tutorial "linked data query processing" part 2 "theoretical foundations" (www...

Linked Data Query ProcessingTutorial at the 22nd International World Wide Web Conference (WWW 2013)

May 14, 2013

http://db.uwaterloo.ca/LDQTut2013/

2.Theoretical Foundations

Olaf HartigUniversity 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 retrievaland result construction

http://mdb.../Paul http://geo.../Berlinhttp://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➢ Theoretical Properties


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 yogaalice 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 yogaalice 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




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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 yogaalice tea

QP(W ) = { μ1 , μ2 , ... }

SPARQL-Based Linked Data Query


Outline




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Full-Web Semantics

QP(W ) :═ eval(P,AllData(W ))


Computability

● (Ordinary) Turing machinesare unsuitable:● Limited data access capabilities

not properly captured

● Web machines● Abiteboul and Vianu, 1997● Mendelzon and Milo, 1997


TM


LD Machine

● Multi-tape Turing machine

➔ Input

➔ Work

➔ Output


LD Machine

# enc(u1) enc(adoc(u1)) # enc(u2) enc(adoc(u2)) # ∙ ∙ ∙

● Multi-tape Turing machine➔ Web Input

➔ Input

➔ Work

➔ Output


LD Machine


● Multi-tape Turing machine➔ Web Input

➔ Input

➔ Work

➔ Output

● Access to Web Input is restricted● Only by performing

a particular procedurein 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

stepk + 2

∙ ∙ ∙∙ ∙ ∙

stepk - 3

stepk - 2

stepk - 1

stepk

stepk + 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

stepk + 2

∙ ∙ ∙∙ ∙ ∙

stepk - 3

stepk - 2

stepk - 1

stepk

stepk + 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.


● Main reason: Set of all URIs is infinite (but countable)

Computability (Full-Web Semantics)



Satisfiability and Monotonicity

Q( ) = { …, ri , … } ≠ Ø

⟹ Q( ) ⊆ Q( )

Proposition [Har12]: Let QP be a SPARQLLD query under full-Websemantics.● 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-Websemantics.● QP is satisfiable ⟺ its query pattern P is satisfiable● QP is monotonic ⟺ its query pattern P is monotonic

Unfortunately: Satisfiability andmonotonicity of SPARQLquery patterns is undecidable

Satisfiability:

Monotonicity:

OPTtp

AND

FILTER

UNION




● Main reason: Set of all URIs is infinite (but countable)

Computability (Full-Web Semantics)



Outline




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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

:=



* ))cNone

:=

(Reachability-Based) cNone-Semantics



* ))cMatch

:=

(Reachability-Based) cMatch-Semantics


Satisfiability and Monotonicity

Q( ) = { …, ri , … } ≠ Ø

⟹ Q( ) ⊆ Q( )

Satisfiability:

Monotonicity:

Proposition [Har12]: Let c be a reachability criterion and QPcS be a

SPARQLLD query under c-semantics.● QP

cS is satisfiable ⟺ its SPARQL expression P is satisfiable

● QPcS is monotonic ⇒ its SPARQL expression P is monotonic

Proposition [Har12]: Let c be a reachability criterion and QPcS be a

SPARQLLD query under c-semantics.● QP

cS is satisfiable ⟺ its SPARQL expression P is satisfiable

● QPcS is monotonic ⇒ its SPARQL expression P is monotonic

Counterexample: Let QPcS be a SPARQLLD query under c-semantics.

If c = cNone and | S | = 1, then QPcS is monotonic.

Counterexample: Let QPcS be a SPARQLLD query under c-semantics.

If c = cNone and | S | = 1, then QPcS 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 Websof Linked Data are finite (i.e., consist of a finitenumber 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




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Next part: 3. Source Selection Strategies ...


These slides have been created byOlaf Hartig

for theWWW 2013 tutorial on

Link Data Query Processing

Tutorial Website: http://db.uwaterloo.ca/LDQTut2013/

This work is licensed under aCreative Commons Attribution-Share Alike 3.0 License

(http://creativecommons.org/licenses/by-sa/3.0/)


http://creativecommons.org/licenses/by-sa/3.0/

tutorial "linked data query processing" part 2 "theoretical foundations" (www...

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queried data

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basic query pattern

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