:

    _:anonymous1 sh:resultMessage "The current value node is not a member of the specified class <https://rdfox.com/examples/shacl#Employer>.";
        sh:value <https://rdfox.com/examples/shacl#Company1>;
        sh:sourceShape <https://rdfox.com/examples/shacl#ClassShape> .

Scope of SHACL support:

• RDFox supports SHACL Core.

• SHACL validation is available during query answering, but not in rules.

• The definitions of SHACL Subclass, SHACL Superclass, and SHACL Type rely on a limited form of taxonomical reasoning. This is not automatically performed during SHACL validation, since the desired consequences can be derived using the standard reasoning facilities of RDFox.

• owl:imports in shapes graph is not supported.

• sh:shapesGraph in data graphs is not supported.

6.5.3. DependencyGraph

The dependency graph of a Datalog program can be inspected using the following tuple tables.

DependencyGraph { NamedGraph [FactDomain = rdfox:all] S P O }
DependencyGraph_N { NamedGraph [FactDomain = rdfox:all] S P O }
DependencyGraph_D { [FactDomain = rdfox:all] S P O }

The tuple tables work with RDF encoded Datalog programs stored as RDF graphs. Tuple tables DependencyGraph and DependencyGraph_N read the encoded program from the named graph NamedGraph, while the tuple table DependencyGraph_D reads the encoded program from the default graph. The RDF encoding of Datalog programs is defined below. The FactDomain argument specifies the domain of the facts in the named graph that will be analyzed. This argument is optional with default value rdfox:all, and possible values rdfox:explicit, rdfox:derived, and rdfox:all, corresponding to the respective fact domain values described in Section 6.2. The last three arguments receive the subject, the predicate and the object of each triple in the RDF encoding of the dependency graph. The arguments NamedGraph and FactDomain, if specified, should be bound at the time of evaluation, while the arguments S, P, and O can be either bound or unbound. The tuple table is available during query answering, but not in rules.

Dependency Graph

Datalog rules have to be evaluated in a specific order due to the presence of negation and aggregation. In particular, a rule can only be evaluated after all of its negated and aggregated atoms have been fully computed, i.e. the rules deriving such atoms and all the rules that they depend on have been fully evaluated. RDFox uses the dependency graph of a Datalog program to determine the evaluation order of its rules.

The dependency graph of an RDFox Datalog program encodes the dependencies between the atoms in the program. The nodes of the dependency graph are the atoms in the program, while the edges of the dependency graph determine the different types of dependencies between the atoms. (Note that RDFox uses an extension of the standard definition of a dependency graph in which the nodes of the graph are atoms rather than predicates. This is because in RDF there is typically only one predicate, i.e. the predicate for all triples in the default graph. Therefore, using the standard definition of a dependency graph, most programs with negation and aggregation would not have a valid rule evaluation order.)

There are three types of dependencies between atoms. Positive dependencies encode the dependencies of head atoms on the body atoms of the rule that are not under aggregation or negation. Negative dependencies encode the dependencies of head atoms of a rule on the body atoms that are under aggregation or negation. Finally, unification dependencies encode that two atoms match a common fact, e.g. [?X, :r, :b] and [:a, :r, ?Y] unify since they both match the triple :a :r :b.

Once the dependency graph of a Datalog program has been constructed, RDFox determines its strongly connected components. A component can be evaluated only if it is stratifiable, i.e. if it contains no atom that negatively depends on another atom from the same component. If all strongly connected components are stratifiable, then the whole program is stratifiable. RDFox groups the strongly connected components by strata. The first stratum contains all components that don’t depend on other components; the second stratum contains all components that depend only on components from the first stratum; and so on. The rules are then evaluated by RDFox according to the stratification of components. Rules that derive facts in the first stratum are evaluated first, rules that derive facts in the second stratum are evaluated next, and so on.

RDF Encoding of Datalog Programs

To extract the dependency graph of a Datalog program, one first has to add its RDF encoding into a named graph. The RDF encoding of a Datalog program is done using the predicate rdfox:rule to specify the rules of the program and the predicate rdfox:prefix to specify the prefixes used in the rule definitions.

Example: Consider for example the following program.

prefix : <https://rdfox.com/examples/>
prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>

:C[?x] :- not :A[?x], :r[?x, ?y].

This program can be encoded using the following RDF triples.

_:p1 rdfox:prefix "prefix : <https://rdfox.com/examples/>".
_:p2 rdfox:prefix "prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>".
_:r1 rdfox:rule ":C[?x] :- not :A[?x], :r[?x, ?y].".

Querying for the Dependency Graph of a Program

Once the RDF encoding of a Datalog program is in a named graph, one can simply query the tuple table DependencyGraph.

Example: Let’s assume that the RDFox encoding of the above Datalog program has been added to the graph :G. To extract the dependency we can simply run the following SPARQL query.

SELECT ?s ?p ?o WHERE { (:G ?s ?p ?o) rdfox:TT "DependencyGraph" }

The result of this query will contain the following triples.

 1) _:p2 rdfox:prefix "prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>".
 2) _:p1 rdfox:prefix "prefix : <https://rdfox.com/examples/>".
 3) _:r1 rdfox:rule ":C[?x] :- not :A[?x], :r[?x, ?y]." .
 4) _:r1 rdfox:headAtom _:atom2 .
 5) _:r1 rdfox:negativeBodyAtom _:atom0 .
 6) _:r1 rdfox:positiveBodyAtom _:atom1 .
 7) _:component0 rdfox:stratumIndex 1 .
 8) _:component0 rdfox:stratifiable true .
 9) _:atom2 rdfox:component _:component0 .
10) _:atom2 rdfox:atom "[*, rdf:type, :C]" .
11) _:atom2 rdfox:dependsPositivelyOn _:atom1 .
12) _:atom2 rdfox:dependsNegativelyOn _:atom0 .
13) _:component1 rdfox:stratumIndex 0 .
14) _:component1 rdfox:stratifiable true .
15) _:atom0 rdfox:component _:component1 .
16) _:atom0 rdfox:atom "[*, rdf:type, :A]" .
17) _:component2 rdfox:stratumIndex 0 .
18) _:component2 rdfox:stratifiable true .
19) _:atom1 rdfox:component _:component2 .
20) _:atom1 rdfox:atom "[*, :r, *]" .

The triples 1-3 encode the input program. The triples 4-6 establish the link between the rule and its atoms. The remaining triples describe the strongly connected components of the dependency graph of the program. There are three components for each of the three atoms in the program. The components of the body atoms [*, :r, *] and [*, rdf:type, :A] are in the first statum (index 0), since they don’t depend on other components. The component for the head atom [*, rdf:type, :C] is in the second stratum (stratum 1), since it depends on the components in stratum 0. The result set also encodes that atom [*, rdf:type, :C] depends negatively on atom [*, rdf:type, :A] and that it depends positively on the atom [*, :r, *]. All three components are stratifiable.

Example: We now give an example of a program that is not stratifiable and therefore cannot be evaluated by RDFox.

prefix : <https://rdfox.com/examples/>
prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>

:B[?x] :- :A[?x].
:C[?x] :- :B[?x], not :A[?x].
:A[?x] :- :C[?x].

Now assume that the following encoding has been added to the named graph :G.

_:p1 rdfox:prefix "prefix : <https://rdfox.com/examples/>".
_:p2 rdfox:prefix "prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>".
_:r1 rdfox:rule ":B[?x] :- :A[?x].".
_:r2 rdfox:rule ":C[?x] :- :B[?x], not :A[?x].".
_:r3 rdfox:rule ":A[?x] :- :C[?x].".

Querying the tuple table DependencyGraph as before will result in the following triples.

 1) _:p2 rdfox:prefix "prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>" .
 2) _:p1 rdfox:prefix "prefix : <https://rdfox.com/examples/>" .
 3) _:r3 rdfox:rule ":A[?x] :- :C[?x]." .
 4) _:r3 rdfox:headAtom _:atom1 .
 5) _:r3 rdfox:positiveBodyAtom _:atom0 .
 6) _:r2 rdfox:rule ":C[?x] :- :B[?x], not :A[?x]." .
 7) _:r2 rdfox:headAtom _:atom0 .
 8) _:r2 rdfox:positiveBodyAtom _:atom2 .
 9) _:r2 rdfox:negativeBodyAtom _:atom1 .
10) _:r1 rdfox:rule ":B[?x] :- :A[?x]." .
11) _:r1 rdfox:headAtom _:atom2 .
12) _:r1 rdfox:positiveBodyAtom _:atom1 .
13) _:component0 rdfox:stratumIndex 0 .
14) _:component0 rdfox:stratifiable false .
15) _:atom1 rdfox:component _:component0 .
16) _:atom1 rdfox:atom "[*, rdf:type, :A]" .
17) _:atom1 rdfox:dependsPositivelyOn _:atom0 .
18) _:atom0 rdfox:component _:component0 .
19) _:atom0 rdfox:atom "[*, rdf:type, :C]" .
20) _:atom0 rdfox:dependsNegativelyOn _:atom1 .
21) _:atom0 rdfox:dependsPositivelyOn _:atom2 .
22) _:atom2 rdfox:component _:component0 .
23) _:atom2 rdfox:atom "[*, rdf:type, :B]" .
24) _:atom2 rdfox:dependsPositivelyOn _:atom1 .

As before, the first two blocks of triples encode the input program and the relationships between rules and their head and body atoms (1-12). The remaining triples describe the dependency graph of the program. In this example, all atoms in the program depend on each other in a recursive fashion. As a result, the dependency graph has exactly one strongly connected component, which contains all the atoms in the program. Since the atom [*, rdf:type, :C] negatively depends on another atom from the same component (i.e. [*, rdf:type, :A]), the component is not stratifiable. Therefore, the program as a whole has no valid rule evaluation order and will thus be rejected by RDFox.

RDF Vocabulary for Dependency Graph Encoding

The following table describes the vocabulary used in the RDF encoding of Datalog programs and their dependency graphs.

Predicate	Description	Example
rdfox:prefix	Specifies a prefix mapping.	_:p1 rdfox:prefix "@prefix : <https://rdfox.com/examples/> ." .
rdfox:rule	Specifies a rule.	_:r1 rdfox:rule ":C[?x] :- :A[?x], :r[?x, ?y]." .
rdfox:atom	Specifies an atom.	_:atom2 rdfox:atom "[*, rdf:type, :C]" .
rdfox:headAtom	Links a rule with a head atom.	_:r1 rdfox:headAtom _:atom2 .
rdfox:positiveBodyAtom	Links a rule with a positive body atom.	_:r1 rdfox:positiveBodyAtom _:atom2 .
rdfox:negativeBodyAtom	Links a rule with a negative body atom.	_:r1 rdfox:negativeBodyAtom _:atom2 .
rdfox:component	Links an atom with its strongly connected component in the dependency graph.	_:atom2 rdfox:component _:component0 .
rdfox:stratumIndex	Links a strongly connected component with its stratum index.	_:component1 rdfox:stratumIndex 0 .
rdfox:stratifiable	Specifies whether a strongly connected component is stratifiable.	_:component1 rdfox:stratifiable true .
rdfox:dependsPositivelyOn	Specifies a positive dependency between two atoms.	_:atom2 rdfox:dependsPositivelyOn _:atom1 .
rdfox:dependsNegativelyOn	Specifies a negative dependency between two atoms.	_:atom2 rdfox:dependsNegativelyOn _:atom1 .
rdfox:unifiesWith	Specifies that two atoms unify.	_:atom1 rdfox:unifiesWith _:atom3 .