- The document proposes a formal context-based semantics for evaluating SPARQL property path queries over the Web of Linked Data. - This semantics defines how to compute the results of such queries in a well-defined manner and ensures the "web-safeness" of queries, meaning they can be executed directly over the Web without prior knowledge of all data. - The paper presents a decidable syntactic condition for identifying SPARQL property path queries that are web-safe based on their sets of conditionally bounded variables.

1. A Context-Based Semantics
for SPARQL Property Paths
over the Web
Olaf Hartig and Giuseppe Pirrò

2. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 2
Puzzle
Give me the URL of every Web document that
mentions its own URL somewhere in its text.

3. A Context-Based Semantics
for SPARQL Property Paths
over the Web
Olaf Hartig and Giuseppe Pirrò
Disclaimer:
This work is not about reasoning.

4. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 4
Property Path Pattern (PP Pattern)
(α, path, β)
variable, IRI, literal (or
bnode), respectively
property path expression

5. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 5
Property Path Expression
(α, path, β)
variable, IRI, literal (or
bnode), respectively
path := p | (IRI)
^path | (inverse)
path / path | (concatenation)
(path | path) | (alternative)
(path)* (recursive concatenation)

6. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 6
path := p | (IRI)
^path | (inverse)
path / path | (concatenation)
(path | path) | (alternative)
(path)* (recursive concatenation)
Example
( ex:bob , foaf:knows / (foaf:made | ^foaf:maker) , ?x )
( ex:bob , foaf:knows , ex:alice )
( ex:alice , foaf:made , a:doc1 )
( e:objA , foaf:maker , ex:alice )
Data:
μ1 = { ?x → a:doc1 }
μ2 = { ?x → e:objA }

7. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 7
Example
( ex:bob , foaf:knows / (foaf:made | ^foaf:maker) , ?x )
( ex:bob , foaf:knows , ex:alice )
( ex:alice , foaf:made , a:doc1 )
( e:objA , foaf:maker , ex:alice )
G =
⟦P⟧G = { μ1, μ2 }
P =
μ1 = { ?x → a:doc1 }
μ2 = { ?x → e:objA }

8. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 8
Example
( ex:bob , foaf:knows / (foaf:made | ^foaf:maker) , ?x )
( ex:bob , foaf:knows , ex:alice )
( ex:alice , foaf:made , a:doc1 )
( e:objA , foaf:maker , ex:alice )
P =

9. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 9
Example
( ex:bob , foaf:knows / (foaf:made | ^foaf:maker) , ?x )
P =
( ex:alice , foaf:made , a:doc1 )
( ex:bob , foaf:knows , ex:alice )
( e:objA , foaf:maker , ex:alice )

10. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 10
Example
( ex:bob , foaf:knows / (foaf:made | ^foaf:maker) , ?x )
P =
⟦P⟧W = ??
What is the expected result of executing P
as a query over the Web of Linked Data?
Can this expected result be computed in practice?

11. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 11
Contributions
● Formal definition of a context-based query
semantics to use PP patterns (and SPARQL
patterns constructed thereof) as queries over
Linked Data in a well-defined manner.
● Web-safeness property of such queries (i.e.,
the fundamental feasibility of executing such a
query in practice over the WWW)
● Decidable syntactic property to identify
queries that are Web safe

12. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 12
⟦(u1, p, ?x)⟧W =
Context-Based Query Semantics
…
(u1, p, u2)
(u1, p, u3)
...
{ μ1, μ2 }
μ1 = { ?x → u2 }
μ2 = { ?x → u3 }

13. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 13
⟦(u1, p / q , ?x)⟧W =
Context-Based Query Semantics
…
(u1, p, u2)
(u1, p, u3)
...
{ μ1, μ2 }
μ1 = { ?x → b }
μ2 = { ?x → a }
…
(u3, q, a)
...
…
(u2, q, b)
...

14. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 14
Context-Based Query Semantics
= { μ={ ?x → u } | for every URI u s.t.
(u, p, a) in ctxt(u) }
⟦(?x, p, a)⟧W

15. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 15
Main Result
Theorem: A PP-based SPARQL pattern P is
Web-safe if cbvars( P | Ø ) = vars(P),
...where cbvars( P | X ) denotes the set of
conditionally bounded variables of P for a
given set X of variables (see paper).
● Open problem:
– Note that the condition is sufficient only.
– Is there an alternative condition that is sufficient
and also necessary (and decidable)?

16. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 16
Summary
● Formal definition of a context-based query
semantics to use PP patterns (and SPARQL
patterns constructed thereof) as queries over
Linked Data in a well-defined manner.
● Web-safeness property of such queries (i.e.,
the fundamental feasibility of executing such a
query in practice over the WWW)
● Decidable syntactic property to identify
queries that are Web safe

17. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 17
Backup Slides

18. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 18
Web-Safeness
A PP-based SPARQL pattern P is Web-safe if
there exists an algorithm that, for any Web of
Linked Data W = (D, adoc), computes ⟦P⟧W
without assuming an a-priori availability of
information about the sets D and dom(adoc).

19. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 19
Conditionally Bounded Variables
● cbvars( P | X ) denotes the set of conditionally
bounded variables of P for a given set X of variables
● Defined recursively for all types of patterns
– If P is ( P1 AND P2 ) s.t. cbvars( P1 | X ) = vars(P1)
and cbvars( P2 | X U sbvars(P1) ) = vars(P2), then
cbvars( P | X ) = vars(P).
– If P is (u, p, β), then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x in X, then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x not in X, then cbvars( P | X ) = Ø.
– etc.

20. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 20
Conditionally Bounded Variables
● cbvars( P | X ) denotes the set of conditionally
bounded variables of P for a given set X of variables
● Defined recursively for all types of patterns
– If P is ( P1 AND P2 ) s.t. cbvars( P1 | X ) = vars(P1)
and cbvars( P2 | X U sbvars(P1) ) = vars(P2), then
cbvars( P | X ) = vars(P).
– If P is (u, p, β), then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x in X, then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x not in X, then cbvars( P | X ) = Ø.
– etc.
Pex = ( (u1, p, ?x) AND (?x, q, ?y) )
cbvars( Pex | Ø ) = ?
Example

21. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 21
Conditionally Bounded Variables
● cbvars( P | X ) denotes the set of conditionally
bounded variables of P for a given set X of variables
● Defined recursively for all types of patterns
– If P is ( P1 AND P2 ) s.t. cbvars( P1 | X ) = vars(P1)
and cbvars( P2 | X U sbvars(P1) ) = vars(P2), then
cbvars( P | X ) = vars(P).
– If P is (u, p, β), then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x in X, then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x not in X, then cbvars( P | X ) = Ø.
– etc.
Pex = ( (u1, p, ?x) AND (?x, q, ?y) )
cbvars( P1 | Ø ) = {?x} cbvars( P2 | Ø ) = Ø
Example

22. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 22
Conditionally Bounded Variables
● cbvars( P | X ) denotes the set of conditionally
bounded variables of P for a given set X of variables
● Defined recursively for all types of patterns
– If P is ( P1 AND P2 ) s.t. cbvars( P1 | X ) = vars(P1)
and cbvars( P2 | X U sbvars(P1) ) = vars(P2), then
cbvars( P | X ) = vars(P).
– If P is (u, p, β), then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x in X, then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x not in X, then cbvars( P | X ) = Ø.
– etc.
Pex = ( (u1, p, ?x) AND (?x, q, ?y) )
cbvars( P1 | Ø ) = {?x} cbvars( P2 | sbvars(P1) )
Example

23. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 23
Conditionally Bounded Variables
● cbvars( P | X ) denotes the set of conditionally
bounded variables of P for a given set X of variables
● Defined recursively for all types of patterns
– If P is ( P1 AND P2 ) s.t. cbvars( P1 | X ) = vars(P1)
and cbvars( P2 | X U sbvars(P1) ) = vars(P2), then
cbvars( P | X ) = vars(P).
– If P is (u, p, β), then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x in X, then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x not in X, then cbvars( P | X ) = Ø.
– etc.
Pex = ( (u1, p, ?x) AND (?x, q, ?y) )
cbvars( P1 | Ø ) = {?x} cbvars( P2 | {?x} ) = {?x,?y}
Example

24. A Context-Based Semantics for SPARQL Property Paths over the Web - Olaf Hartig and Giuseppe Pirrò 24
Conditionally Bounded Variables
● cbvars( P | X ) denotes the set of conditionally
bounded variables of P for a given set X of variables
● Defined recursively for all types of patterns
– If P is ( P1 AND P2 ) s.t. cbvars( P1 | X ) = vars(P1)
and cbvars( P2 | X U sbvars(P1) ) = vars(P2), then
cbvars( P | X ) = vars(P).
– If P is (u, p, β), then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x in X, then cbvars( P | X ) = vars(P).
– If P is (?x, p, β) s.t. ?x not in X, then cbvars( P | X ) = Ø.
– etc.
Pex = ( (u1, p, ?x) AND (?x, q, ?y) )
cbvars( Pex | Ø ) = vars(Pex) = {?x,?y}
Example