Dl2014 slides

Contextualized Knowledge Repositories with
Justifiable 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 reasoning
with 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 contextual
knowledge 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 layers
Global context:
Structure of contexts and object knowledge shared by all contexts
(Local) contexts:
Local object knowledge (with references)

Global context:
(Local) contexts:
Bird ⊑ Fly
Horse ⊑ ¬Fly

Global context:
(Local) contexts:
Bird ⊑ Fly
Horse ⊑ ¬Fly
greek_myths
Horse(pegasus), Fly(pegasus)

Global context:
(Local) contexts:
Bird ⊑ Fly
Horse ⊑ ¬Fly
greek_myths
Horse(pedasus)

Global context:
(Local) contexts:
Bird ⊑ Fly
Horse ⊑ ¬Fly
greek_myths
Horse(pedasus), ¬Fly(pedasus)

Global context:
(Local) contexts:
Bird ⊑ Fly
Horse ⊑ ¬Fly
greek_myths
Horse(pedasus), ¬Fly(pedasus)
Ô We want to specify that certain global axioms are defeasible:
they hold globally, but allow exceptional instances in local contexts

CKR extension for defeasibility
CKR extension for defeasibility:
Syntax and semantics of an extension of CKR with
defeasible axioms in global context
Extend datalog translation for OWL RL based CKR with
rules for the translation of defeasible axioms
Prototype implementation for CKR datalog rewriter

Outline

CKR introduction
A CKR is composed by 2 layers:
Global context
(Local) contexts
Global context Local contexts

CKR introduction
Global context
Metaknowledge:
structure of contexts, context classes,
relations, modules and attributes
(Local) contexts
Event
SportEvent
m_event
m_sport_ev
VolleyMatch
VolleyA1
Competition
A1_2012-13
m_v_match
match1 match2
m_match1 m_match2

CKR introduction
Global context
Metaknowledge:
Global object knowledge:
knowledge shared by all contexts
(Local) contexts
Event
SportEvent
m_event
m_sport_ev
VolleyMatch
VolleyA1
Competition
A1_2012-13
m_v_match
match1 match2
m_match1 m_match2
Country(Italy), City(Trento)...
hasParentLocation(Trento, Italy)...

CKR introduction
Global context
Metaknowledge:
Global object knowledge:
knowledge shared by all contexts
(Local) contexts
Object knowledge with references:
local knowledge with references to
value of predicates in other contexts
Knowledge distributed across
different modules Km
Event
SportEvent
m_event
m_sport_ev
VolleyMatch
VolleyA1
Competition
A1_2012-13
m_v_match
match1 match2
m_match1 m_match2
Country(Italy), City(Trento)...
hasParentLocation(Trento, Italy)...
Kmatch1 Winner(bre_banca_cuneo_volley),
RunnerUp(itas_trentino_volley)...
Kmatch2 Winner(casa_modena_volley),
RunnerUp(itas_trentino_volley)...

SROIQ-RL
SROIQ-RL
Restriction of SROIQ to the syntax of OWL-RL axioms:
C := Aj fag j C1 u C2 j C1 t C2 j 9R.C1 j 9R.fag j 9R.>
D := AjD1 u D2 j :C1 j 8R.D1 j 9R.fag j 6 [0, 1]R.C1 j 6 [0, 1]R.>
TBox axioms: C v D ABox axioms: D(a),R(a, b)

Metalanguage LG
Metavocabulary G: Contexts structure objects
N: context names (match1, volley_season2013)
A: contextual attributes (time, location, topic)
DA attribute values of A 2 A (2013, trento, sport)
M: module names (m_match1, m_event)
with role mod : N M
C: context classes (Event, VolleyMatch)
with Ctx 2 C: class of all contexts
R: contextual relations (hasSubEvent)
Metalanguage LG: DL language over G

Object language LS
Object vocabulary S: domain vocabulary
Eval expression
For X a concept or role expression in S, C a concept expression in G
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 Le
S: LS with eval expressions

Object language LS
Object vocabulary S: domain vocabulary
Eval expression
For X a concept or role expression in S, C a concept expression in G
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
TopTeam(bre_banca_cuneo_volley)
TopTeam(casa_modena_volley)
Object language with references Le
S: LS with eval expressions

Defeasible axioms
Ô We extend the type of axioms appearing in global object knowledge:
Defeasible axiom a of G: D(a) 2 G for a 2 LS
“a propagates to local contexts, but admits exceptional instances”
D(Cheap ⊑ Interesting)
Cheap(fbmatch), Cheap(market)
DL language LDS
LS with defeasibile axioms

Defeasible axioms
cultural_tourist
¬Interesting(fbmatch)
DL language LDS

Defeasible axioms
cultural_tourist
¬Interesting(fbmatch)
Interesting(market)
DL language LDS

Contextualized Knowledge Repository
Contextualized Knowledge Repository (CKR):
K = hG, fKmgm2Mi
G contains
metaknowledge axioms in LG
(defeasible) global object axioms in LDS
for every module name m 2 M,
Km contains object axioms with references in Le
S

CKR interpretation
Idea
CKR interpretations are two layered interpretations
CKR interpretation I = hM, Ii
Mis a DL interpretation over G [ S
For every x 2 CtxM, I(x) is a DL interpretation over S
DI(x) = DM
for a 2 NIS, aI(x) = aM
Interpretation of eval: eval(X,C)I(x) =
[
e2CM
XI(e)

Clashing assumptions
Idea
Exception of axiom instances modelled as clashing assumptions ha, ei
“In context c, ignore instance e in evaluation of a”

Idea
“In context c, ignore instance e in evaluation of a” h(Cheap v Interesting), fbmatchi

Idea
Clashing assumption ha, ei:
assumption that e is exceptional for a
CAS-interpretation ICAS = hM, I, CASi:
CAS(c): set of clashing assumptions of context c

Idea
CAS-model ICAS j= K
ICAS is a CAS-model for K if:
Mj= a, for every a 2 G strict or defeasible

Idea
CAS-model ICAS j= K
I(x) j= Km, if m is a module of context x
I(x) j= a, for every a 2 G strict

Idea
CAS-model ICAS j= K
I(x) j= Km, if m is a module of context x
I(x) j= a, for every a 2 G strict
for every D(a) 2 G, if I(x)6j= a(e), then ha, ei 2 CAS(x)

Justification
Idea
Assumptions must be justified by local assertions in a clashing set S
“In context c, a(e) [ S is unsatisfiable”

Justification
Idea
“In context c, a(e) [ S is unsatisfiable” fCheap(fbmatch), :Interesting(fbmatch)g

Justification
Idea
“In context c, a(e) [ S is unsatisfiable” fCheap(fbmatch), :Interesting(fbmatch)g
Justification
ICAS = hM, I, CASi model of K is justified, if:
for every context x 2 CtxM and clashing assumption ha, ei 2 CAS(x)
some clashing set S exists s.t. I(x) j= S
Ô Justified if, for every clashing assumption ha, ei,
we have a factual evidence S of its local unsatisfiability

CKR model
Idea
CKR models are interpretation where all c. assumptions are justified
CKR model I j= K
I = hM, Ii is a CKR model of K,
if some ICAS = hM, I, CASi is a justified CAS-model of K

Outline

CKR translation to datalog
Datalog translation:
Materialization calculus for instance checking in SROIQ-RL CKR
Extends with defeasible propagation the calculus presented
in [Bozzato and Serafini, 2013]
Idea
Composed by 3 kinds of rule sets:
Input rules I: translation of DL axioms to Datalog atoms
Deduction rules P: forward inference rules
Output 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 S
GL-reduct PS: set of rules obtained from ground(P) by removing
(i). every rule r s.t. Body(r) S6= Æ
(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 j= l iff for every AS S of P, l 2 S

Translation rules
Input rules I
Deduction rules P
Output rules O

Translation rules
Input rules I
Irl: SROIQ-RL input rules
c : A(a) ) finsta(a,A, c)g c : A v B ) fsubClass(A, B, c)g
Deduction rules P
Output rules O

Translation rules
Input rules I
Deduction rules P
Prl: SROIQ-RL deduction rules
instd(x, z, c) subClass(y, z, c), instd(x, y, c).
Output rules O

Translation rules
Input rules I
Iglob: Global input rules
c 2 N ) finsta(c, Ctx,gm)g C 2 C ) fsubClass(C, Ctx,gm)g
Deduction rules P
Output rules O

Translation rules
Input rules I
Iloc: Local input rules
c : eval(A,C) v B ) fsubEval(A,C, B, c)g
Deduction rules P
Ploc: Local deduction rules
instd(x, b, c) subEval(a, c1, b, c), instd(c0, c1,gm), instd(x, a, c0).
Output rules O

Translation rules
Input rules I
Iloc: Local input rules
c : eval(A,C) v B ) fsubEval(A,C, B, c)g
Deduction rules P
Ploc: Local deduction rules
instd(x, b, c) subEval(a, c1, b, c), instd(c0, c1,gm), instd(x, a, c0).
Output rules O
finstd(a,A, c)g ) c : A(a) ftripled(a,R, b, c)g ) c : R(a, b)

Translation rules
ID: Defeasibility input rules (overriding conditions)
D(A v B) )
fovr(subClass, x,A, B, c) :instd(x, B, c), instd(x,A, c), prec(c, g).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).

Translation rules
D(A v B) )
D(Cheap v Interesting) )
fovr(subClass, x, Cheap, Interesting, c) :instd(x, Interesting, c),
instd(x, Cheap, c), prec(c, g).g

Translation rules
D(A v B) )
D(Cheap v Interesting) )
fovr(subClass, x, Cheap, Interesting, c) :instd(x, Interesting, c),
instd(x, Cheap, c), prec(c, g).g
Ô PK(K) j= ovr(subClass, fbmatch, Cheap, Interesting, c) but
PK(K)6j= ovr(subClass, market, Cheap, Interesting, c) thus
PK(K) j= instd(market, Interesting, c)

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

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

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

Translation process
3 Local programs PC(c): translation for local contexts
4 CKR program PK(K): union of global and local programs
K entails a in a context c when PK(K) j= O(a, c)

Correctness (sketch)
Let us fix a set of clashing assumptions CASN(c) for every c 2 N
and the corresponding set OVR(CASN) of ovr atoms:
OVR(CASN) = fovr(p(e)) j ha, ei 2 CASN(c), Irl(a, c) = pg
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 j= O(a, c) iff K j=CASM
N
c : a.

Correctness (sketch)
Properties (sketch)
If ICASM
N
N i justified with K,
= hM, I, CASM
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) 2 S
s.t. ICASM
S
S i is justified for K
= hM, I, CASM
Theorem (“CKR-correctness”)
PK(K) j= O(a, c) iff K j= c : a.

Outline

Prototype structure
CKR Schema Rewriter (on DReW)
DLV system
CQ
query
CKR RL
rules
Global
context
OWL
Knowledge
modules
OWL
Prototype implementation:
Extends basic translation of OWL RL ontologies to 2 layer CKR structure
Input: OWL files for global context and knowledge modules
Output: datalog translation for CKR program

Prototype implementation
Translation process implementation:
DLV system
output.dlv
Global
context
OWL
Knowledge
modules
OWL
Translate
PG(G)
Translate
every
PC(c)
Merge
PG + PC
PG ⊨ 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

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 C
In ALC + Tmin, well founded “generality” order x y
Prototypical elements of C are: C u :C
“all C’s for which there is no more generic element of type C”
Models minimize the set of C
Ô elements are typical unless a contrary assertion exists

Discussion: typicality in DL
We compare to:
Typicality in DLs: ALC + Tmin [Giordano et al., 2013]
Idea: Defeasible membership similar to typical instances of C
In ALC + Tmin, well founded “generality” order x y
Prototypical elements of C are: C u :C
“all C’s for which there is no more generic element of type C”
Models minimize the set of C
Ô elements are typical unless a contrary assertion exists
Ô Similar to our “membership blocking” for D(a)
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 rules
Ô G and each local context as MCS contexts g and ci
Ô Mimic clashing assumptions with open bridge rules:
D(C v D) Ô
c : C u Aa v D g : Ctx(c)
c : Aa(y) g : Ctx(c), not (c : :Aa(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 axioms
Datalog translation based on materialization calculus for instance
checking [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 defeasibility
in DLs and logics of context
Prototype evaluation
comparison to SPARQL based implementation [Bozzato and Serafini, 2013]
Extension for defeasible axioms across local contexts
along explicit order relation (e.g. temporal, extension, revision, . . . )

Thank you for listening
Contextualized Knowledge Repositories with
Justifiable Exceptions
Loris Bozzato, Thomas Eiter, Luciano Serafini
DKM, Fondazione Bruno Kessler – Trento, Italy
Inst. für Informationssysteme, TU Wien – Wien, Austria
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 = hG, fm1gi where:
G :

mod(c1,m1)
D(A v B), D(C v :B)

m1 : f A(a), C(a) g

Example: priority and consequence
Let K = hG, fm1gi where:
G :

mod(c1,m1)
D(A v B), D(C v :B)

m1 : f A(a), C(a) g
It has 2 justified CAS-models s.t.:
1 I1(c1) j= B(a) and CAS1(c1) = fh(C v :B), aig,
with clashing set S = fC(a), B(a)g
2 I2(c1) j= :B(a) and CAS2(c1) = fh(A v B), aig,
with clashing set S = fA(a), :B(a)g
However, consequence is given as “cautious reasoning”, thus:
K6j= c1 : B(a) K6j= c1 : :B(a)

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