Tutorial "Linked Data Query Processing" Part 2 "Theoretical Foundations" (WWW 2013 Ed.)

Linked Data Query Processing
Tutorial at the 22nd International World Wide Web Conference (WWW 2013)
May 14, 2013
http://db.uwaterloo.ca/LDQTut2013/
2.
Theoretical Foundations
Olaf Hartig
University of Waterloo

WWW 2013 Tutorial on Linked Data Query Processing [ Theoretical Foundations ] 2
● Data retrieval approach
● Data source selection
● Data source ranking
(optional, for optimization)
Query-local data
GET http://.../movie2449
● Result construction approach
● i.e., query-local data processing
● Combining data retrieval
and result construction
http://mdb.../Paul http://geo.../Berlin
http://mdb.../Ric http://geo.../Rome
?loc?actor
“Ingredients” for LD Query Execution

Outline
 Basics of the SPARQL Query Language
 Data Model for Linked Data Queries
 Full-Web Query Semantics
➢ Definition
➢ Theoretical Properties
 Reachability-Based Query Semantics
➢ Definition

General Idea of SPARQL
● SPARQL: Declarative query language for RDF data
● Main idea: pattern matching
● Query describes a pattern to be found in RDF data
● Basically, query patterns consist of RDF triples with variables
( ?p , affiliated with , orgaX ) ,
( ?p , interested in , ?i )
Basic Query Pattern:

General Idea of SPARQL
● SPARQL: Declarative query language for RDF data
● Main idea: pattern matching
● Query describes a pattern to be found in RDF data
● Basically, query patterns consist of RDF triples with variables
● Any part of the queried data that matches the query pattern
contributes a solution to the query result
Basic Query Pattern:
( alice , affiliated with , orgaX ) ,
( bob , affiliated with , acme ) ,
( alice , interested in , yoga ) ,
( alice , interested in , tea ) ,
( bob , interested in , tea ) ,
Data:
?p ?i
alice yoga
alice tea
Query Result:

Some Terminology
● Any query result is represented by a set of valuations
Ω = { μ1 , … , μn }
● Valuation: mapping of query variables to RDF terms
μ = { ?p → alice , ?i → yoga }
● Any valuation in a query result is a solution
● Two valuations μ1 and μ2 are compatible if they agree on
the binding for any shared variable
● i.e., μ1(?v) = μ2(?v) for all ?v Є dom(μ1) ∩ dom(μ2)
?p ?i
alice yoga
alice tea

Query Patterns and their Semantics
● Triple pattern: An RDF triple that may have a variable as
subject, predicate, or object, respectively
●
eval( tp, G ) :═ { μ | μ[tp] Є G and dom(μ) = vars(tp) }
● Group graph pattern: Conjunction of two query patterns
●
eval( (P1 AND P2), G ) :═ { μ1 U μ2 | μ1 Є eval(P1,G) and
μ2 Є eval(P2,G) and
μ1 and μ2 are compatible }
● Associative and commutative
● Basic graph pattern (BGP): A set of triple patterns
● B = { tp1 , … , tpn } is equivalent to ( tp1 AND … AND tpn )
● Filter, union, optional, etc.

Outline
➢ Definition
➢ Definition
√

Data Model
● Static view (i.e., no changes during computation)
● Web of Linked Data: W = ( D, adoc, data )
● D – set of symbols that represent Web documents
from which Linked Data can be extracted
● adoc – partial mapping from URIs to D
● data – total mapping from D to finite sets of RDF triples
●
All data in W: AllData(W) :═ Ud є D data(d)
D = adoc( ) = data( ) =

Linked Data Query
Q(W)

?p ?i
alice yoga
alice tea
QP
(W) = { μ1 , μ2 , ... }
SPARQL-Based Linked Data Query

Outline
➢ Definition
➢ Definition
√
√

Full-Web Semantics
QP
(W) :═ eval(P,AllData(W))

Computability
● (Ordinary) Turing machines
are unsuitable:
● Limited data access capabilities
not properly captured
● Web machines
● Abiteboul and Vianu, 1997
● Mendelzon and Milo, 1997
QP
TM

LD Machine
● Multi-tape Turing machine
➔ Input
➔ Work
➔ Output

LD Machine
# enc(u1) enc(adoc(u1)) # enc(u2) enc(adoc(u2)) # ∙ ∙ ∙
➔ Web Input
➔ Input
➔ Work
➔ Output

LD Machine
➔ Web Input
➔ Input
➔ Work
➔ Output
● Access to Web Input is restricted
● Only by performing
a particular procedure
in a particular state

# enc(u1) enc(adoc(u1)) # enc(u2) enc(adoc(u2)) # ∙ ∙ ∙➔ Web Input
➔ Input
➔ Work
➔ Output
● For Q exists an LD machine MQ such that for any W holds:
● MQ halts after a finite number of computation steps, and
● MQ outputs the complete result Q(W)
Finitely Computable LD Queries
step 1 ∙ ∙ ∙ step k - 3 step k - 2 step k – 1 step k
∙ ∙ ∙
# enc(μ1) # enc(μ2) # ∙ ∙ ∙ # enc(μn) #

Eventually Computable LD Queries
step
k + 2
∙ ∙ ∙∙ ∙ ∙
step
k - 3
step
k - 2
step
k - 1
step
k
step
k + 1
# enc(μ1) # enc(μ2)
➔ Web Input
➔ Input
➔ Work
➔ Output
1. Output always encodes a subset of query result Q(W), and
2. Each μ Q(W) eventually appears on the output
✗ No guarantee for termination

Not (LD Machine) Computable
step
k + 2
∙ ∙ ∙∙ ∙ ∙
step
k - 3
step
k - 2
step
k - 1
step
k
step
k + 1
# enc(μ1) # enc(μ2)
➔ Web Input
➔ Input
➔ Work
➔ Output
1. Output always encodes a subset of query result Q(W), and
2. Each μ Q(W) eventually appears on the output
✗ No guarantee for termination

Theorem [Har12]: If a satisfiable SPARQLLD query
QP
under full-Web semantics is monotonic, then it is
eventually computable (but not finitely computable);
otherwise, it is not even eventually computable.
QP
● Main reason: Set of all URIs is infinite (but countable)
Computability (Full-Web Semantics)
QP

Satisfiability and Monotonicity
Q( ) = { …, ri , … } ≠ Ø
⟹ Q( ) ⊆ Q( )
Proposition [Har12]: Let QP
be a SPARQLLD query under full-Web
semantics.
● QP
is satisfiable ⟺ its query pattern P is satisfiable
● QP
is monotonic ⟺ its query pattern P is monotonic
Proposition [Har12]: Let QP
be a SPARQLLD query under full-Web
semantics.
● QP
is satisfiable ⟺ its query pattern P is satisfiable
● QP
is monotonic ⟺ its query pattern P is monotonic
Unfortunately:
Satisfiability and
monotonicity of SPARQL
query patterns is undecidable
Satisfiability:
Monotonicity:
O
PT
tp
A
N
D
FILTER
U
N
IO
N

QP
QP
● Main reason: Set of all URIs is infinite (but countable)
Computability (Full-Web Semantics)
QP

Outline
➢ Definition
➢ Definition
√
√
√

Reachability-Based Query Semantics

●
Seed URIs S ᑕ U

●
Seed URIs S ᑕ U● Reachability criterion c
●
(Turing) computable function c: T ×U × P → {true,false}

WQP,S
( ):= eval(P,AllData(W*
))c

(Reachability-Based) cAll-Semantics
WQP,S
( ) eval(P,AllData(W*
))cAll
:=

WQP,S
))cNone
:=
(Reachability-Based) cNone-Semantics

WQP,S
))cMatch
:=
(Reachability-Based) cMatch-Semantics

Satisfiability and Monotonicity
Q( ) = { …, ri , … } ≠ Ø
⟹ Q( ) ⊆ Q( )
Satisfiability:
Monotonicity:
Proposition [Har12]: Let c be a reachability criterion and QP
c
S
be a
SPARQLLD query under c-semantics.
● QP
c
S
is satisfiable ⟺ its SPARQL expression P is satisfiable
● QP
c
S
is monotonic ⇒ its SPARQL expression P is monotonic
Proposition [Har12]: Let c be a reachability criterion and QP
c
S
be a
SPARQLLD query under c-semantics.
● QP
c
S
is satisfiable ⟺ its SPARQL expression P is satisfiable
● QP
c
S
is monotonic ⇒ its SPARQL expression P is monotonic
Counterexample: Let QP
c
S
be a SPARQLLD query under c-semantics.
If c = cNone and |S| = 1, then QP
c
S
is monotonic.
Counterexample: Let QP
c
S
be a SPARQLLD query under c-semantics.
If c = cNone and |S| = 1, then QP
c
S
is monotonic.

LD Machine-Based Computability
Theorem [Har12]: For any reachability criterion c, any
SPARQL based Linked Data query under c-semantics
is finitely computable.
Theorem [Har12]: For any reachability criterion c, any
SPARQL based Linked Data query under c-semantics
is finitely computable.
● If we restrict our data model such that Webs
of Linked Data are finite (i.e., consist of a finite
number of documents), then:
For results w.r.t. the unrestricted data model (i.e.,
Webs of Linked Data that may be infinitely large),
refer to [Har12].

Outline
➢ Definition
➢ Definition
√
√
√
√
Next part: 3. Source Selection Strategies ...

These slides have been created by
Olaf Hartig
for the
WWW 2013 tutorial on
Link Data Query Processing
Tutorial Website: http://db.uwaterloo.ca/LDQTut2013/
This work is licensed under a
Creative Commons Attribution-Share Alike 3.0 License
(http://creativecommons.org/licenses/by-sa/3.0/)

Tutorial "Linked Data Query Processing" Part 2 "Theoretical Foundations" (WWW 2013 Ed.)

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