presentation

Query Answering over ContextualizedRDF/OWL Knowledge with Forall-Existential

Bridge Rules: Decidable Classes

Mathew Joseph1,2

1DKM, FBK-IRST, Trento, Italy

2DISI, University of Trento, Trento, Italy

PhD Defence Presentation

Mathew Joseph Query Answering over Contextualized RDF/OWL Knowledge

Outline of the talk

1 Introduction

2 Quad-Systems

3 Query Answering over Quad-Systems

4 Decidable Classes of Quad-SystemsContext Acyclic Quad-SystemsCsafe, Msafe, and Safe Quad-systemsRange Restricted Quad-Systems

5 Quad-systems and Forall-Existential rules

6 Related Work

7 Conclusion


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Contextualized Knowledge

The fact “I am giving this presentation” is only true in acertain context.Contextualized RDF knowledge is proliferating:

Recent releases of Billion Triples Challenge Datasets,DBPedia datasets are all in NQuads format.

Triple stores are more and more moving to quad-stores -4store, Openlink Virtuoso, Sesame.RDF 1.1 introduced NQuads as official W3Crecommendation in 2014.

The focus of the thesis work is query answering overcontextualized RDF Knowledge/Quads.


Contextualized Knowledge

The fact “I am giving this presentation” is only true in acertain context.Contextualized RDF knowledge is proliferating:

Recent releases of Billion Triples Challenge Datasets,DBPedia datasets are all in NQuads format.

Triple stores are more and more moving to quad-stores -4store, Openlink Virtuoso, Sesame.RDF 1.1 introduced NQuads as official W3Crecommendation in 2014.

The focus of the thesis work is query answering overcontextualized RDF Knowledge/Quads.


Contexts: Literature review

John McCarthy 1987 - Proposed contexts as a solution toGenerality problem in AI.Multi-context Systems (MCS) - contexts are propositionaltheories and propositional bridge rules enable interoperability.Distributed Description Logics (DDL) - contexts are descriptionlogic KBs and bridge rules are of the form:

c : φ(~x)→ c′ : φ′(~x),

where φ(~x), φ′(~x) are either both concept (role) atoms.Contextualized Knowledge Repository (CKR) - a frameworkdeveloped at DKM group. Its aims are to design and implementeffective algorithms for reasoning and query answering overcontextual knowledge.


Thesis Novelty/Advancement

Key DifferenceBRs, we consider, are more expressive than the BRs in theabove works and contain ∧s and ∃ operators


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Quads and Quad-graphs

Let C be a distinguished set of URIs called context identifiers.

A quad is an expression of the form c : (s,p,o), where c ∈ C,(s,p,o) is a triple.

A quad graph is a set of quads.

Notation:

QC is the quad-graph whose set of context identifiers is C.

For any c ∈ C, graphQC(c) = {(s,p,o) | c : (s,p,o) ∈ QC}


Example: Quad-graph

Example

Let C = {cWC2014, cUEFA2014, cSerieA2014}cWC2014 - context about World cup football 2014.cUEFA2014 - context about UEFA cup football 2014.cSerieA2014 - context about Italian Serie A football 2014.

Quad-graph

QC =

cWC2014 : (Buffon,playsFor , Italy)cWC2014 : (Buffon, captains, Italy)

. . .cUEFA2014 : (Buffon,playsFor , Juventus)

. . .cSerieA2014 : (Buffon,playsFor , Juventus)

. . .


Quad-graph: Visualization

QC can be viewed as a family of RDF graphs

BuffonItaly

graphQC (cWC2014)

playsFor Buffon

Agnelli

Juventus

graphQC (cUEFA2014)

owns

playsFor

superMario

InterMilan

graphQC (cSerieA2014)


Bridge Rules (BRs)

Eg: cUEFA2014 : (x , a, GoodPlayer), cSerieA : (x , a, GoodPlayer )→ cWC2014 : (x ,playsFor , Italy)

A BR is an expression of the form:

∀~x∀~z [

body︷︸︸︷c1: t1(~x , ~z) ∧ ... ∧ cn: tn(~x , ~z)→

∃~y c′1: t ′1(~x , ~y) ∧ ... ∧ c′m: t ′m(~x , ~y)︸︷︷︸head

]

c1 : t1(~x , ~z), ..., cn : tn(~x , ~z) are quad patterns over variablesets {~x} or {~z}.c′1 : t ′1(~x , ~y), ..., c′m : t ′m(~x , ~y) are quad patterns over variablesets {~x} or {~y},where a quad-pattern is a quad that allows variables at s,p,o.Variables in ~x are called frontier variables.


Quad-Systems

Definition (Quad-System)

A quad-system QSC is defined as a pair 〈QC ,R〉, where QC is aquad-graph, and R is a set of bridge rules.


Quad-System Semantics

Semantics of a quad-system QSC is defined on top of adistributed interpretation structure IC = {Ic}c∈C , whereIc = 〈∆c , ·c〉, for each c ∈ C, is a local interpretation structure.

local ∈ { rdf, rdfs, owl-horst, . . .}

Ic |=local graphQ(c), when Ic is a local model of the triples incontext c.


Model of a Quad-system

Definition (Model of a Quad-system (|=))

A distributed interpretation structure IC = {Ic}c∈C satisfies aquad-system QSC = 〈QC ,R〉, in symbols IC |= QSC iff all thefollowing are satisfied:

1 For every c ∈ C, Ic |=local graphQC(c);2 For every BR r ∈ R, for every σ ∈ {~x} ∪ {~z} → ∆C , where

∆C =⋃

c∈C ∆c , if

Ic1 |=local t1(~x , ~z)[σ], ..., Icn |=local tn(~x , ~z)[σ],

then there exists function σ′ ⊇ σ, s.t.

Ic′1 |=local t ′1(~x , ~y)[σ′], ..., Ic′

m |=local t ′m(~x , ~y)[σ′].


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Contextualized Conjunctive Queries

A Contextualized Conjunctive Query (CCQ) is an expression ofthe form:

∃~y c1 : t1(~x , ~y) ∧ ... ∧ cp : tp(~x , ~y)

where qi , for i = 1, ...,p are quad patterns over vectors of freevariables ~x and quantified variables ~y .

ExampleIf context c1 is about Football World Cup 2014 and context c2about Football Euro Cup 2012. Then the CCQ

c1: (x , beat, Italy) ∧ c2: (x , beat, Italy),where x is a variable.

intuitively means “Who beat Italy in both Euro Cup 2012 andWorld Cup 2014”.


Query Answering Decision Problem over QS

CCQ evaluation problem

Decision problem of determining, for any vector of constants ~a,a CCQ CQ(~x) over a quad-system QSC , if QSC |= CQ(~a).

Distributed chase (dChase) of a quad-systemWe extend the standard chase algorithm [Meir et al. 79] toour setting, call its output the distributed chase,abbreviated dChase.The algorithm runs iteratively, for iterations i = 0, . . . ,producing outputs dChase0(QSC), . . . ,, respectively.dChase(QSC) =

⋃i∈N dChasei(QSC)


(Distributed) Chase Algorithm

dChase0 = QC ; dChasek+1 = dChasek ∪ apply(R, dChasek )

BuffonItaly

graphQC (cWC2014)

playsFor Buffon

Agnelli

Juventus

graphQC (cUEFA2014)

owns

playsFor

superMario

InterMilan





BuffonItaly

graphQC (cWC2014)

playsFor Buffon

Agnelli

Juventus

graphQC (cUEFA2014)

owns

playsFor

superMario

InterMilan


c UEF

A201

4:(x,pl

aysF

or,z)

→

∃yc Se

rieA2

014:(x,pl

aysF

or,y)




BuffonItaly

graphQC (cWC2014)

playsFor Buffon

Agnelli

Juventus

graphQC (cUEFA2014)

owns

playsFor

superMario

InterMilan

Buffon

_:b


playsFor

c UEF

A201

4:(x,pl

aysF

or,z)

→

∃yc Se

rieA2

014:(x,pl

aysF

or,y)




BuffonItaly

graphQC (cWC2014)

playsFor Buffon

Agnelli

Juventus

graphQC (cUEFA2014)

owns

playsFor

superMario

InterMilan

Buffon

_:b

Skolemblank node


playsFor

c UEF

A201

4:(x,pl

aysF

or,z)

→

∃yc Se

rieA2

014:(x,pl

aysF

or,y)


Distributed chase of a quad-system

Termination condition: If ∃i s.t. dChasei(QSC) =dChasei+1, then dChase(QSC) = dChasei(QSC).It might be the case that the termination condition is neversatisfied and dChase is infinite, which leads tonon-termination of dChase algorithm.So what? Same problem occurs in DLs DL-Lite, EL etc.,but QA algorithms based on rewritingtechniques [Calvanese et al. 2007] and combinedapproaches [Lutz et al., 2009] exists.Is there an algorithm for deciding CCQ evaluation problemfor quad-systems?









Termination condition: If ∃i s.t. dChasei(QSC) =dChasei+1, then dChase(QSC) = dChasei(QSC).It might be the case that the termination condition is neversatisfied and dChase is infinite, which leads tonon-termination of dChase algorithm.So what? Same problem occurs in DLs DL-Lite, EL etc.,but QA algorithms based on rewritingtechniques [Calvanese et al. 2007] and combinedapproaches [Lutz et al., 2009] exists.Is there an algorithm for deciding CCQ evaluation problemfor quad-systems? Ans: NO


Undecidability of Query Answering for QS

TheoremCCQ evaluation problem is undecidable

Non emptyness checking of intersection of languagesgenerated by two CFGs in undecidable.

Reduction: Each PR of the form S → S1S2 . . .Sn can beencoded as a BR of the form:c : (x1,S1, x2), c : (x2,S2, x3), . . . , c : (xn,Sn, xn+1)→c : (x1,S, xn+1)


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Triple generating context

For any quad-system QSC = 〈QC ,R〉, a context c ∈ C is called atriple generating context (TGC), if there exists a BR r ∈ R, withc : (s,p,o) ∈ head(r) and s or p or o is an existential variable.

Definition (Context dependency graph)

of a quad-system QSC = 〈QC , R〉 is a directed graph 〈V ,E〉,V = context identifiers in C s.t. TGCs are marked with a ∗, andE are s.t.:for each BR r ∈ R {

for each context ci occurring in the body of r {for each context cj occurring in the head of r {

exists edge from ci to cj ;}}}


Example

Consider a quad-system, whose set of BRs R are:

c1 : (x1, x2,U1)→ ∃y1 c2 : (x1, x2, y1),

c3 : (x2,a,rdf:Property)

c2 : (x1, x2, z1)→ c1 : (x1, x2,U1)

c3 : (x1, x2, x3)→ c1 : (x1, x2, x3)

c1

c2

∗c3


Example


c1 : (x1, x2,U1)→ ∃y1 c2 : (x1, x2, y1),


c2 : (x1, x2, z1)→ c1 : (x1, x2,U1)

c3 : (x1, x2, x3)→ c1 : (x1, x2, x3)

c1

c2

∗c3

A quad-system is said to be context acyclic (cAcyclic), iff itscontext dependency graph does not contain cycles involvingTGCs.


Example


c1 : (x1, x2,U1)→ ∃y1 c2 : (x1, x2, y1),


c2 : (x1, x2, z1)→ c1 : (x1, x2,U1)

c3 : (x1, x2, x3)→ c1 : (x1, x2, x3)

c1

c2

∗c3

A quad-system is said to be context acyclic (cAcyclic), iff itscontext dependency graph does not contain cycles involvingTGCs.

Since the cycle (c1, c2, c1) in the quad-system contains c2which is a TGC, the quad-system is not cAcyclic.


Context Acyclic Quad-Systems: Complexity Results

Theorem(i) Combined Complexity of CCQ evaluation is2EXPTIME-complete.

(ii) Data complexity of CCQ evaluation is PTIME-complete

(ii) PTIME-hardness established by the reduction of 3HornSat,i.e. satisfiability of Propositional Horn clauses with at most 3literals.

(i) 2EXPTIME-hardness established by reduction of wordproblem of double exponentially time bounded DeterministicTuring Machine.


2EXPTIME-Hardness of CCQ Evaluation

0 0 1 1 � � · · ·

qI

Figure : Deterministic Turing Machine (DTM)

A 2EXPTIME DTM is a DTM that decides acceptance inatmost double exponential number of transitions w.r.t. inputsize.Computation also uses atmost double exponential numberof cellsReduction of the word problem of 2EXPTIME DTM to CCQevaluation problem of context acyclic quad-systems.


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


C(dChase(QSC)) = U(dChase(QSC) ∪ L(dChase(QSC)) ∪B(dChase(QSC)) can be potentially infinite.

U(dChase(QSC) ⊆ U(QSC), and L(dChase(QSC)) ⊆ L(QSC)are finite sets.

Hence, the real reason of non-finiteness is B(dChase(QSC))and, specifically, the set of Skolem blank nodes.

Intuitively, csafe, msafe, and safe classes restricts the structureof the Skolem blank nodes in the dChase to be DAGs ofbounded depth.

Assumption: Every BR has a unique identifier.


Origin ruleId, Origin vector, Descendants of Skolemblank nodes

Consider the application of an assignment µ on the followingBR ri = body(ri)(~x , ~z)→ head(ri)(~x , ~y)

body(ri)

x1

. . .

xp z1

. . .

zq

head(ri)

x1

. . .

xp y1

. . .

yr

a1 . . . ap c1 . . . cq

µ

ri



body(ri)

x1

. . .

xp z1

. . .

zq

head(ri)

x1

. . .

xp y1

. . .

yr

a1 . . . ap c1 . . . cq

µ

apply

ri



body(ri)

x1

. . .

xp z1

. . .

zq

head(ri)

x1

. . .

xp y1

. . .

yr

a1 . . . ap _ : b1 . . .a1 . . . ap c1 . . . cq

µ

apply

ri



originRuleId(_ : b1) = i

body(ri)

x1

. . .

xp z1

. . .

zq

head(ri)

x1

. . .

xp y1

. . .

yr

a1 . . . ap _ : b1 . . .a1 . . . ap c1 . . . cq

µ

apply

ri



originVector(_ : b1) = 〈a1, . . . ,ap〉

body(ri)

x1

. . .

xp z1

. . .

zq

head(ri)

x1

. . .

xp y1

. . .

yr

a1 . . . ap _ : b1 . . .a1 . . . ap c1 . . . cq

µ

apply

ri



hasChild(_ : b1,a1), . . . ,hasChild(_ : b1,ap)

body(ri)

x1

. . .

xp z1

. . .

zq

head(ri)

x1

. . .

xp y1

. . .

yr

a1 . . . ap _ : b1 . . .a1 . . . ap c1 . . . cq

µ

apply

ri


Origin ruleId, Origin vector of Skolem blank nodes

For any Skolem blank node _ : b generated in the dChase bythe application of the BR ri = body(ri)(~x , ~z)→ head(ri)(~x , ~y)using assignment µ,

we say thatoriginRuleId(_ : b) = i

Also, we say

originVector(_ : b) = ~a = ~x [µ]


Origin Contexts/Descendants of Skolem blank nodes

Origin contexts

of _ : b is the set of contexts in which triples containing _ : b arefirst generated, during the dChase construction. Formally

originContexts(_ : b) = {c | c : (s,p,o) ∈ dChasei(QSC),

s = _ : b or p = _ : b or o = _ : b, and6 ∃j < i with c′ : (s′,p′,o′) ∈ dChasej(QSC),

s′ = _ : b or p′ = _ : b or o′ = _ : b}

Descendants

We call a c = µ(xi), for any xi ∈ ~x , as the childOf _ : b, insymbols hasChild(_ : b, c).

hasDescendant=hasChild+ (transitive closure)


Origin Contexts/Descendants of Skolem blank nodes

Origin contexts

of _ : b is the set of contexts in which triples containing _ : b arefirst generated, during the dChase construction. Formally

originContexts(_ : b) = {c | c : (s,p,o) ∈ dChasei(QSC),

s = _ : b or p = _ : b or o = _ : b, and6 ∃j < i with c′ : (s′,p′,o′) ∈ dChasej(QSC),

s′ = _ : b or p′ = _ : b or o′ = _ : b}

Descendants

We call a c = µ(xi), for any xi ∈ ~x , as the childOf _ : b, insymbols hasChild(_ : b, c).

hasDescendant=hasChild+ (transitive closure)


Example

Consider the quad-system 〈QC ,R〉, where QC = {c1 : (a,b, c)}.Suppose R is the following set:

R =

c1 : (x11, x12, z1)→ c2 : (x11, x12, y1) (r1)c2 : (z21, z22, x2)→ c3 : (y21, y22, x2) (r2)c3 : (z3, x31, x32)→ c2 : (y3, x31, x32) (r3)

dChase iterations are:dChase0(QSC) = {c1 : (a,b, c)}dChase1(QSC) ={c1 : (a,b, c), c2 : (a,b,_ : b1)}dChase2(QSC) ={c1:(a,b, c), c2 : (a,b,_ : b1),

c3 : (_ : b2,_ : b3,_ : b1) }Mathew Joseph Query Answering over Contextualized RDF/OWL Knowledge

Example


R =




Example


R =



c3 : (_ : b2,_ : b3,_ : b1) }

originRuleId( _ : b1) = 1,originVector( _ :b1) = 〈a,b〉,originContexts(_ :b1) = {c2},hasDescendant(_ :b1, a),hasDescendant(_ :b1, b)


Example


R =




Example


R =



c3 : (_ : b2,_ : b3,_ : b1) }

originRuleId( _ : b2) = 2,originVector( _ :b2) = 〈_ : b1〉,originContexts(_ :b2) = {c3},hasDescendant(_ :b2, _ : b1)


Example


R =



c3 : (_ : b2,_ : b3,_ : b1) }

originRuleId( _ : b3) = 2,originVector( _ :b3) = 〈_ : b1〉,originContexts(_ :b3) = {c3},hasDescendant(_ :b3, _ : b1)


Example Contd.


R =


dChase3(QSC) = {c1:(a,b, c),c2 : (a,b,_ : b1), c3 : (_ : b2, _ : b3,_ : b1), c2 : (_ : b4, _ : b3, _ : b1) }dChase4(QSC) = dChase3(QSC)


Example Contd.


R =



originRuleId( _ : b4) = 3,originVector( _ :b4) = 〈_ : b3,_ : b1〉,originContexts(_ :b4) = {c2},hasDescendant(_ :b4, _ : b3),hasDescendant(_ :b4, _ : b1)


Example Contd.


R =




Descendance Graph

Descendance graph for _ :b4 of example above is:

_:b4

3, 〈_:b3,_:b1〉, {c2}

_:b3

2, 〈_:b1〉,{c3}

_:b1

1, 〈a,b〉,{c2}

a b

Figure : Nodes labelled with tuple:originRuleId, originVector, originContexts


Safe, Msafe, and Csafe Quad-systems

Definition (safe, msafe, csafe quad-systems)A quad-system QSC is said to be:

unsafe iff ∃ Skolem blank nodes _ : b 6= _ : b′ indChase(QSC) s.t. _ : b is a descendant of _ : b′,with originRuleId(_ : b) = originRuleId(_ : b′) andoriginVector(_ : b) ∼= originVector(_ : b′),

unmsafe iff ∃ Skolem blank nodes _ : b 6= _ : b′ indChase(QSC) s.t. _ : b is a descendant of _ : b′,with originRuleId(_ : b) = originRuleId(_ : b′),

uncsafe iff ∃ Skolem blank nodes _ : b 6= _ : b′ indChase(QSC) s.t. _ : b is a descendant of _ : b′,with originContexts(_ : b) = originContexts(_ : b′).

A quad-system is safe (resp. msafe, resp. csafe) iff it is notunsafe (resp. unmsafe, resp. uncsafe).


Safe, Msafe, and Csafe Quad-systems: Properties

Theorem

Let CACYCLIC, SAFE,MSAFE, and CSAFE denote the class ofcontext acyclic, safe, msafe, and csafe quad-systems,respectively, then the following holds:

CACYCLIC ⊂ CSAFE ⊂ MSAFE ⊂ SAFE

Lemma (DAG property)For a safe (csafe, msafe) quad-system QSC , and for any blanknode b ∈ Bsk (dChase(QSC)), its descendance graph is a DAGwith bounded depth.


Safe quad-systems: Properties

TheoremFor any safe/msafe/csafe quad-system, the following holds:(i) size of the dChase is double exponential,(ii) dChase can be computed in 2EXPTIME,(iii) when the size of bridge rules are assumed to be a constant,dChase can be computed in PTIME

Theorem

For any safe/msafe/csafe quad-system, the following holds: (i)The data complexity of CCQ evaluation is PTIME-complete (ii)The combined complexity of CCQ evaluation is2EXPTIME-complete.


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Range Restricted (RR) Quad-Systems

Suppose if we disallow the occurrence of existentially quantifiedvariable from our bridge rules, then the resulting BRs are of theform:

∀~x∀~z[c1 : t1(~x , ~z) ∧ . . . ∧ cn : tn(~x , ~z)

→ c′1 : t ′1(~x) ∧ . . . ∧ c′m : t ′m(~x)]

Any such BR can be replaced with the following equivalent setof BRs each of which has exactly one quad-pattern in its head:

∀~x∀~z[c1 : t1(~x , ~z) ∧ . . . ∧ cn : tn(~x , ~z)→ c′1 : t ′1(~x)]

...∀~x∀~z[c1 : t1(~x , ~z) ∧ . . . ∧ cn : tn(~x , ~z)→ c′m : t ′m(~x)]


RR quad-systems: Computational Properties

TheoremFor any RR quad-system QSC = 〈QC ,R〉, the following holds:

Size of dChase(QSC) is a polynomial sized,dChase(QSC) can be computed in EXPTIME,When R is assumed to be constant sized, thendChase(QSC) can be computed in PTIME.

TheoremFor RR quad-systems, the following holds:(i) Combined complexity of CCQ evaluation problem is inEXPTIME,(ii) Data complexity of CCQ evaluation problem isPTIME-complete. P-hardness by reduction of 3HornSat.


RR quad-systems: Computational Properties

TheoremFor any RR quad-system QSC = 〈QC ,R〉, the following holds:

Size of dChase(QSC) is a polynomial sized,dChase(QSC) can be computed in EXPTIME,When R is assumed to be constant sized, thendChase(QSC) can be computed in PTIME.

TheoremFor RR quad-systems, the following holds:(i) Combined complexity of CCQ evaluation problem is inEXPTIME,(ii) Data complexity of CCQ evaluation problem isPTIME-complete. P-hardness by reduction of 3HornSat.


Restricted RR quad-systems

Restricted RR quad-systemis an RR quad-system in which the number of quad-patterns inthe body of each bridge rule is less than or equal to a constantn. For instance,n = 1, we get linear quad-systems,n = 2, we get quadratic quad-systems, etc.

TheoremFor restricted RR quad-systems, the following holds:(i) Data complexity of CCQ evaluation problem isPTIME-complete. P-hardness by reduction of 3HornSat.(ii) Combined complexity of CCQ evaluation problem isNP-complete. NP-hardness by reduction of the graph coloringproblem.


Restricted RR quad-systems

Restricted RR quad-systemis an RR quad-system in which the number of quad-patterns inthe body of each bridge rule is less than or equal to a constantn. For instance,n = 1, we get linear quad-systems,n = 2, we get quadratic quad-systems, etc.

TheoremFor restricted RR quad-systems, the following holds:(i) Data complexity of CCQ evaluation problem isPTIME-complete. P-hardness by reduction of 3HornSat.(ii) Combined complexity of CCQ evaluation problem isNP-complete. NP-hardness by reduction of the graph coloringproblem.


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Quad-systems and Forall-Existential (∀∃) rules

A ternary ∀∃ rule is an expression of the form:

∀~x∀~z[P1(~x , ~z) ∧ . . . ∧ Pn(~x , ~z)→ ∃~y P ′1(~x , ~y) ∧ . . . ∧ P ′m(~x , ~y)],

wherePi(~x , ~z), 1 ≤ i ≤ n, are atoms over variables {~x} or {~z},Pj(~x , ~y), 1 ≤ j ≤ n, are atoms over variables {~x} or {~y},ar(Pi) ≤ 3 and ar(P ′j ) ≤ 3.


Translation from Quad-systems to ∀∃ rules

Let τq be the translation s.t. for any quad (pattern) c : (s,p,o),τq(c : (s,p,o)) = c(s, p, o);

For any quad-graph QC with bnodes _ : b1, . . ., _ : bnτ(QC) =→ ∃y1, . . . , yn

∧qi∈QC

τq(qi)[µB]where µB = {_ : bi → yi}i=1...n;

For any BR r for the form seen before,τ(r) = ∀~x∀~z τq(q1(~x , ~z)) ∧ . . . τq(qn(~x , ~z))→ ∃~y τq(q′1(~x , ~z))∧ . . .∧ τq(q′m(~x , ~z));

For any quad-system QSC = 〈QC ,R〉,τ(QSC) = τ(QC) ∪

⋃r∈R τ(r).


Quad-system and ∀∃ rules

TheoremGiven the translation τ as defined above, for any quad-systemQSC and boolean CCQ CQ(~a), QSC |= CQ(~a) iffτ(QSC) |=fol τ(CQ(~a)).

Note that τ(QSC) is a ∀∃ rule set, τ(CQ(~a)) is a standardconjunctive query, and τ is a PTIME translation.

Inverse translation τ−1

Similarly, PTIME inverse translation τ−1 exists from ternary ∀∃rules (resp. CQs) to quad-systems (resp. CCQs) s.t. for any ∀∃ruleset P and a boolean CQ Q(), P |=fol Q() iff τ−1(P) |=τ−1(Q()).


Quad-systems and Ternary ∀∃ rules

CorollaryCCQ evaluation problem of quad-systems is polynomiallyequivalent to CQ evaluation problem over ternary ∀∃ rules.

This means that the well known techniques for decidabilityguarantees such as Weak acyclicity (WA) [Fagin et al. 2005],Joint acyclicity (JA) [Krötzsch et al. 2011], and Model faithfulacyclicity (MFA) [Cuenca Grau et al. 2013] from the disciplineof ∀∃ rules are also applicable in our settings, and vice versa.

The following relations holds [Cuenca Grau et al. 2013]:

WA ⊂ JA ⊂ MFA,

what are the relations with our decidability approaches to theseexisting notions?






WA ⊂ JA ⊂ MFA,







WA ⊂ JA ⊂ MFA,



Quad-systems and ∀∃ rules

Theorem1 CACYCLIC ⊂ WA,2 If local semantics of contexts is OWL-Horst or its derivative,

then QSC is context acyclic iff τ(QSC) is weakly acyclic.3 WA 6⊆ CSAFE and CSAFE 6⊆ WA,4 JA 6⊆ CSAFE and CSAFE 6⊆ JA,5 MFA ≡ MSAFE,6 MFA ⊂ SAFE. Important! because MFA was the most

expressive of the known classes with finite chase property,so far


Quad-systems and ∀∃ rules

Theorem1 CACYCLIC ⊂ WA,2 If local semantics of contexts is OWL-Horst or its derivative,

then QSC is context acyclic iff τ(QSC) is weakly acyclic.3 WA 6⊆ CSAFE and CSAFE 6⊆ WA,4 JA 6⊆ CSAFE and CSAFE 6⊆ JA,5 MFA ≡ MSAFE,6 MFA ⊂ SAFE. Important! because MFA was the most

expressive of the known classes with finite chase property,so far


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Related Work

∀∃, Datalog+- rules, TgdsBeeri and Vardi, 1981 Proved that reasoning with Tgds isundecidable.Deutch et al., Fagin et al., 2003 Weakly acyclic Tgds: Adecidable class for query answering. Tgds are analyzed usinga dependency graph. Difference: nodes in the dependencygraph contain predicate positions, in place of context identifiersin our approach.(Weakly) (Frontier) Guarded Rules Ensures decidability usingbounded tree width property of underlying models (Courcelle’stheorem)Linear TGDs, Sticky Tgds Ensures decidability using queryrewriting approach.


Outline

1 Introduction

2 Quad-Systems




6 Related Work

7 Conclusion


Complexity ofCCQ Entailment Expressivity Landscape dChase size

UNRESTRICTEDUNDECIDABLE INFINITETERNARY∀∃ RULES

SAFE

MSAFE MFA [Cuenca Grau et al. 2013]

CSAFE JA [Krötzsch et al. 2011]

WA [Fagin et al. 2005]

CACYCLIC

2EXPTIME-COMPLETE

DOUBLEEXPONENTIAL

RREXPTIMEPOLYNOMIAL

REST. RRNP-COMPLETE


Data Complexity & Complexity of Recognition

Quad-System Complexity Data Complexity ofFragment of Recognition CCQ evaluation

Unrestricted PTIME UndecidableSafe 2EXPTIME PTIME-complete

MSafe 2EXPTIME PTIME-completeCSafe 2EXPTIME PTIME-complete

Context Acyclic PTIME PTIME-completeRR PTIME PTIME-complete

Restricted RR PTIME PTIME-complete


Conclusion

We know ways to code bridge rules over quads s.t. queryanswering can be done with termination guarantees andreasonably efficiently.Since SAFE ⊃ MFA, we also have new ways of writingternary ∀∃ rules that allows for termination guaranteedquery answering.The technique of safety can also be ported to general ∀∃rules setting by keeping track of origin ruleId/vector anddescendants.


Articles and Conference ExperiencesM.Joseph, G.Kuper, T. Mossakowski, L.Serafini. QueryAnswering over Contextualized RDF/OWL Knowledge withForall-Existential Bridge Rules: Decidable Finite ExtensionClasses. Semantic Web Journal (Accepted forPublications, To Appear). IOS Press. 2015M.Joseph, G.Kuper, L.Serafini. Query Answering overContextualized RDF/OWL Knowledge withForall-Existential Bridge Rules: Attaining Decidability usingAcyclicity. In Proceedings of International Conference inWeb Reasoning and Rule Systems (RR-2014). 2014M.Joseph, G.Kuper, L.Serafini. Query Answering overContextualized RDF Knowledge with Forall-ExistentialBridge Rules: Attaining Decidability using Acyclicity. InProceedings of Italian Conference in Computational Logic(CILC-2014). 2014


Articles and Conference Experiences: Contd

M.Joseph, L.Serafini. Simple Reasoning for ContextualizedRDF Knowledge. In Proceedings of Workshop on ModularOntologies (WOMO-2011). Ljubljana, Slovenia. 2011A. Tamilin, B. Magnini, L. Serafini, C. Girardi, M. Joseph, R.Zanoli. Context-driven Semantic Enrichment of ItalianNews Archive. In proceedings of Extended Semantic WebConference (ESWC-2010). In use track. 364-378 crete,greece. 2010M. Joseph. A Contextualized Knowledge Framework forSemantic Web. In proceedings of Extended Semantic WebConference (ESWC-2010). PhD symposium track. crete,greece. 2010


THANKS

Thanks for your attention

Questions?


OWL-Horst inference pattern Induced edgesc(x1,owl:equivalentProperty, z), c(x2,z, x3)→ c(x2, x1, x3) 〈c,1〉 → 〈c,2〉c(x ,rdf:type,rdfs:Class)→c(x ,rdfs:subClassOf, x) 〈c,1〉 → 〈c,3〉c(z1, x , z2)→ c(x ,rdf:type,rdf:Property) 〈c,2〉 → 〈c,1〉c(z1, x , z2)→ c(x ,rdf:type,rdf:Property)→ c(x ,rdfs:subPropertyOf, x) 〈c,2〉 → 〈c,3〉c(z1, z2, x)→ c(x ,rdf:type,rdf:Resource) 〈c,3〉 → 〈c,1〉c(z,rdfs:subPropertyOf, x1), c(x2, z, x3)→ c(x2, x1, x3) 〈c,3〉 → 〈c,2〉

Table : Edges induced in the dependency graph due to OWL-Horst inferencing


c1 : (x1, x2,U1)→ ∃y1 c2 : (x1, x2, y1), c3 : (x2,a,rdf:Property)

c2 : (x1, x2, z1)→ c1 : (x1, x2,U1)

c3 : (x1, x2, x3)→ c1 : (x1, x2, x3)

〈c2,1〉〈c2,3〉〈c2,2〉

〈c1,1〉〈c1,3〉〈c1,2〉

〈c3,1〉〈c3,3〉〈c3,2〉

∗ ∗



c2 : (x1, x2, z1)→ c1 : (x1, x2,U1)

c3 : (x1, x2, x3)→ c1 : (x1, x2, x3)

〈c2,1〉〈c2,3〉〈c2,2〉

〈c1,1〉〈c1,3〉〈c1,2〉

〈c3,1〉〈c3,3〉〈c3,2〉

∗ ∗



c2 : (x1, x2, z1)→ c1 : (x1, x2,U1)

c3 : (x1, x2, x3)→ c1 : (x1, x2, x3)

〈c2,1〉〈c2,3〉〈c2,2〉

〈c1,1〉〈c1,3〉〈c1,2〉

〈c3,1〉〈c3,3〉〈c3,2〉

∗ ∗



c2 : (x1, x2, z1)→ c1 : (x1, x2,U1)

c3 : (x1, x2, x3)→ c1 : (x1, x2, x3)

〈c2,1〉〈c2,3〉〈c2,2〉

〈c1,1〉〈c1,3〉〈c1,2〉

〈c3,1〉〈c3,3〉〈c3,2〉

∗ ∗


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[Cuenca Grau et al. 2013]B. Cuenca Grau„ I. Horrocks, M. Krötzsch, C. Kupke,D. Magka, B. Motik, and Z. Wang, “Acyclicity Notions forExistential Rules and Their Application to Query Answeringin Ontologies,” in Journal of Artificial Intelligence Research(JAIR), vol. 47, pp. 741–808, AI Access Foundation, 2013.

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