Andreas Pieris - dtai.cs.kuleuven.beDefault Negation for Datalog§ Andreas Pieris Institute of...
Transcript of Andreas Pieris - dtai.cs.kuleuven.beDefault Negation for Datalog§ Andreas Pieris Institute of...
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Default Negation for Datalog§
Andreas Pieris
Institute of Information Systems, Vienna University of Technology, Austria
GTTV, Lexington, KY, USA, September 27, 2015
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Goal of the Datalog§ Project
Transform Datalog from a first class database query language
to a first class language for knowledge representation
(and other applications)
But first, let say few words about the good old plain Datalog
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Datalog
• Recursive database query language defined in the 1980s
• A useful framework for inductive definitions
• Simple syntax and clear semantics
• Well-understood (query answering and containment, optimisations)
• Large projects and companies are “Datalog-based”
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London
Vienna
Larnaca
Glasgow
Edinburgh
Datalog
Is Glasgow reachable from Vienna?
Flight(X,Y) Reachable(X,Y)
Flight(X,Y), Reachable(Y,Z) Reachable(X,Z)
Reachable(Vienna,Glasgow) Yes()
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Flight(X,Y) Reachable(X,Y)
Flight(X,Y), Reachable(Y,Z) Reachable(X,Z)
Reachable(Vienna,Glasgow) Yes()
Datalog
DATALOG = Select-Project-Join + Recursion
Recursion - FOL or SQL queries are not enough
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
Parent u Malev Father
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
MetalDevicev 8hasPart.Metal
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
brotherOfv relativeOf
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
R inv S R(X,Y) S(Y,X)
parentOf inv childOf
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
R inv S R(X,Y) S(Y,X)
trans(R) R(X,Y), R(Y,Z) R(X,Z)
trans(ancestorOf)
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
R inv S R(X,Y) S(Y,X)
trans(R) R(X,Y), R(Y,Z) R(X,Z)
A v 9R.B A(X) 9Y R(X,Y) R(X,Y) B(Y)
Studentv attends.Course
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
R inv S R(X,Y) S(Y,X)
trans(R) R(X,Y), R(Y,Z) R(X,Z)
A v 9R.B A(X) 9Y R(X,Y) R(X,Y) B(Y)
A v 9·1R.B A(X), R(X,Y), B(Y), R(X,Z), B(Z) Y = Z
Personv 9·1hasPassport.Valid
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Modeling Ontologies
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
R inv S R(X,Y) S(Y,X)
trans(R) R(X,Y), R(Y,Z) R(X,Z)
A v 9R.B A(X) 9Y R(X,Y) R(X,Y) B(Y)
A v 9·1R.B A(X), R(X,Y), B(Y), R(X,Z), B(Z) Y = Z
A disj B A(X), B(X) ?
Student disj Professor
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Modeling Ontologies using Datalog
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
R inv S R(X,Y) S(Y,X)
trans(R) R(X,Y), R(Y,Z) R(X,Z)
A v 9R.B A(X) 9Y R(X,Y) R(X,Y) B(Y)
A v 9·1R.B A(X), R(X,Y), B(Y), R(X,Z), B(Z) Y = Z
A disj B A(X), B(X) ?
Much is possible with Datalog
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Modeling Ontologies using Datalog
DL Axiom Rule-based Representation
A u B v C A(X), B(X) C(X)
A v 8R.B A(X), R(X,Y) B(Y)
R v S R(X,Y) S(X,Y)
R inv S R(X,Y) S(Y,X)
trans(R) R(X,Y), R(Y,Z) R(X,Z)
A v 9R.B A(X) 9Y R(X,Y) R(X,Y) B(Y)
A v 9·1R.B A(X), R(X,Y), B(Y), R(X,Z), B(Z) Y = Z
A disj B A(X), B(X) ?
Much is not possible with Datalog
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Datalog+
• Extend Datalog by allowing in the head:
o Existential quantification (9)
o Equality atoms (=)
o Constant false (?)
…for query answering over databases
Datalog[9,=,?]
highly expressive KR language
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Datalog+ vs. DLs
• Several Horn-DLs (no disjunction) can be expressed via Datalog+ rules
• But, Datalog+ rules can express more
• Higher arity predicates allow for more flexibility
o DLs have only unary and binary predicates - concepts and roles
Boss(X) supervisorOf(X,X)
siblingOf(X,Y) 9Z (parentOf(Z,X), parentOf(Z,Y))
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Datalog+: Other Appications
• Data Exchange
• Data Extraction
• Conceptual Modeling (e.g., UML)
• Querying the Semantic Web (RDF graphs)
• Automated Product Configuration
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Datalog+: Other Appications
• Data Exchange
• Data Extraction
• Conceptual Modeling (e.g., UML)
• Querying the Semantic Web (RDF graphs)
• Automated Product Configuration
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Data Exchange
Source Schema Target Schema
S T
Σst
Σt
person(ID, Name)
employee(Name, Address)
employee(N,A) → ID person(ID,N)
person(ID,N1), person(ID,N2) → N1 = N2
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Data Extraction
PRODUCT
Toshiba_Protege_cx
Dell_25416
Dell_23233
Acer_78987
PRICE
480
360
470
390
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Data Extraction
PRODUCT
Toshiba_Protege_cx
Dell_25416
Dell_23233
Acer_78987
PRICE
480
360
470
390
T1 T2
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Data Extraction
we need object creation...
PRODUCT
Toshiba_Protege_cx
Dell_25416
Dell_23233
Acer_78987
PRICE
480
360
470
390
T1 T2
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PRODUCT
Toshiba_Protege_cx
Dell_25416
Dell_23233
Acer_78987
PRICE
480
360
470
390
Data Extraction
table(T1),
table(T2),
sameColor(T1,T2),
isNeighbourRight(T1,T2) 9T tablebox(T),
contains(T,T1), contains(T,T2)
T1 T2
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Conceptual Modeling
Stock
0..1
Member
Owns
Competes
0..1
1..1
0..1
1..1
1..1
1..1
1..1
Company
Executive Person
IssuesIndex[0..1]:Str
getIndex():List
Company(X) 9Y Issues(X,Y)
Stock(X), Issues(Y,X), Issues(Z,X) Y = Z
Stock(X),Index(X,Y) Str(Y)
Stock(X), getIndex(X,Y) List(Y)
Stock(X) 9Y Issues(Y,X)
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Main Reasoning Service in Datalog+
D
Σ
hD,Σi
D
database
Datalog+ Program
Query = 9X ('(X))
hD,Σi ² Query , D ̂ Σ ² Query
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Datalog§
• Extend Datalog by allowing in the head:
o Existential quantification (9)
o Equality atoms (=)
o Constant false (?)
…for query answering over databases
• But, already Datalog[9] is undecidable
• Datalog[9,=,?] is syntactically restricted ! Datalog§
Datalog[9,=,?]
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Main Decidability Paradigms for Datalog[9]
Finite Treewidth Sets (FTS) Finite Unification Sets (FUS)
Database expansion is tree-like
Forward chaining procedures
Backward resolution terminates
Proof-theoretic procedures
Q
D
Q
D
…but, identifying the above properties is an undecidable problem
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The Main Decidable Datalog[9] Languages
LinearGuarded
Sticky
FUS
FTS
DL-LiteR EL
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Linear Datalog[9]
• Linearity: there exists only one body-atom
• LOGSPACE data complexity & PSPACE-complete combined complexity
• Strictly more expressive than DL-LiteR
person(P) 9F hasFather(P,F), person(F)
[Calì, Gottlob & Lukasiewicz, JWS 2012]
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DL-LiteR into Linear Datalog[9]
DL-Lite: Popular family of DLs - at the basis of the OWL 2 QL profile of OWL
DL-LiteRAxioms Linear Datalog[9]
A v B A(X) B(X)
A v 9R A(X) 9Y R(X,Y)
9R v A R(X,Y) A(X)
9R v 9P R(X,Y) 9Z P(X,Z)
A v 9R.B A(X) 9Y R(X,Y), B(Y)
R v P R(X,Y) P(X,Y)
A v :B A(X), B(X) ?
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Linear Datalog[9]
• Linearity: there exists only one body-atom
• LOGSPACE data complexity & PSPACE-complete combined complexity
• Strictly more expressive than DL-LiteR
• Query answering is first-order rewritable
person(P) 9F hasFather(P,F), person(F)
[Calì, Gottlob & Lukasiewicz, JWS 2012]
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First-Order Rewritability
D
ΣQ
QSQL evaluation
8D : hD, Σi ² Q , D ² QSQL
compilation
first-order query
QFO
SQL query
translation
[Calvanese, De Giacomo, Lembo, Lenzerini & Rosati, JAR 2007]
evaluated and optimized
in the usual way
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Linear Datalog[9]
• Linearity: there exists only one body-atom
• LOGSPACE data complexity & PSPACE-complete combined complexity
• Strictly more expressive than DL-LiteR
• Query answering is first-order rewritable ) low data complexity
person(P) 9F hasFather(P,F), person(F)
[Calì, Gottlob & Lukasiewicz, JWS 2012]
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Guarded Datalog[9]
• Guardedness: a single body-atom contains all the body-variables
• PTIME-c data complexity & 2EXPTIME-c combined complexity
• Strictly more expressive than EL
supervisorOf(S,E), employee(E) employee(S)
[Calì, Gottlob & Lukasiewicz, JWS 2012] & [Calì, Gottlob & Kifer, JAIR 2013]
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EL into Guarded Datalog[9]
EL: Popular DL for biological applications - at the basis of OWL 2 EL profile
EL Axioms Guarded Datalog[9]
A v B A(X) B(X)
A u B v C A(X), B(X) C(X)
A v 9R.B A(X) 9Y (R(X,Y), B(Y))
9R.B v A R(X,Y), B(Y) A(X)
…several extensions of EL are captured by Guarded Datalog[9]
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Guarded Datalog[9]
• Guardedness: a single body-atom contains all the body-variables
• PTIME-c data complexity + 2EXPTIME-c combined complexity
• Strictly more expressive than EL
• Query answering is Datalog rewritable (cannot be first-order rewritable)
supervisorOf(S,E), employee(E) employee(S)
[Calì, Gottlob & Lukasiewicz, JWS 2012] & [Calì, Gottlob & Kifer, JAIR 2013]
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Datalog Rewritability
D
ΣQ
evaluation
8D : hD, Σi ² Q , D ² QDAT
compilation
Datalog query
QDAT
exploit a Datalog engine
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Guarded Datalog[9]
• Guardedness: a single body-atom contains all the body-variables
• PTIME-c data complexity & 2EXPTIME-c combined complexity
• Strictly more expressive than EL
• Query answering is Datalog rewritable ) low data complexity
supervisorOf(S,E), employee(E) employee(S)
[Calì, Gottlob & Lukasiewicz, JWS 2012] & [Calì, Gottlob & Kifer, JAIR 2013]
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The Main Decidable Datalog[9] Languages
LinearGuarded
FUS
FTS
ELDL-LiteR
Sticky
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Why Beyond Tree-like Models?
elephant(X) 9Y hasEAncestor(X,Y), elephant(Y)
cat(X) 9Y hasCAncestor(X,Y), cat(Y)
elephant(X), cat(Y) biggerThan(X,Y)
elephant(e)
elephant(e1)
elephant(e2)
elephant(e3)
elephant(e4)
.
.
.
cat(c)
cat(c1)
cat(c2)
cat(c3)
cat(c4)
.
.
.
£ infinite complete
bipartite graph=
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• Stickiness: join-variables stick to the inferred atoms
• LOGSPACE data complexity & EXPTIME-complete combined complexity
• Strictly more expressive than DL-LiteR
• Query answering is first-order rewritable ) low data complexity
Sticky Datalog[9]
[Calì, Gottlob & P., AIJ 2012]
R(X,Y), P(Y,Z) 9W T(X,Y,W)
T(X,Y,Z) 9W S(Y,W)
R(X,Y), P(Y,Z) 9W T(X,Y,W)
T(X,Y,Z) 9W S(X,W)
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The Main Decidable Datalog[9] Languages
LinearGuarded ELDL-LiteR
Sticky
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Several Interesting Extensions
Field of intense research - e.g., Montpellier,
Dresden, Calabria, Oxford, Vienna, …
Linear
Guarded
Weakly-Guarded Frontier-Guarded
Weakly-Frontier-Guarded
Sticky
Sticky-Join Weakly-Sticky
Weakly-Sticky-Join
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Complexity of the Main Datalog[9] Languages
Data Complexity Bounded Arity Combined Complexity
Linear in AC0 NP-c PSPACE-c
Guarded PTIME-c EXPTIME-c 2EXPTIME-c
Sticky in AC0 NP-c EXPTIME-c
via query rewriting
…can we go beyond positive rules, i.e., Datalog[9,»]?
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Datalog[9,»]
• Rules extended with negative literals in their body
• But, what is the semantics for Datalog[9,»]?
Number(X) 9Y Succ(X,Y), Number(Y)
Number(X), »Even(X) Odd(X)
Number(X), »Odd(X) Even(X)
Well-Founded Semantics (WFS) & Stable Model Semantics (SMS)
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Semantics of Datalog[9,»]
1. Convert the Datalog[9,»] program into a normal LP (via Skolemization)
2. Use the existing WFS and SMS for normal LPs
WFS(D,Σ) := WFS(ΠD,Σ)
SMS(D,Σ) := SMS(ΠD,Σ)
D = {R(a,b), P(a)}
Σ = {R(X,Y) 9Z R(Y,Z)),
R(X,Y), P(X), »S(X) P(Y)),
R(X,Y), »P(X) S(Y)}
ΠD,Σ = {R(a,b), P(a),
R(X,Y) R(Y,f(X,Y)),
R(X,Y), P(X), »S(X) P(Y),
R(X,Y), »P(X) S(Y)}
Skolemization
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Query Answering and Datalog[9,»]
WFS Boolean Conjunctive Query Answering (WFS-BCQ) :
Input: database D, Datalog[9,»] program Σ, BCQ Q
Question: WFS(D,Σ) ² Q?
SMS Boolean Conjunctive Query Answering (SMS-BCQ) :
Input: database D, Datalog[9,»] program Σ, BCQ Q
Question: Μ ² Q, 8Μ 2 SMS(D,Σ)?
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Guarded Datalog[9,»]
tree-likeness of the underlying models is preserved
R(X,Y,Z), P(X,Y), »S(Z,X) 9W R(Y,Z,W), S(W,Z)
[Gottlob, Hernich, Kupke & Lukasiewicz, PODS 2013, KR 2014]
Data Combined
WFS-BCQ PTIME-c 2EXPTIME-c
SMS-BCQ coNP-c 2EXPTIME-c
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Guarded Datalog[9,»]
[Gottlob, Hernich, Kupke & Lukasiewicz, PODS 2013, KR 2014]
Q
D
blocking technique
R(X,Y,Z), P(X,Y), »S(Z,X) 9W R(Y,Z,W), S(W,Z)
Data Combined
WFS-BCQ PTIME-c 2EXPTIME-c
SMS-BCQ coNP-c 2EXPTIME-c
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Guarded Datalog[9,»]
[Gottlob, Hernich, Kupke & Lukasiewicz, PODS 2013, KR 2014]
Guarded Datalog[9,»]
under SMS
Guarded Datalog[9,» ,_]
with stratified negation
·p
R(X,Y,Z), P(X,Y), »S(Z,X) 9W R(Y,Z,W), S(W,Z)
Data Combined
WFS-BCQ PTIME-c 2EXPTIME-c
SMS-BCQ coNP-c 2EXPTIME-c
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Linear Datalog[9,»]
[Gottlob, Hernich, Kupke & Lukasiewicz, PODS 2013, KR 2014]
R(X,Y,Z), »S(Z,X) 9W R(Y,Z,W), S(W,Z)
Data Combined
WFS-BCQ PTIME-c 2EXPTIME-c
SMS-BCQ coNP-c 2EXPTIME-c
(LOGSPACE) (PSPACE-C)
Linear Datalog[9,»] behaves like Guarded Datalog[9,»]
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Sticky Datalog[9,»]
[Alviano & P., PODS 2015]
• Stickiness: join-variables stick to the inferred atoms
• What is the right definition for Sticky Datalog[9,»]?
…either consider or ignore the variables in negative literals
R(X,Y), P(Y,Z) 9W T(X,Y,W)
T(X,Y,Z) 9W S(Y,W)
R(X,Y), P(Y,Z) 9W T(X,Y,W)
T(X,Y,Z) 9W S(X,W)
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Sticky Sticky+
WFS-BCQ EXPTIME-c / in PTIME Undecidable
SMS-BCQ Undecidable Undecidable
variables in negative literals
do not obey the stickiness condition
Sticky Datalog[9,»]
combined / data
[Alviano & P., PODS 2015]
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Sticky Sticky+
WFS-BCQ EXPTIME-c / in PTIME Undecidable
SMS-BCQ Undecidable Undecidable
variables in negative literals
do not obey the stickiness condition
employ a proof-theoretic approach
Sticky Datalog[9,»]
combined / data
Q
D
[Alviano & P., PODS 2015]
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Sticky Sticky+
WFS-BCQ EXPTIME-c / in PTIME Undecidable
SMS-BCQ Undecidable Undecidable
variables in negative literals
do not obey the stickiness condition
combined / data
[Alviano & P., PODS 2015]
existential quantification + cartesian products + guessing
Sticky Datalog[9,»]
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Sticky Datalog[9,»] is Undecidable under SMS
• Each stable model encodes a possible computation of the Turing machine
• The query checks whether at least one stable model represents a valid
halting computation
k-th horizontal row represents the
k-th configuration of the Turing machine
… … …
…
…
…
[Alviano & P., PODS 2015]
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Sticky Sticky+
WFS-BCQ EXPTIME-c / in PTIME Undecidable
SMS-BCQ Undecidable Undecidable
variables in negative literals
do not obey the stickiness condition
Sticky Datalog[9,»]: Sum Up
combined / data
[Alviano & P., PODS 2015]
Stickiness + WFS - proof-theoretic approach
Stickiness + SMS - 9-quantification + cartesian products + guessing
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But…
move(X,Y), »win(Y) win(X)
• Even rules with exactly one positive atom may not be sticky
• Can we do better?
…the second dimension of stickiness
[Alviano & P., PODS 2015]
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1st Dimension: the positive part is sticky
2nd Dimension: negative literals that lose a variable stick to one positive atom
2D-Stickiness
move(X,Y), »win(Y) win(X)
P(X,Y), R(Y), »R(X) 9Z S(Y,Z)
1st dimension
2nd dimension
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1st Dimension: the positive part is sticky
2nd Dimension: negative literals that lose a variable stick to one positive atom
2D-Stickiness
T(X,Y), R(Y,Z) P(X,Y)
P(X,Y), R(Y,Z), »R(X,X) 9Z S(Y,Z)
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1st Dimension: the positive part is sticky
2nd Dimension: negative literals that lose a variable stick to one positive atom
2D-Stickiness
T(X,Y), R(Y,Z) P(X,Y)
P(X,Y), R(Y,Z), »R(X,X) 9Z S(Y,Z)
1st dimension
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1st Dimension: the positive part is sticky
2nd Dimension: negative literals that lose a variable stick to one positive atom
2D-Stickiness
T(X,Y), R(Y,Z) P(X,Y)
P(X,Y), R(Y,Z), »R(X,X) 9Z S(Y,Z)
1st dimension
2nd dimension
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1st Dimension: the positive part is sticky
2nd Dimension: negative literals that lose a variable stick to one positive atom
2D-Stickiness
P(X), P(Y) T(Y,X)
T(X,Y), »R(Y,X) 9Z S(Z)
1st dimension
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1st Dimension: the positive part is sticky
2nd Dimension: negative literals that lose a variable stick to one positive atom
2D-Stickiness
P(X), P(Y) T(Y,X)
T(X,Y), »R(Y,X) 9Z S(Z)
1st dimension
2nd dimension
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2D-Sticky Datalog[9,»]
2D-Sticky 2D-Sticky+2
WFS-BCQ 2EXPTIME-c / PTIME-c Undecidable
SMS-BCQ Undecidable Undecidable
negative literals may stick
to two positive atoms
combined / data
[Alviano & P., PODS 2015]
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2D-Sticky Datalog[9,»]
2D-Sticky 2D-Sticky+2
WFS-BCQ 2EXPTIME-c / PTIME-c Undecidable
SMS-BCQ Undecidable Undecidable
negative literals may stick
to two positive atoms
combined / data
employ a proof-theoretic approach
Q
D
[Alviano & P., PODS 2015]
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Datalog[9,»]: An Overview
LinearGuarded
Sticky
2D-Sticky
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Conclusions and Future Work
Thank you!
Transform Datalog from a first class database query language to a
first class language for knowledge representation
(and other applications)
Problems under investigation:
• Stickiness + stable model semantics
• Deal with equality - Datalog[9,» ,=]
• New semantics without applying Skolemization - follow the
approach on stable models by Ferraris, Lee & Lifschitz