Towards a Semantic Web Unifying Logic...Matthias Knorr - CENTRIA, UNL Towards a Semantic Web...

Towards a Semantic WebUnifying Logic

Matthias Knorr

CENTRIA, Universidade Nova de Lisboa

ICCL Summer School 2013

Combining non-monotonic rules and DLsTop-down querying

Generalized unifying language

About the Lecturer

• PostDoc at CENTRIA, Universidade Nova de Lisboa (UNL)• One of the partners in the programs EMCL and EPCL• Strong history in Logic Programming• Two ECCAI fellows (Luís M. Pereira and José J. Alferes)

• Research focus: Non-monotonic Reasoning in theSemantic Web; currently in the project ERRO

ERRO. efficient reasoning with rules and ontologies

• More info (including these slides) athttp://centria.di.fct.unl.pt/~mknorr/

Matthias Knorr - CENTRIA, UNL Towards a Semantic Web Unifying Logic 2/81

http://centria.di.fct.unl.pt/~mknorr/



Recap on Ontologies and Rules

• Towards a Unifying Logic• Building on the contents of

“Ontologies and Rules”• Focusing on

Non-monotonic Extensions




Why Non-monotonic Extensions?

Example: matching patients to clinical trial criteria

• Open World Assumption (OWA) in general preferable onthe Web

• Without clinical test, no assumptions can be made onoutcome

• But with complete knowledge, Closed World Assumption(CWA) is better

• Patient’s medication is fully known

• Requirement for local closure of certain information





Person v HeartLeft tHeartRightHeartLeft uHeartRight v ⊥

Person v ∃has.SpinalColumn∃has.SpinalColumn v V ertebrate

Person(Bob)

⇒ V ertebrate(Bob), Person v V ertebrate, and∃x.SpinalColumn(x) derivable







Person(Bob)

HeartLeft(x)← V ertebrate(x),notHeartRight(x)SSN_OK(x)← hasSSN(x, y)

f ← Person(x),notSSN_OK(x)

⇒ HeartLeft(Bob); hasSSN(Bob, y) for ground y required

Model defaults and exceptions, and integrity constraints







Person(Bob)

HeartLeft(x)← V ertebrate(x),notHeartRight(x)SSN_OK(x)← hasSSN(x, y)

f ← Person(x),notSSN_OK(x)

⇒ HeartLeft(Bob); hasSSN(Bob, y) for ground y required

Model defaults and exceptions, and integrity constraintsMatthias Knorr - CENTRIA, UNL Towards a Semantic Web Unifying Logic 6/81



Outline

1 Combining non-monotonic rules and DLs2 Top-down querying in such combinations3 Generalized unifying language




Combining non-monotonic rulesand DLs




Why first focus on Rules?

• Direct non-monotonic extensions exist for DLs, based on,e.g., default logic, epistemic logic, and circumscription

• But non-trivial with few results in terms of implementations• Rules easily extended to non-monotonic features• Well-studied field Logic Programming including fast

reasoners• Leverage the knowledge and reasoners available




How to Combine DLs and Rules?

Non-trivial because of non-monotonicity

• Rules “on top” of ontologies• First define concepts then define rules “on top”• Rules deal with ontologies as external code• Separate semantics for rule and ontology predicates

• Tight (full) integration• Allow for “defining” joint predicates both in the ontology and

the rule layer and use them (almost) freely• Modular combination (loose coupling)

• Trades some expressiveness for an easier to implementinterface integration








the rule layer and use them (almost) freely

• Modular combination (loose coupling)• Trades some expressiveness for an easier to implement

interface integration








the rule layer and use them (almost) freely• Modular combination (loose coupling)

• Trades some expressiveness for an easier to implementinterface integration




Hybrid MKNF Knowledge Bases

• Seamless, tight integration; reflexive, expressive, yetcompetitive w.r.t. computational complexity

• Introduced by Motik and Rosati in 2007 (extended in [Motikand Rosati, JACM10])

• Based on Logics of Minimal Knowledge and Negation asFailure by Lifschitz in 1991: first-order logic with equalityand modal operators K and not




Hybrid MKNF Knowledge Bases

• Consist of a DL knowledge base O and a finite set of rules,P, of the form

KH1 ∨ . . . ∨KHl ← KA1, . . . ,KAn,notB1, . . . ,notBm

• Hi, Aj , and Bk are generalized atoms - arbitrary first-orderformulas - to capture, e.g., conjunctive queries

KinterestingCity(x )← K(∃y : Has(x , y) ∧ Recreational(y)),notRainyCity(x )




Decidability(1)Ensured by two technical restrictions:

1 DL-safety of rules: every variable appears in a positivenon-DL-atom in the body (restriction of application of rulesto known individuals when grounding)

convention: capitalized predicates yield DL-atoms

KinterestingCity(x )← K(∃y : Has(x , y) ∧ Recreational(y)),notRainyCity(x ) ×

KinterestingCity(x )← K(∃y : Has(x , y) ∧ Recreational(y)),notRainyCity(x ), o(x)

+ o(x) for all individuals x appearing in the KB X




Decidability(2)Ensured by two technical restrictions:

2 Decidability of DL reasoning in combination withgeneralized atoms; related to the following:

KinterestingCity(x )← K(∃y : Has(x , y) ∧ Recreational(y)),notRainyCity(x ), o(x)

corresponds to

∃Has.Recreational v RecreationalCityKinterestingCity(x )← KRecreationalCity(x ),notRainyCity(x ), o(x)




Reasoning – Example(1)

Collect all modal atoms KA in ground hybrid MKNF KB KG andadd KB for each notB in KG – obtain the set KA(KG):

A v B B v F KA(a) Kc(a)← KB(a),notd(a)

• KA(KG) = {KA(a),KB(a),Kc(a),Kd(a)}





Guess a partition (P,N) of the collected set (partition into true(P ) and false (N ) modal atoms) and

1 Check if KG is satisfiable w.r.t. (P,N):


• (P1, N1) = ({KA(a),KB(a),Kc(a)}, {Kd(a)}) X

• (P2, N2) = ({KA(a),KB(a),Kc(a),Kd(a)}, ∅) X

• (P3, N3) = ({KA(a),KB(a)}, {Kc(a),Kd(a)}) ×





Guess a partition (P,N) of the collected set (partition into true(P ) and false (N ) modal atoms) and

2 Fixing N (used for the evaluation of not), check if P isminimal:


• (P1, N1) = ({KA(a),KB(a),Kc(a)}, {Kd(a)}) X

• (P2, N2) = ({KA(a),KB(a),Kc(a),Kd(a)}, ∅) ×

Reasoning on O ∪ P1: KF (a) is also derivable




Properties of Hybrid MKNF

• Generalizes/captures (sometimes not entirely) quite anumber of different approaches

• Faithful w.r.t. Stable Models for empty O and w.r.t. OWL forempty P

• Data complexity of instance checking in MKNF:

rules DL = ∅ DL ∈ P DL ∈ coNPdefinite P P coNPstratified P P ∆p

2

normal coNP coNP Πp2

disjunctive Πp2 Πp

2 Πp2




Problems of two-valued Hybrid MKNF

• Models have to be guessed and checked• Unrestricted rules increase computational complexity• Queries for particular information require computation of

the entire model• Limited robustness, e.g., w.r.t. merging of KBs

(Ku← notu)

[Knorr et al., AI11] provides alternative based on well-foundedsemantics for non-disjunctive logic programs




Stable Models vs. Well-FoundedModel in LP

p← notq q ← notp a← notb b←

has two stable models {p, b} and {q, b}, while the uniquewell-founded model assigns t to b, f to a, and u to both p and q.

For

p← notp q ← notq a← notb b←

the well-founded model is the same, but there are no stablemodels!




Stable vs. Well-Founded Semantics inLogic Programming

Stable Models/Answer Sets• More expressive language• More derivable information• Fast ASP solvers available

Well-founded Model• Lower computational complexity• always exists• top-down derivations possible

Similar for combinations of rules and ontologiesMatthias Knorr - CENTRIA, UNL Towards a Semantic Web Unifying Logic 21/81



Well-Founded MKNF semantics

• Restricted to non-disjunctive rules over simple atoms• Partitions divided into true, undefined, and false atoms• Computation of the unique model in a bottom-up fashion




Immediate Consequence OperatorDefinitionWe define on subsets S of KA(KG) for ground, positive KG:

RKG(S) ={KH | PG contains a rule of the form

KH ← KA1, . . . ,KAn

such that, for all i, 1 ≤ i ≤ n,KAi ∈ S}

DKG(S) ={Kξ | Kξ ∈ KA(KG) and OBO,S |= ξ}

TKG(S) =RKG

(S) ∪DKG(S)

OBO,S - first-order representation of O and assumed set S




Example Iteration TKG

R v SKR(a)←Kq(a)← KS(a)

TK ↑ 0 = ∅TK ↑ 1 = {KR(a)}TK ↑ 2 = {KR(a),KS(a)}TK ↑ 3 = {KR(a),KS(a),Kq(a)}




MKNF Transform

DefinitionLet KG = (O,PG) be a ground hybrid MKNF knowledge baseand S ⊆ KA(KG). The MKNF transform KG/S is defined asKG/S = (O,PG/S), where PG/S contains all rules

KH ← KA1, . . . ,KAn

for which there exists a rule

KH ← KA1, . . . ,KAn,notB1, . . . ,notBm

in PG with KBj 6∈ S for all 1 ≤ j ≤ m.




Example Transformation

consider S = {KS(a),KR(a)} (highlighted part is removed):

R v SKR(a)← notR(b)KR(b)←notR(a)Kq(a)← KS(a),notq(a)

Compute least model of the resulting KB:

ΓKG(S) = TKG/S ↑ ω.




Coherence

Problem: classical negation from O does not imply defaultnegation in rules

R v ¬SKS(a)← nott(a)Kt(a)← notS(a)

KR(a)←

Operators so far yield undefined for KS(a) and Kt(a)




MKNF-coherent transform

DefinitionLet KG = (O,PG) be a ground hybrid MKNF knowledge baseand S ⊆ KA(KG). The MKNF-coherent transform KG//S isdefined as KG//S = (O,PG//S), where PG//S contains allrules

KH ← KA1, . . . ,KAn

for which there exists a rule

KH ← KA1, . . . ,KAn,notB1, . . . ,notBm

in PG with KBj 6∈ S for all 1 ≤ j ≤ m and OBO,S 6|= ¬H.




Example Coherence

MKNF-coherent transform for S = {KR(a)}:

R v ¬SKS(a)← nott(a)Kt(a)← notS(a)

KR(a)←

Compute least model of the resulting KB:

Γ′KG(S) = TKG//S ↑ ω




Inconsistencies

R v ¬SKS(a)← nott(a)Kt(a)← notR(a)

KR(a)←

By Γ′KG, we simply derive that KS(a) is false.




Alternating Iteration

P0 = ∅ N0 = KA(KG)Pn+1 = ΓKG

(Nn) Nn+1 = Γ′KG(Pn)

Pω =⋃

Pi Nω =⋂

Ni

two fixpoints for (finite) ω:• Pω – everything that is true• Nω – everything that is not false




Properties of Well-Founded MKNF

• Sound w.r.t. two-valued MKNF semantics• Faithful w.r.t. first-order semantics for empty P and w.r.t.

the Well-Founded Semantics for empty O• Given complexity C for instance checking in O we obtain a

data complexity PC ; for C =P, polynomial data complexity• Inconsistencies can be detected by a simple additional

computation• Automated computation yields model (if it exists)




Summary

• Sophisticated reasoning possible with combinations ofontologies and non-monotonic rules

• MKNF-based framework links information based onderivable information

• Framework itself very general and flexible• Algorithms for fast computation available

What if we do only want to know a small of piece of information?




Top-down querying




Top-down-Procedure SLG(O)

• SLG(O) [Alferes et al., ACM TOCL13] combines a DLreasoner with the top-down query engine XSB Prolog

• Instead of computing the whole model, only compute thepart relevant for the model

• XSB realizes well-founded semantics in LogicProgramming utilizing tabling (re-using previous results,avoiding infinite loops)

• Special oracle resolves queries to the ontology




DL Oracle

• Builds on derivation relation for O• Returns (possibly empty) set of atoms L (potentially

derivable from the set of rules P) that together with O allowto derive the (ground) queried atom S

• If subsequent queries for atoms in L succeed, then Ssucceeds




MKNF KB ExamplePortCity(Barcelona) OnSea(Barcelona,Mediterranean)PortCity(Hamburg) NonSeaSideCity(Hamburg)

RainyCity(Manchester) Has(Manchester ,AquaticsCenter)Recreational(AquaticsCenter)

SeaSideCity v ∃Has.BeachBeach v Recreational

∃Has.Recreational v RecreationalCityKSeaSideCity(x )← KPortCity(x ),notNonSeaSideCity(x )

KinterestingCity(x )← KRecreationalCity(x ),notRainyCity(x )KhasOnSea(x )← KOnSea(x , y)

Kfalse ← KSeaSideCity(x ),nothasOnSea(x )KsummerDestination(x )← KinterestingCity(x ),KOnSea(x , y)




Barcelona

Interesting?




MKNF KB Example Query(1)PortCity(Barcelona) OnSea(Barcelona,Mediterranean)

PortCity(Hamburg) NonSeaSideCity(Hamburg)

RainyCity(Manchester) Has(Manchester ,AquaticsCenter)

Recreational(AquaticsCenter)

SeaSideCity v ∃Has.Beach

Beach v Recreational

∃Has.Recreational v RecreationalCity

KSeaSideCity(x )← KPortCity(x ),notNonSeaSideCity(x )

KinterestingCity(x )← KRecreationalCity(x ),notRainyCity(x )

KhasOnSea(x )← KOnSea(x , y)

Kfalse ← KSeaSideCity(x ),nothasOnSea(x )

KsummerDestination(x )← KinterestingCity(x ),KOnSea(x , y)

Query: KinterestingCity(Barcelona)










∃Has.Recreationalv RecreationalCity






Subquery: KRecreationalCity(Barcelona)Has(Barcelona, x ),Recreational(x ) would fail








SeaSideCityv ∃Has.Beach

Beachv Recreational







Subquery: KRecreationalCity(Barcelona)









Beachv Recreational







Subquery: KSeaSideCity(Barcelona)









Beachv Recreational







Subquery: KPortCity(Barcelona) succeeds









Beachv Recreational


KSeaSideCity(x )←KPortCity(x ),notNonSeaSideCity(x)





Subquery: KNonSeaSideCity(Barcelona) fails









Beachv Recreational







notNonSeaSideCity(Barcelona) and KSeaSideCity(Barcelona)succeed and subsequently KRecreationalCity(Barcelona)









Beachv Recreational



KinterestingCity(x )←KRecreationalCity(x ),notRainyCity(x)




Subquery: KRainyCity(Barcelona) fails









Beachv Recreational



KinterestingCity(x )←KRecreationalCity(x ),notRainyCity(x)




Main query: KInterestingCity(Barcelona) succeeds










∃Has.Recreational v RecreationalCity






KinterestingCity(Barcelona) is storedsubsequently two look-ups yield summerDestination(Barcelona)




General Properties of SLG(O)

• Answers match the well-founded MKNF model if it exists• Otherwise, for consistent O, answers match a

paraconsistent approximation• Data complexity PC maintained (C complexity of instance

checking in O) provided the number of answers returnedfrom the oracle is polynomial

• three polynomial oracles, one for each OWL 2 profile




Reasoner NoHR for OWL 2 EL

• first non-monotonic plug-in for the ontology editor Protégé• presented in [Vadim et al., ISWC13]• unlike OWL reasoners (such as ELK, Fact, Hermit, Racer)

that focus on DL reasoning tasks, realizes top-downquerying on ontologies and non-monotonic rules

• answers (safe) conjunctive queries, returning answer tuplesfor non-ground queries or true/undefined/false/inconsistent

• https://code.google.com/p/nohr-reasoner/


https://code.google.com/p/nohr-reasoner/



NoHR Protégé plug-in




Pre-processing Algorithm

• Compute implicit inferences using ELK reasoner• Discard axioms/information not usable when querying,

such asSeaSideCity v ∃Has.Beach

• Translate the remainder into rules and merge withnon-monotonic rules (applying to both a special doubling incase DisjointWith axioms occur in O, to detect potentialinconsistencies)

• Input the result to XSB Prolog




NoHR Architecture

XSBJava Virtual MachineProtégé

NoHR Plugin

GUI

ELK

Query Processor

InterProlog

NoHR Rules Tab

OWL File

NM RulesFile

XSB Knowledge

Base

Query Answering

Tables

Tracer/Debugger

NoHR Query Tab

Translator

Ontology

NM Rules

ProtégéOntology

NM Rules Base




Test Translation ontologies

! # Axioms! ELK! Translator! XSB! Total!ChEBI! 67184! 3,92! 7,41! 1,75! 9,16!EMAP! 13730! 2,94! 5,14! 0,00! 5,14!Fly Anatomy! 19211! 2,70! 4,99! 0,83! 5,82!FMA! 126548! 7,56! 14,67! 3,58! 18,25!GALEN-OWL! 36547! 5,00! 8,76! 2,46! 11,22!GALEN7! 44461! 5,88! 9,67! 4,56! 14,23!GALEN8! 73590! 21,90! 42,76! 28,12! 70,88!GO1! 28897! 3,74! 6,06! 1,09! 7,15!GO2! 73590! 4,57! 8,90! 5,02! 13,92!Molecule Role! 9629! 3,41! 5,14! 0,15! 5,29!SNOMED CT! 294480! 20,61! 43,32! 24,69! 68,01!




Test Translation SNOMED CT withRules

0!

10!

20!

30!

40!

50!

60!

0! 44! 88! 132! 176! 220! 264! 308! 352! 396! 440!

Tim

e (s

)!

NM Rules + Facts (×1000)!

ELK!Translator!XSB!




Query Time

• Queries of varying size, containing atoms of different depthin the ontology hierarchy with varying number ofspecialized classes, and differing connectivity betweenvariables posed to real ontologies with added rules

• Interactive response time usually considerably below asecond; only higher if the number of answers (e.g., due tomany subclasses of a queried atom) grows very high, orarbitrarily generated answers create too links, essentiallycomputing the entire model in the worst case




Summary

• Top-down querying over ontologies and non-monotonicrules (under well-founded MKNF) can be efficientlyimplemented

• NoHR for OWL 2 EL under active development• current challenge: realize use-cases, develop KBs• try it out at: https://code.google.com/p/nohr-reasoner/


https://code.google.com/p/nohr-reasoner/



Non-monotonic DL Extension withMKNF

ALCKNF [Donini et al., ACM TOCL02]

ALC with MKNF logic-style modal operators K – minimalknowledge – and A – autoepistemic assumption (correspondsto ¬not)

K can be used to derive new information, A to verify ifinformation is already known

Different expressiveness compared to Hybrid MKNF




Non-monotonic Features of ALCKNFfrom [Donini et al., ACM TOCL02]

• Defaults:

KI uK(employee u ∃belongsTo.programmingDept)u¬Amanager v K(engineer tmathematican)

• Integrity Constraints:

Kemployee v (Amale tAfemale)

Kemployee v ∃ASSN.Avalid




Non-monotonic Features of ALCKNF

• Concept and Role Closure

¬UScitizen(Paula) Manages(Ann,Marc)¬UScitizen(Carl) UScitizen(Marc)

adding (∀KManages.KUScitizen)(Ann) closes the role

adding ∃KManages.A¬UScitizen(Ann) closes theconcept




Can We find a joint formalism for bothMKNF extensions?

• Contribute towards a unifying logic• Reconcile OWL and Datalog together with CWA extensions

(on both sides)• Usage of one (DL-style) syntax in opposite to common

hybrid languages• Coverage of many different previous approaches




SROIQV(Bs,×)KNF

• OWL 2 DL (SROIQ) with concept products (× - [Krötzsch,SSW10]) and Boolean constructors over simple roles (Bs -[Rudolph et al., JELIA08])

• Nominal schemas (V - [Krötzsch et al., WWW11]) - variablenominals that can only bind to known individuals

• MKNF logic-style modal operators K – minimal knowledge– and A – autoepistemic assumption – (KNF - fromALCKNF [Donini et al., ACM TOCL02])

• SROIQVK for short




Syntaxsignature Σ = 〈NI , NC , NR, NV 〉

DefinitionThe set of SROIQV(Bs,×)KNF concepts C and(simple/non-simple) SROIQV(Bs,×)KNF roles R (Rs/Rn) aredefined by the following grammar.

Rs ::=N sR | (N s

R)− | U | NC ×NC | ¬Rs | Rs u Rs | Rs t Rs |KRs | ARs

Rn ::=NnR | (Nn

R)− | U | NC ×NC | KRn | ARn

R ::=Rs | Rn

C ::=> | ⊥ | NC | {NI} | {NV } | ¬C | C u C | C t C |∃R.C | ∀R.C | ∃Rs.Self | 6k Rs.C | >k Rs.C | KC | AC




Semantics – Principal Notions

• Based on interpretations I = (∆I , ·I) plus variableassignments (for nominal variables) mapping each variableto the interpretation of one element in NI

• Variant of Standard Name Assumption applied: essentiallyI is a bijective function on NI while still allowing thatelements of NI may be identified (→ only one ∆)

An MKNF structure is a triple (I,M,N ) where I is aninterpretation,M and N are sets of interpretations, and I andall interpretations inM and N are defined over ∆. For any such(I,M,N ) and assignment Z, the function ·(I,M,N ),Z is defined.




Function ·(I,M,N ),Z (parts of it)

Syntax Semanticsa aI ∈ ∆x Z(x) ∈ ∆

¬C ∆ \ C(I,M,N ),Z

{t} {a | a ≈ t(I,M,N ),Z}KC

⋂J∈MC(J ,M,N ),Z

AC⋂J∈N C

(J ,M,N ),Z

KR⋂J∈MR(J ,M,N ),Z

AR⋂J∈N R

(J ,M,N ),Z

C v D C(I,M,N ),Z ⊆ D(I,M,N ),Z




(Monotonic) Semantics

Definition(I,M,N ) satisfies axiom α, written (I,M,N ) |= α, if(I,M,N ),Z |= α for all variable assignments Z.

A (non-empty) set of interpretationsM satisfies α, writtenM |= α, if (I,M,M) |= α holds for all I ∈ M.

M satisfies a SROIQV(Bs,×)KNF knowledge base KB,writtenM |= KB, ifM |= α for all axioms α ∈ KB.




(Non-monotonic) Semantics

DefinitionGiven a SROIQV(Bs,×)KNF knowledge base KB, a(non-empty) set of interpretationsM is an MKNF model of KBif(1) M |= KB, and(2) for eachM′ withM⊂M′, (I ′,M′,M) 6|= KB for someI ′ ∈M′.




Example

C Persons whose parents are married

HasParent(mary, john) (1)(∃HasParent.∃Married.{john})(mary) (2)

∃HasParent.{z} u ∃HasParent.∃Married.{z} v C (3)

We can substitute (3) by

K∃HasParent.{z} uK∃HasParent.∃Married.{z} v KC




Example




We can also substitute (3) by

K∃HasParent.{z} uK∃HasParent.∃Married.{z} v AC

We now require that all elements of class C are explicitlymentioned




Example




We can also substitute (3) by

∃HasParent.{z} u ∃HasParent.∃¬AMarried.{z} v C

Now C are Persons that are known to be not married




Decidability

• First, reduce reasoning in SROIQV(Bs,×)KNF toreasoning in SROIQ(Bs)KNF by grounding and bysimulating concept products

• Then, follow approach for ALCKNK:• each model of a knowledge base in SROIQ(Bs)KNF is

cast into a SROIQ(Bs) KB. Consequently, reasoning inSROIQ(Bs)KNF is reduced to a number of reasoningtasks in the non-modal SROIQ(Bs)

• For simplicity, appearance of modal operators restricted tosimple KBs as in ALCKNF (finitely many, finiterepresentations of models)




(Monotonic) Coverage

• SROIQ (a.k.a. OWL 2 DL);• The tractable profiles OWL 2 EL, OWL 2 RL, OWL 2 QL;• RIF-Core, i.e., n-ary Datalog, interpreted as DL-safe Rules

(general case new result in [Knorr et al., ECAI12]);• DL-safe SWRL [Motik et al., JWS05], AL-log [Donini et al.,

JIIS98], and CARIN [Levy and Rousset, AI98].




(Non-monotonic) Coverage

• ALCKNF [Donini et al., ACM TOCL02]; includes notions ofconcept and role closure present in this formalism;

• Closed Reiter defaults covered through the coverage ofALCKNF ; includes coverage of DLs extended with defaultrules [Baader and Hollunder, JAR95];

• Hybrid MKNF [Motik and Rosati, JACM10];• Answer Set Programming, i.e., disjunctive Datalog with

classical negation and non-monotonic negation under theanswer set semantics; follows from the coverage of HybridMKNF.




N -nary DatalogNR = NP,2 ∪ {U} ∪ S, where S is a special set of roles: IfP ∈ NP,>2 has arity k, then P1, . . . , Pk ∈ S are unique binarypredicates associated with P ;

Translation: dl(P (t1, . . . , tk)) := ∃U.(∃P1.{t1} u . . . u ∃Pk.{tk});

Family of interpretations of J for interpretation I of Datalog RB:

(a) To each (d1, . . . , dk) ∈ P I , assign a unique element e in ∆(i.e., we define a total, injective function from the set oftuples to ∆).

(d) For each P ∈ NP,>2, if (d1, . . . , dk) ∈ P I , then (e, di) ∈ PJi ,where e is the element assigned to (d1, . . . , dk) in point (a).




Hybrid MKNF

Seamless integration of DL ontology O and rules of the form

KH1 ∨KHl ← KA1, . . . ,KAn,notB1, . . . ,notBm

Based on the n-nary Datalog embedding, additionally:

dl(KH1 ∨KHl ← KA1, . . . ,KAn,notB1, . . . ,notBm) :=Kdl(A1) u . . . uKdl(An) u ¬Adl(B1) u . . . u ¬Adl(Bm)v Kdl(H1) t . . . tKdl(Hl)




Example

KC(x)←KHasParent(x, y),KHasParent(x, z),K(y 6≈ z),notMarried(y, z).

can be translated into

K∃U.({x} u ∃HasParent.{y}) uK∃U.({x} u∃HasParent.{z}) uK∃U.({y} u ∃ 6≈ .{z}) u ¬A∃U.({y} u∃Married.{z}) v K∃U.({x} u C)




Summary

• Very expressive and general language with an advancedsemantics

• Large coverage makes it an interesting candidate for aunifying logic

• challenges• Improve reasoning algorithms• Find non-trivial fragments within the full language




The End

Thank you for your attention!




References(1)

• Description logics of minimal knowledge and negation asfailure. Francesco M. Donini, Daniele Nardi, and RiccardoRosati, ACM TOCL, 3(2), 2002, 227–252.

• Reconciling Description Logics and Rules. Boris Motik andRiccardo Rosati, Journal of the ACM, 57(5), 2010, 1–62.

• Local closed world reasoning with description logics underthe well-founded semantics. Matthias Knorr, José J.Alferes, and Pascal Hitzler, Artificial Intelligence,175(9-10), 2011, 1528–1544.




References(2)

• Reconciling OWL and non-monotonic rules for theSemantic Web. Matthias Knorr, Pascal Hitzler, andFrederick Maier, ECAI 2012, 474–479.

• Query-driven procedures for hybrid MKNF knowledgebases. José J. Alferes, Matthias Knorr, and Terrance Swift,ACM TOCL, 14(2), 2013, 1–43.

• A query tool for EL with non-monotonic rules. VadimIvanov, Matthias Knorr, and João Leite, ISWC 13, toappear.


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