Virtuoso SPARQL supports RDF Schema subclasses and subproperties.

The predicates rdfs:subClassOf and rdfs:subPropertyOf are recognized when they appear in graphs included in a rule set. When such a rule set is specified as a context for a SPARQL query, the following extra triples are generated as needed.

For every ?s rdf:type ?class , a triple ?s rdf:type ?superclass is considered to exist, such that ?superclass is a direct or indirect superclass of ?class . Direct superclasses are declared with the rdfs:subClassOf predicate in the rule set graph. Transitivity of superclasses is automatically taken into account, meaning that if a is a superclass of b and b a superclass of c, then a is a superclass of c also. Cyclic superclass relations are not allowed. If such occur in the rule set data, the behavior is undefined but will not involve unterminating recursion.

For every ?s ?subpredicate ?o , a triple ?s ?superpredicate ?o is considered to exist if the rule context declares ?superpredicate to be a superpredicate of ?predicate . This is done by having the triple ?subpredicate rdfs:subPropertyOf ?superpredicate as part of the graphs making up the rule context. Transitivity is observed, thus if a is a subpredicate of b and b a subpredicate of c, then a is also a subpredicate of c.

Two methods can be used for typical recursions, transitivity on inference and plain transitive patterns (or subqueries).

The advantage of inference is that queries are short and one inference rule set may be maintained for numerous queries.

If queries are about trees of classes or properties, or about equivalences of nodes, consider using inference rule sets.

Transitive patterns are inconvenient and may easily result in queries that runs too long or hard to debug, but they're unavoidable in traversing social networks or plain querying of RDF lists.

So consider a rule set, a handful of nodes with classes from the rule set and a couple of RDF Lisp-style lists defined on demo.openlinksw.com:

SQL> SPARQL CLEAR GRAPH <http://example.com/2/owl>;
callret-0
VARCHAR
_______________________________________________________________________________

Clear <http://example.com/2/owl>  -- done

1 Rows. -- 0 msec.

SQL> TTLP (' @prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>  .
  @prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>  .
  @prefix owl: <http://www.w3.org/2002/07/owl#>  .
  @prefix e: <http://example.com/e/>  .
  e:c1 rdfs:subClassOf e:c1or2 .
  e:c2 rdfs:subClassOf e:c1or2 .
  e:c1-10 rdfs:subClassOf e:c1 .
  e:c1-20 rdfs:subClassOf e:c1 .
  e:c2-30 rdfs:subClassOf e:c2 .
  e:c2-40 rdfs:subClassOf e:c2 .
  ', 'http://example.com/2/owl', 'http://example.com/2/owl' );

Done. -- 0 msec.

You can also use the SPARUL equivalent variant:

SPARQL
PREFIX rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>
PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#>
PREFIX owl: <http://www.w3.org/2002/07/owl#>
PREFIX e: <http://example.com/e/>

INSERT IN GRAPH <http://example.com/2/owl>
  {
    e:c1 rdfs:subClassOf e:c1or2 .
    e:c2 rdfs:subClassOf e:c1or2 .
    e:c1-10 rdfs:subClassOf e:c1 .
    e:c1-20 rdfs:subClassOf e:c1 .
    e:c2-30 rdfs:subClassOf e:c2 .
    e:c2-40 rdfs:subClassOf e:c2 .
  } ;

Define the inference rule:

SQL> rdfs_rule_set ('http://example.com/2/owl', 'http://example.com/2/owl');

Done. -- 0 msec.

SQL> SPARQL CLEAR GRAPH <http://example.com/2/data> ;
callret-0
VARCHAR
_______________________________________________________________________________

Clear <http://example.com/2/data>  -- done

1 Rows. -- 0 msec.

SQL>  TTLP ('
  @prefix e: <http://example.com/e/>  .
  @prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>  .
  @prefix owl: <http://www.w3.org/2002/07/owl#>  .
  e:s1 a e:c1 ; e:p1 "Value of p1 for s1" .
  e:s2 a e:c2 ; e:p1 "Value of p1 for s2" .
  e:s1-10 a e:c1-10 ; e:p1 "Value of p1 for s1-10" .
  e:s1-20 a e:c1-20 ; e:p1 "Value of p1 for s1-20" .
  e:s2-30 a e:c2-30 ; e:p1 "Value of p1 for s2-30" .
  e:s2-40 a e:c2-40 ; e:p1 "Value of p1 for s2-40" .
  e:lists
       rdf:_1 ( e:list1-item1 e:list1-item2 e:list1-item3 ) ;
       rdf:_2 (
           [ e:p2 "Value of p2 of item1 of list2" ; e:p3 "Value of p3 of item1 of list2" ]
           [ e:p2 "Value of p2 of item2 of list2" ; e:p3 "Value of p3 of item2 of list2" ]
           [ e:p2 "Value of p2 of item3 of list2" ; e:p3 "Value of p3 of item3 of list2" ] ) .
  ', 'http://example.com/2/data', 'http://example.com/2/data' );

Done. -- 0 msec.

You can also use the SPARUL equivalent variant:

SPARQL
PREFIX e: <http://example.com/e/>
PREFIX rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>
PREFIX owl: <http://www.w3.org/2002/07/owl#>

INSERT IN GRAPH <http://example.com/2/data>
  {
    e:s1 a e:c1 ; e:p1 "Value of p1 for s1" .
    e:s2 a e:c2 ; e:p1 "Value of p1 for s2" .
    e:s1-10 a e:c1-10 ; e:p1 "Value of p1 for s1-10" .
    e:s1-20 a e:c1-20 ; e:p1 "Value of p1 for s1-20" .
    e:s2-30 a e:c2-30 ; e:p1 "Value of p1 for s2-30" .
    e:s2-40 a e:c2-40 ; e:p1 "Value of p1 for s2-40" .
    e:lists
      rdf:_1 ( e:list1-item1 e:list1-item2 e:list1-item3 ) ;
      rdf:_2 (
           [ e:p2 "Value of p2 of item1 of list2" ; e:p3 "Value of p3 of item1 of list2" ]
           [ e:p2 "Value of p2 of item2 of list2" ; e:p3 "Value of p3 of item2 of list2" ]
           [ e:p2 "Value of p2 of item3 of list2" ; e:p3 "Value of p3 of item3 of list2" ] )
  };

SPARQL DESCRIBE works fine with inference, deriving additional type information:

DEFINE input:inference <http://example.com/2/owl>
DESCRIBE <http://example.com/e/s1>
FROM <http://example.com/2/data>

fmtaggret-
LONG VARCHAR
_______________________________________________________________________________

@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>  .
@prefix ns1: <http://example.com/e/>  .
ns1:s1 rdf:type ns1:c1or2 ,
ns1:c1 ;
ns1:p1 "Value of p1 for s1" .

1 Rows. -- 0 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs:

DEFINE input:inference <http://example.com/2/owl>
DESCRIBE <http://example.com/e/s2>
FROM <http://example.com/2/data>
fmtaggret-
LONG VARCHAR
_______________________________________________________________________________

@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>  .
@prefix ns1: <http://example.com/e/>  .
ns1:s2 rdf:type ns1:c1or2 ,
ns1:c2 ;
ns1:p1 "Value of p1 for s2" .

1 Rows. -- 0 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs:

Querying is simple as well:

SQL>SPARQL DEFINE input:inference <http://example.com/2/owl>
PREFIX e:<http://example.com/e/>
SELECT *
FROM <http://example.com/2/data>
WHERE
  {
    ?s a e:c1or2 ;
          e:p1 ?o
  }

s                            o
VARCHAR                      VARCHAR
___________________________

http://example.com/e/s1      Value of p1 for s1
http://example.com/e/s1-10   Value of p1 for s1-10
http://example.com/e/s1-20   Value of p1 for s1-20
http://example.com/e/s2-30   Value of p1 for s2-30
http://example.com/e/s2-40   Value of p1 for s2-40
http://example.com/e/s2      Value of p1 for s2

6 Rows. -- 0 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs:

SQL>SPARQL DEFINE input:inference <http://example.com/2/owl>
PREFIX e:<http://example.com/e/>
SELECT * FROM <http://example.com/2/data>
WHERE
  {
    ?s a e:c1 ;
       e:p1 ?o
  }

s                            o
VARCHAR                      VARCHAR
___________________________

http://example.com/e/s1      Value of p1 for s1
http://example.com/e/s1-10   Value of p1 for s1-10
http://example.com/e/s1-20   Value of p1 for s1-20

3 Rows. -- 0 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs:

However you should care about duplicates if both types and properties are queried: the join will result in all combinations of types and property values.

SQL>SPARQL DEFINE input:inference <http://example.com/2/owl>
PREFIX e:<http://example.com/e/>
SELECT * FROM <http://example.com/2/data>
WHERE
  {
    ?s a ?t ;
     e:p1 ?o
  }

s                            t                            o
VARCHAR                      VARCHAR                      VARCHAR
___________________________

http://example.com/e/s1      http://example.com/e/c1      Value of p1 for s1
http://example.com/e/s1      http://example.com/e/c1or2   Value of p1 for s1
http://example.com/e/s1-10   http://example.com/e/c1-10   Value of p1 for s1-10
http://example.com/e/s1-10   http://example.com/e/c1      Value of p1 for s1-10
http://example.com/e/s1-10   http://example.com/e/c1or2   Value of p1 for s1-10
http://example.com/e/s1-20   http://example.com/e/c1-20   Value of p1 for s1-20
http://example.com/e/s1-20   http://example.com/e/c1      Value of p1 for s1-20
http://example.com/e/s1-20   http://example.com/e/c1or2   Value of p1 for s1-20
http://example.com/e/s2-30   http://example.com/e/c2-30   Value of p1 for s2-30
http://example.com/e/s2-30   http://example.com/e/c2      Value of p1 for s2-30
http://example.com/e/s2-30   http://example.com/e/c1or2   Value of p1 for s2-30
http://example.com/e/s2-40   http://example.com/e/c2-40   Value of p1 for s2-40
http://example.com/e/s2-40   http://example.com/e/c2      Value of p1 for s2-40
http://example.com/e/s2-40   http://example.com/e/c1or2   Value of p1 for s2-40
http://example.com/e/s2      http://example.com/e/c2      Value of p1 for s2
http://example.com/e/s2      http://example.com/e/c1or2   Value of p1 for s2

16 Rows. -- 0 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs:

Transitive queries are convenient as SPARQL 1.1 "predicate+" equivalent. The equivalent of "predicate*" requires the use of a union:

SQL>SPARQL PREFIX e:<http://example.com/e/>
SELECT ?item
FROM <http://example.com/2/data>
WHERE
  {
    {
      ?lists rdf:_1 ?node
    }
    UNION
    {
      ?lists rdf:_1 ?l .
      ?l rdf:rest ?node option (transitive) .
    }
    ?node rdf:first ?item
  }

item
VARCHAR
_______________________________________________________________________________

http://example.com/e/list1-item1
http://example.com/e/list1-item2
http://example.com/e/list1-item3

3 Rows. -- 0 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs:

SQL> SPARQL PREFIX e:<http://example.com/e/>
SELECT ?p ?o
FROM <http://example.com/2/data>
WHERE
  {
    {
      ?lists rdf:_2 ?node
    }
    UNION
    {
      ?lists rdf:_2 ?l .
      ?l rdf:rest ?node option (transitive) .
    }
    ?node rdf:first ?item .
    ?item ?p ?o
  }

p                         o
VARCHAR                   VARCHAR
________________________

http://example.com/e/p2   Value of p2 of item1 of list2
http://example.com/e/p3   Value of p3 of item1 of list2
http://example.com/e/p2   Value of p2 of item2 of list2
http://example.com/e/p3   Value of p3 of item2 of list2
http://example.com/e/p2   Value of p2 of item3 of list2
http://example.com/e/p3   Value of p3 of item3 of list2

6 Rows. -- 0 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs:

Note that the result set can be in order of items in the list, but it don't have to. If the order should be preserved, then fix the direction of transitive scan, get step number as a variable, order by that variable.

-- Line 82:
SQL> SPARQL PREFIX e:<http://example.com/e/>
SELECT ?p ?o bif:coalesce(?step_no, 0)
FROM <http://example.com/2/data>
WHERE
  {
    {
      ?lists rdf:_2 ?node
    }
    UNION
    {
      ?lists rdf:_2 ?l .
      ?l rdf:rest ?node OPTION (transitive, t_direction 1, t_step("step_no") as ?step_no) .
    }
    ?node rdf:first ?item .
    ?item ?p ?o
  }
ORDER BY ASC (?step_no)

p                         o                               callret-2
VARCHAR                   VARCHAR                         VARCHAR
________________________

http://example.com/e/p2   Value of p2 of item1 of list2   0
http://example.com/e/p3   Value of p3 of item1 of list2   0
http://example.com/e/p2   Value of p2 of item2 of list2   1
http://example.com/e/p3   Value of p3 of item2 of list2   1
http://example.com/e/p2   Value of p2 of item3 of list2   2
http://example.com/e/p3   Value of p3 of item3 of list2   2

6 Rows. -- 7 msec.

Example links against Virtuoso Demo Server SPARQL Endpoint with SPARQL Protocol URLs: