Datalog+-Track Introduction & Reasoning on UML Class Diagrams via Datalog+-

Datalog§ Track Introduction

&

Reasoning on UML Class Diagrams via Datalog§

Georg Gottlob

Department of Computer Science

University of Oxford

Joint work with G. Orsi, A. Pieris et al.

09:00 <Datalog+/- Track>

09:00 - Datalog Track Introduction

Tue, 4 August, 09:00 – 09:30

Description Georg Gottlob, Giorgio Orsi, and Andreas Pieris: Consistency Checking of Re-Engineered UML

Class Diagrams via Datalog+/- (Invited)

09:30 - Graal: A Toolkit for Query Answering with Existential Rules

Tue, 4 August, 09:30 – 10:00

DescriptionJean-François Baget, Michel Leclère, Marie-Laure Mugnier, Swan Rocher and Clément Sipieter

10:00 - Binary Frontier-guarded ASP with Function Symbols

Tue, 4 August, 10:00 – 10:30

DescriptionMantas Simkus

11:00 - Ontology-Based Multidimensional Contexts with Applications to Quality Data Specification and

Extraction

WhenTue, 4 August, 11:00 – 11:30

DescriptionMostafa Milani and Leopoldo Bertossi

11:30 - Existential rules and Bayesian Networks for Probabilistic Ontological Data Exchange

Tue, 4 August, 11:30 – 12:00

DescriptionThomas Lukasiewicz, Maria Vanina Martinez, Livia Predoiu, Gerardo Ignacio Simari

more details» copy to my calendar

Wed, 09:00 – 09:10

Datalog+, RuleML and OWL 2: Formats and Translations for Existential Rules

JF Baget, A Gutierrez, M Leclère, ML Mugnier, S Rocher and C Sipieter

https://www.google.com/calendar/event?eid=b2luY2o0YWJqaTZlczA4cGtoMXE0c3FndHMgOGhmZzN2OXZ1dGhvNm02cDFoNXMzZXY3ZmtAZw&ctz=Europe/Berlin

https://www.google.com/calendar/event?action=TEMPLATE&hl=en_GB&text=-%20Existential%20rules%20and%20Bayesian%20Networks%20for%20Probabilistic%20Ontological%20Data%20Exchange&dates=20150804T113000%2F20150804T120000&location&ctz=Europe%2FBerlin&details=Thomas%20Lukasiewicz%2C%20Maria%20Vanina%20Martinez%2C%20Livia%20Predoiu%2C%20Gerardo%20Ignacio%20Simari

Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person

IssuesIndex[0..1]:Str

getIndex():List

The Need for Modeling Data and Objects

UML Class Diagrams

F-Logic

Description Logics

Speaker v Person u 9gives.Talk

Tutorial v 8attendedBy.Student

attendedBy v attended-1

Speaker u Student v ?

student::person

person[age ¤) number]

person[age {0:1} ¤) number]

person[name {1:¤} ¤) string]

PREFIX abc: hhttp://example.com/exampleOntology#i

SELECT ?capital ?country

WHERE {

?x abc:cityname ?capital ;

?x abc:isCapitalOf ?y .

?x abc:countryname ?country ;

?x abc:isInContinent abc:Africa . }

SPARQL

Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List

A Unifying Framework for Reasoning over Data

UML Class Diagrams

F-Logic

Description Logics





student::person






WHERE {





SPARQL

Datalog±

• Logical foundations, semantics and decidability

• Complexity results for ontological query answering

• Identification of tractable fragments

The Datalog§ Family

• Extend Datalog by allowing in the head:

• But query answering under Datalog[9] is already undecidable

see, e.g., [Beeri & Vardi, ICALP 1981] and [Calì, G. & Kifer, KR 2008]

• Datalog[9,=,?] is syntactically restricted ! Datalog§

• Datalog: 8X8Y (X,Y) R(X)

• Existential quantification (TGDs) – 8X8Y (X,Y) 9Z (X,Z)

• Equality predicate (EGDs) – 8X (X) Xi = Xj

• Constant false (Negative Constraints) – 8X (X) ?

Ontological Query Answering

D

Ο

hD,Oi

D

database

ontology

Query

knowledge base

hD,Οi ² Query , D Æ Ο ² Query

Guarded Datalog[9]

8X8Y8Z R(X,Y,Z) ^ S(Y) ^ P(X,Z) 9W Q(X,W)

• All 8-variables occur in one body atom – guard atom

guard

• Query answering is PTIME-complete in data complexity

• Models of finite treewidth ) decidability of query answering

related to [Andréka, Németi & van Benthem, J. Philosophical Logic 1998] and

[Grädel, J. Symb. Log. 1999]

[Calì, G. & Kifer, KR 2008]

Linear Datalog[9]

• Rules with just one body-atom

• Query answering is first-order rewritable

8X supervisorOf(X,X) manager(X)

[Calì, G. & Lukasiewicz, J. Web. Sem. 2012]

First-Order Rewritability

8D (D [ O ² Q , D ² Q*)

Query answering is in AC0 in data complexity

D

translationevaluation

SQLpositive

first-order query

Q Ο

QFO Q*

compilation

Linear Datalog[9]

• Rules with just one body-atom

• … and thus, query answering is in AC0 in data complexity

• Query answering is first-order rewritable

8X supervisorOf(X,X) manager(X)

[Calì, G. & Lukasiewicz, J. Web. Sem. 2012]

Datalog[9,=,?]

• Every decidable Datalog[9] language can be enriched with:

• Non-Conflicting EGDs – no interaction with TGDs

• Negative constrains

• Unsatisfiability can be reduced to query answering

• If the theory is satisfiable, then consider only the TGDs

see, e.g., [Calì, G. & Pieris, Artif. Intell. 2012]

Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List


UML Class Diagrams

F-Logic

Description Logics





student::person






WHERE {





SPARQL

Guarded Datalog[9,=,?]




Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List


UML Class Diagrams

F-Logic

Description Logics





student::person






WHERE {





SPARQL

UML Class Diagrams

• Modeling the intensional structure of a software system

• Reasoning services limited to diagram satisfiability checking

• Verify properties at runtime by reasoning over a (possibly

incomplete) state of the system – query answering

Querying UML Class Diagrams: Example I

Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person

Executive(john)

Member(john,LU)

Stock(BAY)

Issues(BA,BAY)

Owns(john,BAY)

Competes(LU,BA)

Index[0..1]:Str

getIndex():List

Issues

Does anybody have a potential conflict of interest?


Stock

0..1

Member

Issues

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person

Executive(john)

Member(john,LU)

Stock(BAY)

Issues(BA,BAY)

Owns(john,BAY)

Competes(LU,BA)

Index[0..1]:Str

getIndex():List

Conflict Person(P), Company(C1), Company(C2), Stock(S),

Owns(P,S), Member(P,C1), Issues(C2,S), Competes(C1,C2)


Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person

Executive(john)

Member(john,LU)

Stock(BAY)

Issues(BA,BAY)

Owns(john,BAY)

Competes(LU,BA)

Person(john)


getIndex():List




Executive(john)

Member(john,LU)

Stock(BAY)

Issues(BA,BAY)

Owns(john,BAY)

Competes(LU,BA)

Person(john)

Company(LU)

Company(BA)

Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List




Executive(john)

Member(john,LU)

Stock(BAY)

Issues(BA,BAY)

Owns(john,BAY)

Competes(LU,BA)

Person(john)

Company(LU)

Company(BA)

Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person

{P ! john, C1 ! LU, C2 ! BA, S ! BAY}


getIndex():List



Querying UML Class Diagrams: Example II

Group(DB)

Is there a professor who works in the database group?

WorksIn

Leads

1..1

3..1

Student Professor

Member

{disjoint} Group

1..1

0..1

CLeads

since: Date

Group(DB)


WorksIn

Leads

1..1

3..1

Student Professor

Member

{disjoint} Group

1..1

0..1

CLeads

Ans Professor(P), WorksIn(P,DB)

since: Date

Group(DB)

Leads(z1,DB)

Professor(z1)

glue(z1,DB,z2)

CLeads(z2)


WorksIn

Leads

1..1

3..1

Student Professor

Member

{disjoint} Group

1..1

0..1

CLeads

since: Date


Group(DB)

Leads(z1,DB)

Professor(z1)

WorksIn(z1,DB)


WorksIn

Leads

1..1

3..1

Student Professor

Member

{disjoint} Group

1..1

0..1


SINCE

since: Date

Group(DB)

Leads(z1,DB)

Professor(z1)

WorksIn(z1,DB)

…

{P ! z1, DB ! DB}


WorksIn

Leads

1..1

3..1

Student Professor

Member

{disjoint} Group

1..1

0..1

SINCE

since: Date


From Diagrams to First-Order Logic (Datalog§)

Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List


Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List

8X8Y Member(X,Y) Company(X) ^ Executive(Y)

8X Company(X) 9Y9Z Member(X,Y) ^ Member(X,Z) ^ Y ≠ Z

8X Executive(X) 9Y Member(Y,X)

[Berardi et al., Artif. Intell. 2005]


Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List

8X Company(X) 9Y Issues(X,Y)

8X8Y8Z Stock(X) ^ Issues(Y,X) ^ Issues(Z,X) Y = Z

8X8Y Stock(X) ^ Index(X,Y) Str(Y)

8X8Y Stock(X) ^ getIndex(X,Y) List(Y)

8X Stock(X) 9Y Issues(Y,X)



Stock

0..1

Member

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List

8X Executive(X) Person(X)


Complexity of Query Answering

• EXPTIME-complete in combined complexity

• coNP-complete in data complexity

• Undecidable when diagrams are combined with arbitrary OCL

(Object Constraint Language) constraints

implicit in [Berardi et al., Artif. Intell. 2005] and [Lutz, IJCAR 2008]

implicit in [Ortiz, Calvanese & Eiter, AAAI 2006]

[folklore]

Research Challenge: Reduce High Complexity

• Diagrams often have very large instantiations

• Some applications require very large diagrams

• OCL constraints, which are not expressible diagrammatically,

lead to undecidability or high complexity

Aims and Objectives

• Restrict UML class diagrams to achieve tractability of query

answering in data complexity

• Better understanding of combined complexity

• Add relevant OCL constraints without losing tractability of

query answering in data complexity

1. For each attribute assertion Attribute[ i..j ]:Type j 2 {1,1}

Lean UML Class Diagrams

[Calì, G., Orsi & Pieris, FOSSACS 2012]


2. For each association A:

• upper bounds mU , nU 2 {1,1},

• if A generalizes some other association, then mU = nU = 1


C1 C2mL..mU nL..nUA



2. For each association A:

• upper bounds mU , nU 2 {1,1},

• if A generalizes some other association, then mU = nU = 1

3. Completeness constraints are forbidden


C1 C2

C

{complete}

8X C(X) C1(X) _ C2(X)

C1 C2mL..mU nL..nUA


Lean UML Class Diagrams: Example I

0..1

Member

Issues

Owns

Competes

0..1

1..1

0..1

1..1

1..1

1..1

2..1

CompanyStock

Index[0..1]:Str

getIndex():List

Executive Person

Lean UML Class Diagrams: Example II

WorksIn

Leads

1..1

3..1

Student Professor

Member

{disjoint}Group

1..1

0..1

CLeads

since: Date

Non-Diagrammatic Constraints (OCL)

C1 C2

C

C3

disjoint classes

8X C2(X) ^ C3(X) ?

We need negative constraints of the form

8X C1(X) ^ … ^ Cn(X) ?

C1 C2

C

most-specific class

We need most-specific class constraints of the form

8X C1(X) ^ … ^ Cn(X) C(X)

8X C1(X) ^ C2(X) C(X)


type the domain of Enrolled

We need domain-type constraints of the form

8X8Y C(X) ^ Attribute(Y,X) T(Y)

8X8Y CS-Course(X) ^ Enrolled(Y,X) CS-Student(Y)

Enrolled [0..1]:CS-Course

CS-Student

Enrolled [0..1]:Course

Student


OCL (Object Constraint Language)

Context C1 inv:

C1.allInstances -> forAll ( x1: C1 |


x1=x2 implies x2.oclIsTypeOf(C)

)

)

Context C1 inv:



x1<>x2

)

)

Context Object inv:

Object.allInstances -> forAll ( y: Object |

y.A.oclIsTypeOf(C) implies y.oclTypeOf(T)

)

8X C1(X) ^ C2(X) ?

8X C1(X) ^ C2(X) C(X)

8X8Y C(X) ^ A(Y,X) T(Y)

8X8Y C(X) ^ Attr(X,Y) T(Y)

8X C(X) 9Y1…9Yn Attr(X,Y1) ^ … ^ Attr(X,Yn)

8X C1(X) C2(X)

8X18X2 A(X1,X2) C1(X1) ^ C2 (X2)

8X18X28Y A(X1,X2) ^ glue(X1,X2,Y) CA(Y)

8X18X2 A(X1,X2) 9Y glue(X1,X2,Y)

8X C(X) 9Y1…9Yn A(X,Y1) ^ … ^ A(X,Yn)

8X C(X) 9Y1…9Yn A(Y1,X) ^ … ^ A(Yn,X)

8X18X2 A1(X1,X2) A2(X1,X2)

8X C1(X) ^ … ^ Cn(X) C(X)

8X8Y C(X) ^ Attr(Y,X) T(Y)

Lean UML Class Diagrams as Rules

additional constraints

associations

classes

8X C1(X) ^ … ^ Cn(X) C(X)

8X8Y C(X) ^ Attr(Y,X) T(Y)

8X8Y C(X) ^ Attr(X,Y) T(Y)

8X C(X) 9Y1…9Yn Attr(X,Y1) ^ … ^ Attr(X,Yn)

8X C1(X) C2(X)

8X18X2 A(X1,X2) C1(X1) ^ C2 (X2)

8X18X28Y A(X1,X2) ^ glue(X1,X2,Y) CA(Y)

8X18X2 A(X1,X2) 9Y glue(X1,X2,Y)

8X C(X) 9Y1…9Yn A(X,Y1) ^ … ^ A(X,Yn)

8X C(X) 9Y1…9Yn A(Y1,X) ^ … ^ A(Yn,X)

8X18X2 A1(X1,X2) A2(X1,X2)

Lean UML Class Diagrams as Rules

additional constraints

associations

classes

Data Complexity of Query Answering


Theorem: Query answering under Lean UML class diagrams + negative

& most-specific class & domain-type constraints is PTIME-complete

Data Complexity of Query Answering

Proof:

• in PTIME: reduction to query answering under guarded TGDs

• PTIME-hardness (even without domain-type constraints): reduction

from Path System Accessibility

Theorem: Query answering under Lean UML class diagrams + negative

& most-specific class & domain-type constraints is PTIME-complete


Combined Complexity of Query Answering

Theorem: Query answering under Lean UML class diagrams +

negative & most-specific class constraints is PSPACE-complete


Proof:

• in PSPACE: there exists a tree-like universal model (chase procedure)

A(y,z)

C(x)

D


Proof:


A(y,z)

…can be computed from A(y,z) [ type(A(y,z))

atoms whose arguments are among {y,z}

8X C(X) ^ A(X,Y) T(Y)

8X8Y A(X,Y) C1(X) ^ C2(Y)

8X8Y A(X,Y) A0(X,Y)

C(x)

D


Proof:


A(y,z)



• type(A(y,z)) can be computed from A(y,z) [ type(C(x)) in NP

C(x)

D

8X C(X) ^ A(X,Y) T(Y)

8X8Y A(X,Y) C1(X) ^ C2(Y)

8X8Y A(X,Y) A0(X,Y)


Proof:


A(y,z)



• type(A(y,z)) can be computed from A(y,z) [ type(C(x)) in NP

• construct non-deterministically a finite part with at most |Q| atoms

C(x)

D

8X C(X) ^ A(X,Y) T(Y)

8X8Y A(X,Y) C1(X) ^ C2(Y)

8X8Y A(X,Y) A0(X,Y)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:


A(y,z)

C(x)

D

R(a,v) S(v,w)

B(y,z) T(z)

Q Ã R(X1,X2),S(X2,X3),B(X4,X5),T(X5)


Proof:

• PSPACE-hardness: simulation of the computation of a PSPACE Turing machine

on input I = α0α1...αn-1, assuming that it uses m = nk cells

• Initialization rules 8X initial(X) initial-state(X)

8X initial(X) cell0[α0,1](X)

8X initial(X) celli[αi,0](X), for each i 2 {1,…,n-1}

8X initial(X) celli[0,0](X), for each i 2 {n,…,m-1}

initial-state cell0[α0,1] celli[αi,0] celln-1[αn-1,0]… celln[0,0] cellm-1[0,0]…

initial

8X config(X) 9Y succ(X,Y)

8X8Y config(X) ^ succ(X,Y) config(Y)

Proof:



• Configuration generation rules

8X initial(X) config(X)


initial

config

succ[1..1]:config

Proof:



• Rules to describe the transition from one configuration to another,

e.g., state transition rules


8X8Y s1-celli [α1,1](X) ^ succ(X,Y) state-s2(Y), for each i 2 {0,…,m-1}

for each δ(hs1,α1i) = hs2,α2,di:

in configuration X, which has state s1, the i-th

cell contains α1, and the cursor is over cell is1-celli [α1,1]

succ[0..1]:state-s2

Proof:



• Acceptance rule 8X accept-state(X) accept(X)

• Initial database D = {initial(c)} – c is the initial configuration

• Boolean CQ yes accept(c)

• Turing machine accepts I iff D [ ² yes


accept

accept-state


Theorem: Query answering under Lean UML class diagrams + negative &

most-specific class & domain-type constraints is EXPTIME-complete


Theorem: Query answering under Lean UML class diagrams + negative &

most-specific class & domain-type constraints is EXPTIME-complete

Proof:

• in EXPTIME: reduction to guarded TGDs of bounded arity


Proof:

• EXPTIME-hardness: simulation of an alternating PSPACE Turing machine

• Acceptance rules – q 2 {9,} and i 2 {1,2}:

8X accept-stateq(X) accept(X)

8X8Y accept(X) ^ succi(Y,X) accepti(Y)

8X state9(X) ^ accepti(X) accept(X)

8X state(X) ^ accept1(X) ^ accept2(X) accept(X)

accept

accept-stateq

domain-type

most-specific class

Further Restrictions – Restricted Lean (RLean)

different classes have disjoint sets

of attributes and operations

instead of 8X8Y Student(X) ^ Enrolled(X,Y) Course(Y)

we have 8X8Y Student-Enrolled(X,Y) Course(Y)

Data complexity in AC0 and combined complexity NP-complete

Enrolled [0..1]:CS-Course

CS-Student

Enrolled [0..1]:Course

Student

Overview

Linear

Guarded

in AC0

PTIME-c

Lean UML

RLean UML

Complexity of Querying Lean UML Class Diagrams

UML

Formalism

Additional

Constraints

Data

Complexity

Combined

Complexity

Lean

negative

specific-class

domain-type

PTIME-c EXPTIME-c

Leannegative

specific-classPTIME-c PSPACE-c

RLeannegative

specific-classin AC0 NP-c

Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List


UML Class Diagrams

F-Logic

Description Logics





student::person






WHERE {





SPARQL

Description Logics (DLs)

• Logics that model the domain of interest in terms of:

• Concepts ! sets of objects

• Roles ! binary relations on sets of objects

• Used for conceptual reasoning in the Semantic Web context






p1

p2

p3

f1 p2

f2 p3

f3 p1

person father

person v 9father¡1 each person has a father

9father v person each father is a person






p1

p2

p3

f1 p2

f2 p3

f3 p1

person father








p1

p2

p3

f1 p2

f2 p3

f3 p1

person father



The DL-Lite Family

B ::= A | 9R | 9R-1 (basic concept)

C ::= B | :B (general concept)

R ::= P | P-1 (basic role)

E ::= R | :R (general role)

DL-Litecore

DL-LiteF DL-LiteR (OWL 2 QL)

B v C

R v E(funct R)

Popular family of DLs with AC0 data complexity

[Calvanese, De Giacomo, Lembo, Lenzerini & Rosati, J. Autom. Reasoning 2007]

The DL-Lite Family

Popular family of DLs with AC0 data complexity

DL-Lite TBox First-Order Representation (Datalog§)

DL-Litecore

professor v 9teachesTo

professor v :student

DL-LiteR

hasTutor¡ v teachesTo

DL-LiteF

funct(hasTutor)

8X professor(X) 9Y teachesTo(X,Y)

8X professor(X) Æ student(X) ?

8X8Y hasTutor(X,Y) teachesTo(Y,X)

8X8Y8Z hasTutor(X,Y) Æ hasTutor(X,Z) Y = Z

The EL Family

Popular family of DLs with PTIME data complexity (biological applications)

see, e.g., [Baader, IJCAI 2003], [Rosati, DL 2007] and [Pérez-Urbina et al., J. Applied Logic 2010]

C1 v C2

R v P

EL

ELH

ELHI

ELHI :

C ::= > | A | C1 u C2 | 9P.C

P1 v P2

R ::= P | P-1

R v E E ::= R | :R

The EL Family

Popular family of DLs with PTIME data complexity (biological applications)

ELHI : TBox First-Order Representation (Datalog§)

A u B v C

A v 9R

8X A(X) Æ B(X) C(X)

8X A(X) 9Y R(X,Y)

8X A(X) 9Y R(X,Y) Æ B(Y)A v 9R.B

9R v A 8X8Y R(X,Y) A(X)

9R.A v B 8X8Y R(X,Y) Æ A(X) B(X)

A v :B 8X A(X) Æ B(X) ?

R v : P 8X8Y R(X,Y) Æ P(X,Y) ?

DLs vs. Datalog[9,=,?]

• Theorem: For query answering,

DL-Lite ·L Linear

ELHI : ·L Guarded

• Remark: the data complexity is preserved

• Theorem: Linear/Guarded TGDs are strictly more expressive

8X supervisorOf(X,X)manager(X)

Overview

Linear

Guarded

in AC0

PTIME-c

Lean UML and ELHI :DL-Lite and RLean UML

Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List


UML Class Diagrams

F-Logic

Description Logics





student::person






WHERE {





SPARQL

Frame Logic (F-Logic)

• Originally developed for object-oriented deductive databases

• Now used as an ontology language – Semantic Web

• F-Logic is in general undecidable

see, e.g., [Kifer, Lausen & Wu, J. ACM 1995]

F-Logic Lite

• An expressive decidable fragment of F-Logic

• Obtained from F-Logic by:

• Excluding negation and default inheritance

• Allowing only a limited form of cardinality constraints

• Query answering is PTIME-complete in data complexity

[Calì & Kifer, VLDB 2006]

From F-Logic Lite to First-Order Logic (Datalog§)


data(O,A,V) Æ data(O,A,W),funct(A,O) V = W

type(O,A,T) Æ data(O,A,V) member(V,T)

sub(C1,C3) Æ sub(C3,C2) sub(C1,C2)

member(O,C) Æ sub(C,C1) member(O,C1)

mandatory(A,O) 9V data(O,A,V)

member(O,C) Æ type(C,A,T) type(O,A,T)

sub(C,C1) Æ type(C1,A,T) type(C,A,T)

type(C,A,T1) Æ sub(T1,T) type(C,A,T)

sub(C,C1) Æ mandatory(A,C1) mandatory(A,C)

member(O,C) Æ mandatory(A,C) member(A,O)

sub(C,C1) Æ funct(A,C1) funct(A,C)

member(O,C) Æ funct(A,C) funct(A,O)



data(O,A,V) Æ data(O,A,W),funct(A,O) V = W














More expressive

Datalog[9] language?












Non-guarded rules

Weakly-Guarded Datalog[9]


8X8Y S(X,Y) Æ P(X,Y) 9Z P(Y,Z)

8X8Y8W P(X,Y) Æ P(W,X) S(Y,X)

Affected positions = ?

• All 8-variables at affected positions occur in one body atom

null can occur

during the chase





Affected positions = {P[2]


null can occur

during the chase





Affected positions = {P[2], S[1]


null can occur

during the chase





Affected positions = {P[2], S[1]}


null can occur

during the chase

• Models of finite treewidth ) decidability of query answering

related to [Andréka, Németi & van Benthem, J. Philosophical Logic 1998] and

[Grädel , J. Symb. Log. 1999]




null can occur

during the chase

• Query answering is EXPTIME-complete in data complexity












F-Logic Lite is Weakly-Guarded


Affected Positions

data[1]

data[3]

type[1]

member[1]

mandatory[2]

funct[2]












F-Logic Lite is Weakly-Guarded


Affected Positions

data[1]

data[3]

type[1]

member[1]

mandatory[2]

funct[2]

Tractability?

Polynomial Cloud Criterion (PCC)


cloud(a): all atoms in the chase

with arguments in a

and domain constants

• For every database, only polynomially many clouds in the chase (up

to isomorphism), and can be computed in PTIME

Polynomial Cloud Criterion (PCC)


• Theorem: Query answering under fixed weakly-guarded TGDs is

in PTIME in data complexity

• A useful tool for identifying tractable cases – F-Logic Lite

cloud(a): all atoms in the chase

with arguments in a

and domain constants

• For every database, only polynomially many clouds in the chase (up

to isomorphism), and can be computed in PTIME

Overview

Linear

Guarded

Weakly-Guarded

PCC

Lean UML and ELHI :

F-Logic Lite

in AC0

PTIME-c

EXPTIME-c

DL-Lite and RLean UML

Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List


UML Class Diagrams

F-Logic

Description Logics





student::person






WHERE {





SPARQL

SPARQL Protocol and RDF Query Language

• RDF – data model for representing information in the Web

• … in fact, is a finite set of triples (subject, predicate, object) – or

a relational database for the schema {triple(.,.,.)}

• SPARQL – the standard language for querying RDF data

Some SPARQL Queries

• P = (?X, name, ?Y) – list of pairs (o1,o2) such as o2 is the name of o1

• P = (?X, name, B) – list of elements that have a name

• P = (?X, name, ?Y) OPT (?X, phone, ?Y) – for every object o, return

the object o, the name of o, and the phone number of o, if the phone

number is available; otherwise, return the object o and its name

From SPARQL to First-Order Logic (Datalog§)

P = (?X, name, ?Y) – list of pairs (o1,o2) such as o2 is the name of o1

8X8Y triple(X,name,Y) queryP(X,Y)


P = (?X, name, B) – list of elements that have a name

8X8Y triple(X,name,Y) queryP(X)


P = (?X, name, ?Y) OPT (?X, phone, ?Y) – for every object o, return the object o, the

name of o, and the phone number of o, if the phone number is available;

otherwise, return the object o and its name

8X8Y triple(X,name,Y) Æ triple(X,phone,Z) queryP(X,Y,Z) Æ compatibleP(Z)

8X8Y triple(X,name,Y) Æ :compatibleP(X) queryP,3(X,Y)

list of individuals with phone number

the third argument (i.e., the phone no.) is missing




Non-guarded rules, and also negation is needed







Non-guarded rules, and also negation is needed

Stratified Weakly-Guarded Datalog[9,:,?]




Additional Functionalities

• Reasoning capabilities – deal with RDFS and OWL vocabularies

• Navigational capabilities – exploit the graph structure of RDF data

• General form of recursion – express natural queries

Additional Functionalities

• Reasoning capabilities – deal with RDFS and OWL vocabularies

• Navigational capabilities – exploit the graph structure of RDF data

• General form of recursion – express natural queries

Theorem: Stratified weakly-guarded Datalog[9,:,?] is strictly more

expressive than SPARQL enriched with the above functionalities

(under the the OWL 2 QL and OWL 2 RL profiles of OWL 2)

Overview

Linear

Guarded

Weakly-Guarded

PCC

Lean UML and ELHI :

F-Logic Lite

in AC0

PTIME-c

EXPTIME-c

DL-Lite and RLean UML

SPARQL

Complexity of Datalog[9,=,:,?]

Linear Guarded Weakly-Guarded

Arbitrary

QueryPSPACE-c 2EXPTIME-c 2EXPTIME-c

Arbitrary

ProgramFixed/Atomic


Arbitrary

QueryNP-c NP-c EXPTIME-c

Fixed

ProgramFixed/Atomic

Queryin AC0 PTIME-c EXPTIME-c


Data Complexity


Arbitrary


Arbitrary

ProgramFixed/Atomic


Arbitrary

QueryNP-c NP-c EXPTIME-c

Fixed

ProgramFixed/Atomic

Queryin AC0 PTIME-c EXPTIME-c


Polynomial Cloud Criterion


Arbitrary


Arbitrary

ProgramFixed/Atomic


Arbitrary

QueryNP-c NP-c NP-c

Fixed

ProgramFixed/Atomic

Queryin AC0 PTIME-c PTIME-c

Competes

Stock

0..1

Member

Owns

0..1

1..1

0..1

1..1

1..1

1..1

2..1

Company

Executive Person


getIndex():List


UML Class Diagrams

F-Logic

Description Logics





student::person






WHERE {





SPARQL

Datalog±




Thank you!