dl2014 slides

Contextualized Knowledge Repositories withJustifiable Exceptions

1Loris Bozzato 2Thomas Eiter 1Luciano Serafini

1DKM, Fondazione Bruno Kessler – Trento, Italy

2Inst. für Informationssysteme, TU Wien – Wien, Austria

27th International Workshop on Description Logics (DL2014)

July 17-20, 2014 – Vienna, Austria

L. Bozzato (DKM - FBK) DL2014 1 / 34

Outline

1 Introduction and motivation

2 Contextualized Knowledge Repository (CKR)

3 Datalog translation (materialization calculus)

4 Datalog rewriter prototype

5 Comparison to approaches for defeasibility in DLs

6 Conclusion and future directions


Outline








Introduction and motivation

Need for context in Semantic Web:Validity of Semantic Web data related to specific context(time, location, topic...)

No explicit support for modelling and reasoningwith context sensitive knowledge in SW

Ô Need for well-defined theory of contexts

Contextualized Knowledge Repository (CKR)DL based framework for representation and reasoning with contextualknowledge in the Semantic Web

Theory: DL formalization based on AI theories of context[McCarthy, 1993, Lenat, 1998, Ghidini and Giunchiglia, 2001]

Implementation: built over state of the art Semantic Web tools


Need for defeasibility in contexts

CKR structure: two layersGlobal context:Structure of contexts and object knowledge shared by all contexts

(Local) contexts:Local object knowledge (with references)

Bird ⊑ FlyHorse ⊑ ¬Fly

Ô We want to specify that certain global axioms are defeasible:they hold globally, but allow exceptional instances in local contexts






greek_myths

Horse(pegasus), Fly(pegasus)







greek_myths

Horse(pegasus), Fly(pegasus)Horse(pedasus)







greek_myths

Horse(pegasus), Fly(pegasus)Horse(pedasus), ¬Fly(pedasus)



CKR extension for defeasibility

CKR extension for defeasibility:Syntax and semantics of an extension of CKR withdefeasible axioms in global contextExtend datalog translation for OWL RL based CKR withrules for the translation of defeasible axiomsPrototype implementation for CKR datalog rewriter


Outline








CKR introduction

A CKR is composed by 2 layers:

Global context

Metaknowledge:structure of contexts, context classes,relations, modules and attributesGlobal object knowledge:knowledge shared by all contexts

(Local) contexts

Object knowledge with references:local knowledge with references tovalue of predicates in other contextsKnowledge distributed acrossdifferent modules Km

Glo

bal c

onte

xt

Local c

onte

xts


CKR introduction


Global contextMetaknowledge:structure of contexts, context classes,relations, modules and attributes

Global object knowledge:knowledge shared by all contexts

(Local) contexts


Event

SportEvent

VolleyMatch VolleyA1

Competition

A1_2012-13

match1 match2

m_sport_ev

m_event

m_v_match

m_match1 m_match2

Glo

bal c

onte

xt

Local c

onte

xts


CKR introduction


Global contextMetaknowledge:structure of contexts, context classes,relations, modules and attributesGlobal object knowledge:knowledge shared by all contexts

(Local) contexts


Event

SportEvent


Competition

A1_2012-13

match1 match2

m_sport_ev

m_event

m_v_match

m_match1 m_match2

Country(Italy), City(Trento)...

hasParentLocation(Trento, Italy)...

Glo

bal c

onte

xt

Local c

onte

xts


CKR introduction


Global contextMetaknowledge:structure of contexts, context classes,relations, modules and attributesGlobal object knowledge:knowledge shared by all contexts

(Local) contexts


Event

SportEvent


Competition

A1_2012-13

match1 match2

m_sport_ev

m_event

m_v_match

m_match1 m_match2

Kmatch1 Winner(bre_banca_cuneo_volley),

RunnerUp(itas_trentino_volley)...

Kmatch2 Winner(casa_modena_volley),

RunnerUp(itas_trentino_volley)...

Country(Italy), City(Trento)...

hasParentLocation(Trento, Italy)...

Glo

bal c

onte

xt

Local c

onte

xts


SROIQ-RL

SROIQ-RLRestriction of SROIQ to the syntax of OWL-RL axioms:

C := A | {a} |C1 u C2 |C1 t C2 | ∃R.C1 | ∃R.{a} | ∃R.>D := A |D1 uD2 | ¬C1 | ∀R.D1 | ∃R.{a} | 6 [0, 1]R.C1 | 6 [0, 1]R.>

TBox axioms: C v D ABox axioms: D(a), R(a, b)


Metalanguage LΓ

Metavocabulary Γ: Contexts structure objects

N: context names (match1, volley_season2013)

A: contextual attributes (time, location, topic)DA attribute values of A ∈ A (2013, trento, sport)

M: module names (m_match1, m_event)with role mod : N×M

C: context classes (Event, VolleyMatch)with Ctx ∈ C: class of all contexts

R: contextual relations (hasSubEvent)

Metalanguage LΓ: DL language over Γ


Object language LΣ

Object vocabulary Σ: domain vocabulary

Eval expressionFor X a concept or role expression in Σ, C a concept expression in Γ

eval(X, C)

“The interpretation of X in all the contexts of type C”

VolleyTopMatch

match1 match2

Winner(bre_banca_cuneo_volley)

Winner(casa_modena_volley)

sports_news

eval(Winner,VolleyTopMatch) ⊑ TopTeam

Object language with references LeΣ: LΣ with eval expressions


Object language LΣ

Object vocabulary Σ: domain vocabulary

Eval expressionFor X a concept or role expression in Σ, C a concept expression in Γ

eval(X, C)

“The interpretation of X in all the contexts of type C”

VolleyTopMatch

match1 match2

Winner(bre_banca_cuneo_volley)

Winner(casa_modena_volley)

sports_news

eval(Winner,VolleyTopMatch) ⊑ TopTeamTopTeam(bre_banca_cuneo_volley)TopTeam(casa_modena_volley)

Object language with references LeΣ: LΣ with eval expressions


Defeasible axioms

Ô We extend the type of axioms appearing in global object knowledge:

Defeasible axiom α of G: D(α) ∈ G for α ∈ LΣ

“α propagates to local contexts, but admits exceptional instances”

D(Cheap ⊑ Interesting)Cheap(fbmatch), Cheap(market)

DL language LDΣ LΣ with defeasibile axioms


Defeasible axioms





cultural_tourist

¬¬¬¬Interesting(fbmatch)



Defeasible axioms





cultural_tourist

¬¬¬¬Interesting(fbmatch)Interesting(market)



Contextualized Knowledge Repository

Contextualized Knowledge Repository (CKR):

K = 〈G, {Km}m∈M〉

G containsmetaknowledge axioms in LΓ(defeasible) global object axioms in LD

Σ

for every module name m ∈ M,Km contains object axioms with references in Le

Σ


CKR interpretation

IdeaCKR interpretations are two layered interpretations

CKR interpretation I = 〈M, I〉M is a DL interpretation over Γ ∪ Σ

For every x ∈ CtxM, I(x) is a DL interpretation over Σ∆I(x) = ∆M

for a ∈ NIΣ, aI(x) = aM

Interpretation of eval: eval(X, C)I(x) =⋃

e∈CMXI(e)


Clashing assumptions

Idea

Exception of axiom instances modelled as clashing assumptions 〈α, e〉“In context c, ignore instance e in evaluation of α”

〈(Cheap v Interesting), fbmatch〉

Clashing assumption 〈α, e〉:assumption that e is exceptional for α

CAS-interpretation ICAS = 〈M, I , CAS〉:CAS(c): set of clashing assumptions of context c

CAS-model ICAS |= K

ICAS is a CAS-model for K if:M |= α, for every α ∈ G strict or defeasibleI(x) |= Km, if m is a module of context xI(x) |= α, for every α ∈ G strictfor every D(α) ∈ G, if I(x) 6|= α(e), then 〈α, e〉 ∈ CAS(x)



Idea

Exception of axiom instances modelled as clashing assumptions 〈α, e〉“In context c, ignore instance e in evaluation of α” 〈(Cheap v Interesting), fbmatch〉



CAS-model ICAS |= K




Idea




CAS-model ICAS |= K



Justification

IdeaAssumptions must be justified by local assertions in a clashing set S“In context c, α(e) ∪ S is unsatisfiable”

{Cheap(fbmatch),¬Interesting(fbmatch)}

JustificationICAS = 〈M, I , CAS〉 model of K is justified, if:

for every context x ∈ CtxM and clashing assumption 〈α, e〉 ∈ CAS(x)some clashing set S exists s.t. I(x) |= S

Ô Justified if, for every clashing assumption 〈α, e〉,we have a factual evidence S of its local unsatisfiability


Justification

IdeaAssumptions must be justified by local assertions in a clashing set S“In context c, α(e) ∪ S is unsatisfiable” {Cheap(fbmatch),¬Interesting(fbmatch)}

JustificationICAS = 〈M, I , CAS〉 model of K is justified, if:

for every context x ∈ CtxM and clashing assumption 〈α, e〉 ∈ CAS(x)some clashing set S exists s.t. I(x) |= S

Ô Justified if, for every clashing assumption 〈α, e〉,we have a factual evidence S of its local unsatisfiability


CKR model

IdeaCKR models are interpretation where all c. assumptions are justified

CKR model I |= K

I = 〈M, I〉 is a CKR model of K,if some ICAS = 〈M, I , CAS〉 is a justified CAS-model of K


Outline








CKR translation to datalog

Datalog translation:Materialization calculus for instance checking in SROIQ-RL CKRExtends with defeasible propagation the calculus presentedin [Bozzato and Serafini, 2013]

IdeaComposed by 3 kinds of rule sets:

Input rules I: translation of DL axioms to Datalog atomsDeduction rules P: forward inference rulesOutput rules O: translation for DL proved ABox assertion

Ô In I and P, “overriding” rules to treat defeasible propagation


Rules syntax and semantics

Translation produces general LPs interpreted under answer set semantics

Syntax: programs are finite set of rules:

a← b1, . . . , bk,not bk+1, . . . ,not bm.

with a, b1, . . . , bm literals

Semantics: given a program P and set of ground literals SGL-reduct PS: set of rules obtained from ground(P) by removing

(i). every rule r s.t. Body−(r) ∩ S 6= ∅(ii). the NAF part from bodies of remaining rules

S answer set of P: S least set of ground literals closed under PS

Literal l consequence of P: P |= l iff for every AS S of P, l ∈ S


Translation rules

Input rules I

Irl: SROIQ-RL input rulesc : A(a)⇒ {insta(a, A, c)} c : A v B⇒ {subClass(A, B, c)}Iglob: Global input rulesc ∈ N⇒ {insta(c, Ctx, gm)} C ∈ C⇒ {subClass(C, Ctx, gm)}Iloc: Local input rulesc : eval(A, C) v B⇒ {subEval(A, C, B, c)}

Deduction rules P

Prl: SROIQ-RL deduction rulesinstd(x, z, c)← subClass(y, z, c),instd(x, y, c).

Ploc: Local deduction rulesinstd(x, b, c)← subEval(a, c1, b, c),instd(c′, c1, gm),instd(x, a, c′).

Output rules O

{instd(a, A, c)} ⇒ c : A(a) {tripled(a, R, b, c)} ⇒ c : R(a, b)


Translation rules

Input rules IIrl: SROIQ-RL input rulesc : A(a)⇒ {insta(a, A, c)} c : A v B⇒ {subClass(A, B, c)}

Iglob: Global input rulesc ∈ N⇒ {insta(c, Ctx, gm)} C ∈ C⇒ {subClass(C, Ctx, gm)}Iloc: Local input rulesc : eval(A, C) v B⇒ {subEval(A, C, B, c)}

Deduction rules P

Prl: SROIQ-RL deduction rulesinstd(x, z, c)← subClass(y, z, c),instd(x, y, c).


Output rules O



Translation rules

Input rules IIrl: SROIQ-RL input rulesc : A(a)⇒ {insta(a, A, c)} c : A v B⇒ {subClass(A, B, c)}

Iglob: Global input rulesc ∈ N⇒ {insta(c, Ctx, gm)} C ∈ C⇒ {subClass(C, Ctx, gm)}Iloc: Local input rulesc : eval(A, C) v B⇒ {subEval(A, C, B, c)}

Deduction rules PPrl: SROIQ-RL deduction rulesinstd(x, z, c)← subClass(y, z, c),instd(x, y, c).


Output rules O



Translation rules

Input rules IIrl: SROIQ-RL input rulesc : A(a)⇒ {insta(a, A, c)} c : A v B⇒ {subClass(A, B, c)}Iglob: Global input rulesc ∈ N⇒ {insta(c, Ctx, gm)} C ∈ C⇒ {subClass(C, Ctx, gm)}

Iloc: Local input rulesc : eval(A, C) v B⇒ {subEval(A, C, B, c)}



Output rules O



Translation rules

Input rules IIrl: SROIQ-RL input rulesc : A(a)⇒ {insta(a, A, c)} c : A v B⇒ {subClass(A, B, c)}Iglob: Global input rulesc ∈ N⇒ {insta(c, Ctx, gm)} C ∈ C⇒ {subClass(C, Ctx, gm)}Iloc: Local input rulesc : eval(A, C) v B⇒ {subEval(A, C, B, c)}



Output rules O



Translation rules

Input rules IIrl: SROIQ-RL input rulesc : A(a)⇒ {insta(a, A, c)} c : A v B⇒ {subClass(A, B, c)}Iglob: Global input rulesc ∈ N⇒ {insta(c, Ctx, gm)} C ∈ C⇒ {subClass(C, Ctx, gm)}Iloc: Local input rulesc : eval(A, C) v B⇒ {subEval(A, C, B, c)}



Output rules O{instd(a, A, c)} ⇒ c : A(a) {tripled(a, R, b, c)} ⇒ c : R(a, b)


Translation rules

ID: Defeasibility input rules (overriding conditions)D(A v B)⇒{ovr(subClass, x, A, B, c)← ¬instd(x, B, c),instd(x, A, c),prec(c, g).}

PD: Defeasibility deduction rules (defeasible propagation)instd(x, z, c)← subClass(y, z, g),instd(x, y, c),prec(c, g),

not ovr(subClass, x, y, z, c).

D(Cheap v Interesting)⇒{ovr(subClass, x, Cheap, Interesting, c)← ¬instd(x, Interesting, c),

instd(x, Cheap, c),prec(c, g).}

Ô PK(K) |= ovr(subClass, fbmatch, Cheap, Interesting, c) butPK(K) 6|= ovr(subClass, market, Cheap, Interesting, c) thusPK(K) |= instd(market, Interesting, c)


Translation process

1 Global program PG(G): translation for global context

2 Computation of local knowledge bases Kc for each context c in G

3 Local programs PC(c): translation for local contexts4 CKR program PK(K): union of global and local programs

K entails α in a context c when PK(K) |= O(α, c)


Translation process

1 Global program PG(G): translation for global context2 Computation of local knowledge bases Kc for each context c in G




Translation process


3 Local programs PC(c): translation for local contexts

4 CKR program PK(K): union of global and local programs



Translation process





Correctness (sketch)

Let us fix a set of clashing assumptions CASN(c) for every c ∈ Nand the corresponding set OVR(CASN) of ovr atoms:

OVR(CASN) = {ovr(p(e)) | 〈α, e〉 ∈ CASN(c), Irl(α, c) = p}

Let PK(K)OVR be the reduct of PK(K) w.r.t. OVR(CASN)(i.e. positive program with resolved overridings)

Lemma (“CAS-correctness”)PK(K)OVR |= O(α, c) iff K |=CASMN

c : α.


Correctness (sketch)

Properties (sketch)

If ICASMN= 〈M, I , CASMN 〉 justified with K,

then there is an answer set S of PK(K)s.t. its ovr facts equals OVR(CASN)

If S answer set of PK(K),then we can build a map CASS(c) from ovr(p) ∈ Ss.t. ICASMS

= 〈M, I , CASMS 〉 is justified for K

Theorem (“CKR-correctness”)PK(K) |= O(α, c) iff K |= c : α.


Outline








Prototype structure

DLV system

CKR Schema Rewriter (on DReW)

CQ

query

CKR RL

rules

Global

context

OWLKnowledge

modules

OWL

Prototype implementation:Extends basic translation of OWL RL ontologies to 2 layer CKR structureInput: OWL files for global context and knowledge modulesOutput: datalog translation for CKR program


Prototype implementation

Translation process implementation:

DLV system

output.dlv

Global

context

OWLKnowledge

modules

OWL

Translate

PG(G)

Translate

every

PC(c)

Merge

PG + PCPG ⊨ hasMod(x,y)?

PK ⊨ c: A(a)?

Prototype and examples available at:http://dkm.fbk.eu/resources/ckr/ckr-datalog-rewriter-d-1.1.zip


http://dkm.fbk.eu/resources/ckr/ckr-datalog-rewriter-d-1.1.zip

Outline








Discussion: typicality in DL

We compare to:Typicality in DLs: ALC + Tmin [Giordano et al., 2013]

Idea: Defeasible membership similar to typical instances of CIn ALC + Tmin, well founded “generality” order x < yPrototypical elements of C are: Cu ¬♦C“all C’s for which there is no more generic element of type C”

Models minimize the set of ♦CÔ elements are typical unless a contrary assertion exists

Ô Similar to our “membership blocking” for D(α)

Ô Idea for encoding in CKR: CT v C, D(C v CT)

Circumscription in DLs [Bonatti et al., 2006]:similar notion of abnormality under model based minimization


Discussion: non-monotonic MCS

We compare to:Non-monotonic multi-context systems[Brewka and Eiter, 2007, Bikakis and Antoniou, 2010]

Idea: translate CKR to MCS with open bridge rulesÔ G and each local context as MCS contexts g and ci

Ô Mimic clashing assumptions with open bridge rules:

D(C v D) Ôc : CuAα v D← g : Ctx(c)c : Aα(y)← g : Ctx(c),not (c : ¬Aα(y))

Ô Equilibria (stable global belief states) then similar to CKR-models


Outline








Conclusion and future directions

Conclusions:CKR framework extension with defeasibility for global axiomsDatalog translation based on materialization calculus for instancechecking [Bozzato and Serafini, 2013]

Nonmonotonicity expressed using answer set semantics:instance checking as cautious inference from all answer sets of PK(K)

Current and future directions:Formal comparison to known approaches for defeasibilityin DLs and logics of contextPrototype evaluationcomparison to SPARQL based implementation [Bozzato and Serafini, 2013]

Extension for defeasible axioms across local contextsalong explicit order relation (e.g. temporal, extension, revision, . . . )


Thank you for listening

Contextualized Knowledge Repositories withJustifiable Exceptions

Loris Bozzato, Thomas Eiter, Luciano Serafini

DKM, Fondazione Bruno Kessler – Trento, ItalyInst. für Informationssysteme, TU Wien – Wien, Austria

https://dkm.fbk.eu/index.php/CKR


https://dkm.fbk.eu/index.php/CKR

References I

Bikakis, A. and Antoniou, G. (2010).Defeasible contextual reasoning with arguments in ambient intelligence.IEEE Trans. Knowl. Data Eng., 22(11):1492–1506.

Bonatti, P. A., Lutz, C., and Wolter, F. (2006).Description logics with circumscription.In KR, pages 400–410.

Bozzato, L. and Serafini, L. (2013).Materialization Calculus for Contexts in the Semantic Web.In DL2013, CEUR-WP. CEUR-WS.org.

Brewka, G. and Eiter, T. (2007).Equilibria in heterogeneous nonmonotonic multi-context systems.In Proceedings of the Twenty-Second Conference on Artificial Intelligence (AAAI-07), pages 385–390, Vancouver,Canada.

Ghidini, C. and Giunchiglia, F. (2001).Local models semantics, or contextual reasoning = locality + compatibility.Artificial Intelligence, 127.

Giordano, L., Gliozzi, V., Olivetti, N., and Pozzato, G. L. (2013).A non-monotonic description logic for reasoning about typicality.Artif. Intell., 195:165–202.

Lenat, D. (1998).The Dimensions of Context Space.Technical report, CYCorp.Published online http://www.cyc.com/doc/context-space.pdf (accessed June 21, 2009).


References II

McCarthy, J. (1993).Notes on formalizing context.In IJCAI.


Example: priority and consequence

Let K = 〈G, {m1}〉 where:

G :{mod(c1, m1)D(A v B), D(C v ¬B)

}m1 : { A(a), C(a) }

It has 2 justified CAS-models s.t.:1 I1(c1) |= B(a) and CAS1(c1) = {〈(C v ¬B), a〉},

with clashing set S = {C(a), B(a)}2 I2(c1) |= ¬B(a) and CAS2(c1) = {〈(A v B), a〉},

with clashing set S = {A(a),¬B(a)}

However, consequence is given as “cautious reasoning”, thus:

K 6|= c1 : B(a) K 6|= c1 : ¬B(a)


dl2014 slides

Internet

local knowledge

local object knowledge

structure of contexts

bozzato dkm fbk dl2014

contextsckr structure

layersglobal context

ckr introductiona ckr

context classes