Querying UML Class Diagrams - FoSSaCS 2012

Querying UML Class Diagrams

Georg Gottlob
Department of Computer Science
University of Oxford

joint work with Andrea Calì, Giorgio Orsi and Andreas Pieris

Major Modeling Formalisms for Data and Objects
Competes
person: SSN  Name
0..1
Stock employee[1]  person[1]
Company Issues
Index[0..1]:Str
0..1 1..1 1..1
getIndex():List
Relational Data Dependencies
1..1
0..1
Member

Owns
2..1

Executive Person
1..1

UML Class Diagrams XML Schemas

student v member
m_name since g_name leads¡ v works
member
(1,1) 1
works
2 (1,N)
group
student v :professor
(1,1)
[1,2]
Description Logics
(1,N)
student professor leads
1 2
Context C1 inv: C1.allInstances -> forAll ( x1: C1 | C2.allInstances ->
forAll ( x2: C2 | x1=x2 implies x2.oclIsTyeOf(C)))
ER Diagrams
Object Constraint Language (OCL)

Datalog± : A Unifying Logical Framework

Datalog±

Description Logics
(DL-Lite, EL,…) Relational Constraints
(IDs, FKDs,…)

Datalog
Conceptual Models
(UML, ER,…)

… providing: Logical foundations, semantics, decidability and
complexity results for reasoning and query-answering,
identification of tractable fragments, …

Datalog§
8X8Y (X,Y)  (X)

• Extend Datalog with additional features such as:
• Existential quantification (9): TGDs 8X8Y (X,Y)  9Z (X,Z)

• Equality atoms (=): EGDs 8X (X)  Xi = Xj

• Constant false (?): Negative constraints 8X (X)  ?

• But query answering under Datalog[9] is undecidable
[see, e.g., Beeri & Vardi, ICALP 81]

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

Restriction: Guardedness

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

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

guard

• Models of finite treewidth ) decidability of query answering
[Calì, G. & Kifer, KR 08] related to work by [Andréka, Németi & van Benthem] and [Grädel]

• Query answering is PTIME-complete in data complexity
[Calì, G. & Lukasiewicz, PODS 09]

Reasoning over UML Class Diagrams

Competes
0..1 Stock
Company Issues
0..1 1..1
Index[0..1]:Str
1..1
1..1 getIndex():List
0..1
Member
Owns
2..1
Executive Person
1..1

Satisfiability: Does the diagram admit at least one instantiation?


• Satisfiability - the diagram has a (possibly infinite) non-empty
instantiation

• Full Satisfiability - the diagram has a (possibly infinite) instantiation
where each class and association is non-empty

• Finite Satisfiability - the diagram has a finite instantiation


Person

{disjoint}

Student Worker

finitely satisfiable, e.g., {Worker(john),Person(john)}

but not fully - student class is necessarily empty

Complexity of Reasoning over UML Class Diagrams

• Satisfiability is EXPTIME-complete
[Berardi, Calvanese & De Giacomo, Artificial Intelligence 05]

• Full Satisfiability is EXPTIME-complete
[Artale, Calvanese & Ibánez-García, ER 10]

• Finite Satisfiability is EXPTIME-complete
[implicit in Berardi, Calvanese & De Giacomo, Artificial Intelligence 05]


Executive(john) Competes
Member(john,LU) 0..1
Stock
Stock(BAY) Company Issues
0..1 1..1 1..1
Index[0..1]:Str
Issues(BA,BAY)
Owns(john,BAY) 0..1
Competes(LU,BA) Member
Owns
2..1
Executive Person
1..1

Which persons have a potential conflict of interest?


Stock
0..1 1..1 1..1
Index[0..1]:Str
Issues(BA,BAY)
Owns(john,BAY) 0..1
Owns
2..1
Executive Person
1..1

Conflict(P)  Person(P), Company(C1), Company(C2), Stock(S),
Owns(P,S), Member(P,C1), Issues(C2,S), Competes(C1,C2)


Stock
0..1 1..1 1..1
Index[0..1]:Str
Issues(BA,BAY)
Owns(john,BAY) 0..1
Owns
2..1
Executive Person
1..1

Does anybody have a potential conflict of interest?


Stock
0..1 1..1 1..1
Index[0..1]:Str
Issues(BA,BAY)
Owns(john,BAY) 0..1
Owns
2..1
Executive Person
1..1

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


Stock
0..1 1..1 1..1
Index[0..1]:Str
Issues(BA,BAY)
Owns(john,BAY) 0..1
Person(john) Owns
2..1
Executive Person
1..1



Stock
0..1 1..1 1..1
Index[0..1]:Str
Issues(BA,BAY)
Owns(john,BAY) 0..1
Person(john) Owns
2..1
Company(LU) Executive Person
Company(BA) 1..1



Stock
0..1 1..1 1..1
Index[0..1]:Str
Issues(BA,BAY)
Owns(john,BAY) 0..1
Person(john) Owns
2..1
Company(LU) Executive Person
Company(BA) 1..1

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


Querying UML Class Diagrams: Existential Rules

Group(DB)
Member 3..1 WorksIn

1..1
{disjoint} Group

0..1
Student Professor
1..1 Leads

CLeads
since: Date

Is there a professor who works in the database group?


Group(DB)
Member 3..1 WorksIn

1..1
{disjoint} Group

0..1
Student Professor
1..1 Leads

CLeads
since: Date

Ans  Professor(P), WorksIn(P,DB)


Group(DB)
Leads(z1,DB) Member 3..1 WorksIn
Professor(z1) 1..1
{disjoint} Group
R*(z1,DB,z2)
CLeads(z2) 0..1
Student Professor
1..1 Leads

CLeads
since: Date



Group(DB)
Professor(z1) 1..1
{disjoint} Group
R*(z1,DB,z2)
CLeads(z2) 0..1
WorksIn(z1,DB) Student Professor
1..1 Leads

CLeads
since: Date



Group(DB)
Professor(z1) 1..1
{disjoint} Group
R*(z1,DB,z2)
CLeads(z2) 0..1
WorksIn(z1,DB) Student Professor
1..1 Leads
…
CLeads
since: Date

{P ! z1, DB ! DB}



D



Q

D[²Q , 8M (M ² D [  ! M ² Q)


D



Q

D[²Q , 8M (M ² D [  ! M ² Q)

M¶D Æ M²

From Diagrams to First-Order Logic (Datalog§)
Competes
0..1 Stock
Company Issues
0..1 1..1
Index[0..1]:Str
1..1
0..1
Member
Owns
2..1
Executive Person
1..1

Competes
0..1 Stock
Company Issues
0..1 1..1
Index[0..1]:Str
1..1
0..1
Member
Owns
2..1
Executive Person
1..1

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)
[Satoh & Kaneiwa, TCS 10
and Berardi et al., AI 05]

Competes
0..1 Stock
Company Issues
0..1 1..1
Index[0..1]:Str
1..1
0..1
Member
Owns
2..1
Executive Person
1..1

8X Company(X)  9Y Issues(X,Y) 8X Stock(X)  9Y Issues(Y,X)
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) and Berardi et al., AI 05]

Competes
0..1 Stock
Company Issues
0..1 1..1
Index[0..1]:Str
1..1
0..1
Member
Owns
2..1
Executive Person
1..1

8X Executive(X)  Person(X)

and Berardi et al., AI 05]

Complexity of Query Answering

• EXPTIME-complete in combined complexity (everything is part of the input)
[implicit in Berardi et al., AI 05 and Lutz, IJCAR 08]

• coNP-complete in data complexity (the diagram and the query are fixed)
[implicit in Ortiz, Calvanese & Eiter, AAAI 06]

• Undecidable when diagrams are combined with arbitrary OCL
(Object Constraint Language) constraints
[folklore]

Research Challenge: Reduce High Complexity

• Diagrams often have very large instantiations

• Some applications require very large diagrams

• OCL constraints, that are not expressible diagrammatically,
lead to undecidability or high complexity

Our Goals

• 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

Lean UML Class Diagrams

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



mL..mU A nL..nU
• For each association A: C1 C2

- upper bounds mU ,nU 2 {1,1},

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



mL..mU A nL..nU
• For each association A: C1 C2

- upper bounds mU ,nU 2 {1,1},

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

C
• Completeness constraints are forbidden
{complete}

8X C(X)  C1(X) _ C2(X) C1 C2

Lean UML Class Diagrams: Example

Competes
0..1 Stock
Company Issues
0..1 1..1
Index[0..1]:Str
1..1
0..1
Member
Owns
2..1
Executive Person
1..1

Lean UML Class Diagrams: Example

3..1 WorksIn
Member
1..1
{disjoint} Group

0..1
Student Professor
1..1 Leads

CLeads
since: Date

Add Some Non-Diagrammatic Constraints (OCL)

C

C1 C2 8X C2(X) ^ C3(X)  ?
C3

disjoint classes

We need negative constraints of the form

8X C1(X) ^ … ^ Cn(X)  ?


C1 C2

C 8X C1(X) ^ C2(X)  C(X)

most-specific class

We need most-specific class constraints of the form

8X C1(X) ^ … ^ Cn(X)  C(X)


Student
Enrolled[0..1]:Course

type the domain of Enrolled

CS-Student
Enrolled[0..1]:CS-Course

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

We need domain-type constraints of the form

8X8Y C(X) ^ Attr(Y,X)  T(Y)


Student


CS-Student




pullback rule


Student


CS-Student




pullback rule will make a difference!

OCL (Object Constraint Language)

Context C1 inv:
C1.allInstances -> forAll ( x1: C1 |
8X C1(X) ^ C2(X)  ? C2.allInstances -> forAll ( x2: C2 |
x1<>x2
)
)

Context C1 inv:
8X C1(X) ^ C2(X)  C(X) x1=x2 implies x2.oclIsTypeOf(C)
)
)

Context Object inv:
8X8Y C(X) ^ a(Y,X)  T(Y) Object.allInstances -> forAll ( y: Object |
y.a.oclIsTypeOf(C) implies y.oclTypeOf(T)
)

Lean UML Class Diagrams as Rules
8X8Y C(X) ^ Attr(X,Y)  T(Y)
8X C(X)  9Y1…9Yn Attr(X,Y1) ^ … ^ Attr(X,Yn) 1 ≤ i < j ≤ n Yi ≠ Yj
8X8Y8Z C(X) ^ Attr(X,Y) ^ Attr(X,Z)  Y = Z
8X8Y1…8Yn8Z C(X) ^ Op(X,Y1, …,Yn,Z)  T1(Y1) ^ … ^ Tn(Yn) ^ T(Z)
8X8Y1…8Yn8Z18Z2 C(X) ^ Op(X,Y1, …,Yn,Z1) ^ Op(X,Y1, …,Yn,Z2)  Z1 = Z2
8X C1(X)  C2(X)
8X C1(X) ^ … ^ Cn(X)  ?
8X1…8Xn A(X1,…,Xn)  C1(X1) ^ … ^ Cn(Xn) 8X18X2 A(X1,X2)  C1(X1) ^ C2(X2)
8X1…8Xn8Y A(X1,…,Xn) ^ R*(X1,…,Xn,Y)  CA(Y) 8X18X28Y A(X1,X2) ^ R*(X1,X2,Y)  CA(Y)
8X1…8Xn A(X1,…,Xn)  9Y R*(X1,…,Xn,Y) 8X18X2 A(X1,X2)  9Y R*(X1,X2,Y)
8X1…8Xn8Y8Z A(X1,…,Xn) ^ R*(X1,…,Xn,Y) ^ R*(X1,…,Xn,Z)  Y = Z
8X1…8Xn8Y1…8Yn8Z R*(X1,…,Xn,Z) ^ R*(Y1,…,Yn,Z) ^ CA(Z)  X1 = Y1 ^ … ^ Xn = Yn
8X C(X)  9Y1…9Yn A(X,Y1) ^ … ^ A(X,Yn) 1 ≤ i < j ≤ n Yi ≠ Yj
8X8Y8Z C(X) ^ A(X,Y) ^ A(X,Z)  Y = Z
8X C(X)  9Y1…9Yn A(Y1,X) ^ … ^ A(Yn,X) 1 ≤ i < j ≤ n Yi ≠ Yj
8X8Y8Z C(X) ^ A(Y,X) ^ A(Z,X)  Y = Z
8X1…8Xn A1(X1,…,Xn)  A2(X1,…, Xn) 8X18X2 A1(X1,X2)  A2(X1,X2)

8X C(X)  9Y1…9Yn Attr(X,Y1) ^ … ^ Attr(X,Yn) 1 ≤ i < j ≤ n Yi ≠ Yj
pre-process
8X8Y8Z C(X) ^ Attr(X,Y) ^ Attr(X,Z)  Y = Z
8X8Y1…8Yn8Z C(X) ^ Op(X,Y1, …,Yn,Z)  T1(Y1) ^ … ^ Tn(Yn) ^ T(Z)
8X8Y1…8Yn8Z18Z2 C(X) ^ Op(X,Y1, …,Yn,Z1) ^ Op(X,Y1, …,Yn,Z2)  Z1 = Z2 check via query
8X C1(X)  C2(X)
8X C1(X) ^ … ^ Cn(X)  ?
8X1…8Xn A(X1,…,Xn)  C1(X1) ^ … ^ Cn(Xn) 8X18X2 A(X1,X2)  C1(X1) ^ C2(X2)
8X1…8Xn8Y A(X1,…,Xn) ^ R*(X1,…,Xn,Y)  CA(Y) 8X18X28Y A(X1,X2) ^ R*(X1,X2,Y)  CA(Y)
8X1…8Xn A(X1,…,Xn)  9Y R*(X1,…,Xn,Y) 8X18X2 A(X1,X2)  9Y R*(X1,X2,Y)
8X1…8Xn8Y8Z A(X1,…,Xn) ^ R*(X1,…,Xn,Y) ^ R*(X1,…,Xn,Z)  Y = Z
8X1…8Xn8Y1…8Yn8Z R*(X1,…,Xn,Z) ^ R*(Y1,…,Yn,Z) ^ CA(Z)  X1 = Y1 ^ … ^ Xn = Yn
8X C(X)  9Y1…9Yn A(X,Y1) ^ … ^ A(X,Yn) 1 ≤ i < j ≤ n Yi ≠ Yj
8X8Y8Z C(X) ^ A(X,Y) ^ A(X,Z)  Y = Z
8X C(X)  9Y1…9Yn A(Y1,X) ^ … ^ A(Yn,X) 1 ≤ i < j ≤ n Yi ≠ Yj
8X8Y8Z C(X) ^ A(Y,X) ^ A(Z,X)  Y = Z
8X1…8Xn A1(X1,…,Xn)  A2(X1,…, Xn) 8X18X2 A1(X1,X2)  A2(X1,X2)



8X C(X)  9Y1…9Yn Attr(X,Y1) ^ … ^ Attr(X,Yn) classes
8X C1(X)  C2(X)

8X18X2 A(X1,X2)  C1(X1) ^ C2 (X2)

8X18X28Y A(X1,X2) ^ R*(X1,X2,Y)  CA(Y)

8X18X2 A(X1,X2)  9Y R*(X1,X2,Y)
associations
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)
additional constraints



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) ^ R*(X1,X2,Y)  CA(Y)

8X18X2 A(X1,X2)  9Y R*(X1,X2,Y)
guard atoms
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)


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

Theorem: Query answering under Lean UML class diagrams + negative
& most-specific class & domain-type constraints is PTIME-complete

Proof:

• in PTIME: reduction to assertions 8X8Y body(X,Y)  9Z head(X,Z),
where body(X,Y) has a guard-atom [Calì, G. & Lukasiewicz, PODS 09]

• PTIME-hardness (even without domain-type constraints): reduction
from Path System Accessibility

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

D
8X8Y C(X) ^ A(X,Y)  T(Y)
C(x)
8X8Y A(X,Y)  C1(X) ^ C2(Y)
polynomial depth
Relevant part:

8X8Y A(X,Y)  B(X,Y)
A(y,z)

B(y,z) T(z)
R(a,v) S(v,w)

Proof:

D
8X8Y C(X) ^ A(X,Y)  T(Y)
C(x)
8X8Y A(X,Y)  C1(X) ^ C2(Y)
polynomial depth
Relevant part:

A(y,z)

B(y,z) T(z)
R(a,v) S(v,w)

Q= R(X1,X2) ^ S(X2,X3) ^ B(X4,X5) ^ T(X5)

Proof:
• in PSPACE: there exists a tree-like universal model (Chase)

D
8X8Y C(X) ^ A(X,Y)  T(Y)
C(x)
8X8Y A(X,Y)  C1(X) ^ C2(Y)
polynomial depth
Relevant part:

A(y,z)

B(y,z) T(z)
R(a,v) S(v,w)

Q= R(X1,X2) ^ S(X2,X3) ^ B(X4,X5) ^ T(X5)

Proof:

D
8X8Y C(X) ^ A(X,Y)  T(Y)
C(x)
8X8Y A(X,Y)  C1(X) ^ C2(Y)
A(y,z)

B(y,z) T(z)
R(a,v) S(v,w)

Q= R(X1,X2) ^ S(X2,X3) ^ B(X4,X5) ^ T(X5)

With each current atom A we need to compute and memorize its type(A). This is in NP

Proof:

D

C(x)

A(y,z)

B(y,z) T(z)
R(a,v) S(v,w)

8X8Y C(X) ^ A(X,Y)  T(Y)

With each current atom A we need to compute 8X8Y A(X,Y)  C1(X) ^ C2(Y)
and memorize its type(A). This is in NP 8X8Y A(X,Y)  B(X,Y)

Proof:

D

C(x)

A(y,z)

B(y,z) T(z)
R(a,v) S(v,w)

8X8Y C(X) ^ A(X,Y)  T(Y)
Q= R(X1,X2) ^ S(X2,X3) ^ B(X4,X5) ^ T(X5)
With each current atom A we need to compute 8X8Y A(X,Y)  C1(X) ^ C2(Y)
and memorize its type(A). This is in NP 8X8Y A(X,Y)  B(X,Y)


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] cell1[α1,0] … celln-1[αn-1,0] celln[0,0] … cellm-1[0,0]

initial


Proof:

• Configuration generation rules

config
8X initial(X)  config(X) succ[1..1]:config

8X config(X)  9Y succ(X,Y)

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


Proof:

• Rules to describe the transition from one configuration to another,
e.g., state transition rules

for each δ(hs1,α1i) = hs2,α2,di:

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

in configuration X, which has state s1, the i-th
cell contains α1, and the cursor is over cell i s1-celli [α1,1]
succ[0..1]:state-s2


Proof:

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

accept

accept-state

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

• Boolean CQ Q= accept(X)

• Turing machine accepts I iff D [  ² Q


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 assertions 8X8Y body(X,Y)  9Z head(X,Z), where
body(X,Y) has a guard-atom, and all the predicates are of bounded arity
[Calì, G. & Kifer, KR 08]

Proof:
• EXPTIME-hardness: simulation of an alternating PSPACE Turing machine
• Acceptance rules - q 2 {9,} and i 2 {1,2}:

8X q-accept-state(X)  accept(X) accept

q-accept-state

8X8Y accept(X) ^ succ1(Y,X)  accept1(Y) domain-type constraints
pullback rules
8X8Y accept(X) ^ succ2(Y,X)  accept2(Y)

8X 9-state(X) ^ accept1(X)  accept(X)
most-specific class
8X 9-state(X) ^ accept2(X)  accept(X) constraints
8X -state(X) ^ accept1(X) ^ accept2(X)  accept(X)

Further Restrictions

Student
Enrolled [0..1]:Course

CS-Student
Enrolled [0..1]:CS-Course

Further Restrictions

Student
Enrolled [0..1]:Course

different classes have disjoint sets
of attributes and operations

CS-Student
Enrolled [0..1]:CS-Course

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

Complexity of Querying UML Class Diagrams

UML Additional Data Combined
Formalism Constraints Complexity Complexity

Full none coNP-complete EXPTIME-complete

+ negative
Lean + specific-class PTIME-complete EXPTIME-complete
+ domain-type

+ negative
Lean PTIME-complete PSPACE-complete
+ specific-class

Restricted + negative
in AC0 NP-complete
Lean + specific-class

Datalog± : A Unifying Logical Framework

Datalog±

Description Logics
(DL-Lite, EL,…) Relational Constraints
(IDs, FKDs,…)

Datalog
Conceptual Models
(UML, ER,…)

… without losing tractable data complexity

Ontological Reasoning and Datalog

DL Assertion Datalog Rule
Concept Inclusion
emp v person emp(X)  person(X)
Concept Product
sen-emp £ emp v moreThan sen-emp(X) ^ emp(Y)  moreThan(X,Y)
(Inverse) Role Inclusion
reports¡ v mgr reports(X,Y)  mgr(Y,X)
Role Transitivity
trans(mgr) mgr(X,Y) ^ mgr(Y,Z)  mgr(X,Z)

Ontological Reasoning and Datalog

DL Assertion Datalog Rule
Concept Inclusion
emp v person emp(X)  person(X)
Concept Product
sen-emp £ emp v moreThan sen-emp(X) ^ emp(Y)  moreThan(X,Y)
(Inverse) Role Inclusion
reports¡ v mgr reports(X,Y)  mgr(Y,X)
Role Transitivity
trans(mgr) mgr(X,Y) ^ mgr(Y,Z)  mgr(X,Z)

Participation
emp v 9report emp(X)  9Y report(X,Y)
Disjointness
emp u customer v ? emp(X) ^ customer(X)  ?
Functionality
funct(reports) reports(X,Y) ^ reports(X,Z)  Y = Z

Guardedness

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

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

guard

• Models of finite treewidth ) decidability of query answering
[Calì, G. & Kifer, KR 08]

• Query answering is PTIME-complete in data complexity
[Calì, G. & Lukasiewicz, PODS 09]

• Properly extends ELH (same data complexity)

Ontology Querying

ELH: Popular DL with PTIME data complexity
[Baader, IJCAI 03 and Rosati, DL 07]

ELH TBox Datalog§ Representation

AvB 8X A(X)  B(X)

Au Bv C 8X A(X) ^ B(X)  C(X)

A v 9R.B 8X A(X)  9Y R(X,Y) ^ B(Y)

9R.A v B 8X8Y R(X,Y) ^ A(Y)  B(X)

RvP 8X8Y R(X,Y)  P(X,Y)

First-Order Rewritable Datalog± Fragments

Q 

compilation
Q


Q 

D


Q 

compilation
Q Q*
FO+
translation SQL

D


Q 

compilation
Q Q*
evaluation
translation SQL

D

8D: D [  ² Q , D ² Q*

Linearity

• Just one atom in the body 8X8Y R(X,Y) ! 9Z (X,Z)

guard

• Linear TGDs are trivially guarded

• Linear TGDs are first-order rewritable [Calì, G. & Lukasiewicz, PODS 09]
 Query answering in AC0 data complexity

• Polynomial-size rewriting (SQL DL = NR Datalog) [G. & Schwentick, KR 12]

• Properly extends DL-Lite (same data complexity)

Ontology Querying

DL-Lite: Popular family of DLs with AC0 data complexity (OWL 2 QL)
[Calvanese, De Giacomo, Lembo, Lenzerini & Rosati, JAR 07]

DL-Lite TBox Datalog§ Representation

AvB 8X A(X)  B(X)

A v 9R 8X A(X)  9Y R(X,Y)

9R v A 8X8Y R(X,Y)  A(X)

RvP 8X8Y R(X,Y)  P(X,Y)

Finite Controllability

?
D[²Q , D [  ²fin Q

• Holds for inclusion dependencies
[Rosati, PODS 06]

• Holds for guarded Datalog± (in fact, for the guarded fragment)
[Bárány, G. & Otto, LICS 10]

• Different from finite-model property of the guarded fragment:
If D [  [ Q has a model, then D [  [ Q it has a finite one.

Finite Controllability and Lean UML Class Diagrams

• For each attribute assertion Attribute[ i..j ]:Type: j = 1

C1 mL..mU A nL..nU
• For each association A: mU = nU = 1 C2

• Disjointness and negative constraints are forbidden

• Most-specific class and domain-type constraints are allowed

set of guarded TGDs ) finite controllability holds

Datalog§: Overview

Guarded Linear

8X8Y R(X,Y,Y) ! 9Z P(X,Z)

Datalog§: Overview

Guarded Linear Sticky-join

8X8Y R(X,Y,Y) ! 9Z P(X,Z) 8X8Y8Z R(X,Y) ^ S(Y,Z) ! 9W P(Y,W)

Datalog§: Overview
DL-Lite

PTIME-complete FO-rewritable

Guarded Linear Sticky Sticky-join

ELH
IDs + FKDs
Lean UCDs

Datalog§: Summary of Complexity Results

Data Fixed  Combined

Guarded PTIME-complete NP-complete 2EXPTIME-complete

Linear in AC0 NP-complete PSPACE-complete

Sticky-join in AC0 NP-complete EXPTIME-complete

Same complexity with negative constraints and non-conflicting EGDs

Datalog§: Next Steps

PTIME-complete

PTIME-complete FO-rewritable

Guarded Linear Sticky-join

?
…with disjunction in the head

… finite-model reasoning

But…
• What about joins in rule bodies?
8A8D8P runs(D,P) ^ area(P,A)  9E employee(E,D,P,A)

• What about the DL assertion concept product?
8E8M elephant(E) ^ mouse(M)  biggerThan(E,M)

But…
• What about joins in rule bodies?

• What about the DL assertion concept product?

No tree-like models guaranteed

8X8Y R(X,Y)  9Z R(Y,Z)
Infinitely many symbols in S
8X8Y R(X,Y)  S(X)

8X8Y S(X) ^ S(Y)  P(X,Y) P forms an infinite clique

Stickiness



Stickiness

8X8Y8Z R(X,Y) ^ P(Y,Z)  9W T(X,Y,W)

8X8Y8Z T(X,Y,Z)  9W S(Y,W)

8X8Y8Z R(X,Y) ^ P(Y,Z)  9W T(X,Y,W)

8X8Y8Z T(X,Y,Z)  9W S(X,W)

Stickiness

• Properly generalize inclusion dependencies

• Backward-resolution terminates

• Query answering is in AC0 in data complexity (first-order rewritability)
[Calì, G. & Pieris, VLDB 10]

• Properly extends DL-Lite (same data complexity)

Additional Features
• EGDs, e.g., 8X8Y8Z reports(X,Y) ^ reports(X,Z) ! Y = Z
Non-Conflicting EGDs: do not interact with TGDs
Preliminary check without adding complexity

• Negative constraints, e.g., 8X emp(X) ^ customer(X) ! ?
Check without adding complexity

Finite controllability does not hold
D = {R(a,b)}
D[²Q
8X8Y R(X,Y)  9Z R(Y,Z) but
=
8X8Y8Z R(Y,X) ^ R(Z,X)  Y = Z D [  ²fin Q

Q  R(A,a)

Additional Features
• EGDs, e.g., 8X8Y8Z reports(X,Y) ^ reports(X,Z) ! Y = Z
Non-Conflicting EGDs: do not interact with TGDs
Preliminary check without adding complexity

• Negative constraints, e.g., 8X emp(X) ^ customer(X) ! ?
Check without adding complexity

Comparison with ER§ Schemata
ER§: Extended ER formalism with AC0 data complexity
[Calì, G. & Pieris, Information Systems 2012]

Lean UML ER§

IS-A among classes

IS-A among associations

¸n participation n¸0 n 2 {0,1}

·1 participation

Permutation on IS-A

Values of attributes complex atomic
Operations

Attribute re-use

The Chase Procedure

Input: Database D, set of TGDs 

Output: A model of D [ 

D
person(john)


person(P)  9F father(F,P) father(F,P)  person(F)

chase(D,) = D [ ?

The Chase Procedure



D
person(john)



chase(D,) = D [ {father(z1,john)

The Chase Procedure



D
person(john)



chase(D,) = D [ {father(z1,john), person(z1)

The Chase Procedure



D
person(john)



chase(D,) = D [ {father(z1,john), person(z1), father(z2,z1)

The Chase Procedure



D
person(john)



chase(D,) = D [ {father(z1,john), person(z1), father(z2,z1), …}

The Chase Procedure



D
person(john)



chase(D,) = D [ {father(z1,john), person(z1), father(z2,z1), …}

infinite instance

Query Answering via Chase
Q h

C = chase(D,)

D

h1 h2

h2(C)
h1(C) . . .
M1
M2

D[²Q , chase(D,) ² Q
[see, e.g., Deutsch, Nash & Remmel, PODS 08]

Bounded Derivation-Depth Property (BDDP)

D
Q

constant depth
w.r.t. D

P

chase(D,)

chase(D,) ² Q ) P²Q

[Calì, G. & Lukasiewitcz, PODS 09]

Bounded Derivation-Depth Property (BDDP)

D
Q

constant depth
w.r.t. D

P

chase(D,)

BDDP ) First-Order Rewritability

[Calì, G. & Lukasiewitcz, PODS 09]

First-Order Rewritable TGDs

 Q

rewriting

Q

8D: D [  ² Q , D ² Q

Query answering is in AC0 in data complexity [Vardi, PODS 95]

OMG UML (Unified Modeling Language)
Standard conceptual modeling tool for software design

Competes
0..1 Stock
Company Issues
Index[0..1]:Str
0..1 1..1 1..1
0..1
Member
Owns
2..1
Executive Person
1..1

Class Diagrams

Reasoning over UML models
Model checking
Specification recovery
Software maintenance

Querying UML Class Diagrams - FoSSaCS 2012

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