Ontologies Benefit from Formal Concept Analysis Bar ış Sertkaya SAP Research Center Dresden Germany FCA and DLs what are they Formal Concept Anal ysis FCA field of mathematics based on ID: 333885
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Slide1
How Description Logic Ontologies Benefit fromFormal Concept Analysis
Bar
ış Sertkaya
SAP Research Center Dresden
GermanySlide2
FCA and DLs, what are they?
Formal
Concept
Analysis (FCA) field of mathematics based onlattice theory analyze data and derive a conceptual structuring medicine, psychology, ontologies,linguistic databases, software engineering, musicology, …
Description Logics (DLs) logical languages that are fragmentsof First Order Logic represent conceptual knowledge of an application domain semantic web, ontologies, life sciences, bio-medical computer science, software engineering, …
Concept: collection of objects sharing certain propertiesSlide3
FCA vs. DLs
Formal
Concept
Analysis (FCA) data algorithms formal concepts: concept latticeDescription Logics (DLs)
atomic concepts, roles:logical constructors: concept descriptions: classification algorithm subsumption hierarchyab
c1X2X
X3XX4
X
XSlide4
FCA vs. DLs
DLs
intensional definition of a concept given independent of a specific domain rich language for describing concepts(negation, exists, forall, number restrictions…)individuals partially described (open world semantics)FCAintensional knowledge derived from the extensional knowledge concept definitions are conjunctions ofatomic concepts (attributes) objects fully described(closed world semantics)Slide5
Knowledge Representation (KR)
Develop formalisms
for representing conceptual knowledge of an application domain,
that have a well-defined syntax,formal, unambigious semantics, and practical methods for reasoning / efficient implementations. Conceptual KnowledgeClasses: country, ocean-country, …Relations: has border to, has neighbor, …Individuals: Spain, Mediterranean, Atlantic, …Slide6
Description Logics (DLs)family of
logic-based knowledge representation formalisms
describe an application domain in terms of
concepts (classes): like Country, Ocean, …roles (relations): like hasBorderTo, hasNeighbour, …individuals like Spain, Atlantic, …logical constructors: well-defined formal semantics, decidable fragments of First Order Logic
Slide7
The DL
: The smallest propositionally closed description logic
atomic concepts:
A
,
B
, …
(unary predicates)
atomic roles:
r
,
s
, …
(binary predicates)
constructors:
(negation)
(conjunction)
(disjunction)
(existential restriction)
(value restriction)
Examples:
Slide8
Semantics of Based on interpretation consisting of:
a
domain
(a non-empty set), and an interpretation function Concept and role names: (concept names interpreted as subsets of the domain) (role names interpreted as binary relations)Complex concept descriptions: is a model of if Slide9
Example of an interpretation
Interpretation
domain
Concept
names
BodyOfWater
Sea
Ocean
Individual
names
Mediterranean
Atlantic
Country
OceanCountry
Roles
hasBorderTo
hasNeighbourPacific
SpainPortugal
Austria
LandlockedCountry
Interpretation function
Interpretation functionSlide10
Example of an interpretation
Interpretation
domain
Ocean
Atlantic
Country
hasBorderTo
hasNeighbour
Spain
PortugalSlide11
Reasoning
Main reasoning task:
Concept
subsumption: Is subsumed by ? (written )(Does hold for all )Concept subsumption for computing the subsumption hierarchy (classification)
BodyOfWater
Ocean
Sea
LandMass
Country
OceanCountry
LandLockedCountry
ReasonerSlide12
DL Knowledge Bases (Ontologies) DL Knowledge Base (Ontology) = TBox + ABox TBox defines the terminology of the application domain
ABox states facts about a specific world TBox: a set of concept definitions ABox: concept and role assertions General TBox:General concept inclusion axiomsSlide13
Bridging the gap between FCA and DLsExisting work mainly under 2 categories:
enriching FCA by borrowing constructors from DLs
theory-driven logical scaling
[Prediger,Stumme’99] terminological attribute logic [Prediger’00] relational concept analysis [Rouane,Huchard,Napoli,Valtchev’07] logical concept analysis [Ferré, Ridoux’01] employing FCA methods in DL knowledge basesComputation of an extended subsumption hierarchy [Baader’95] Subsumption hierarchy of conjunctions and disjunctions of DL concepts [Stumme’96] Subsumption hierarchy of least common subsumers [Baader,Molitor’00] Relational exploration [Rudolph’04,06] Supporting bottom-up construction of DL knowledge bases [Baader,Turhan,Sertkaya’07] Knowledge Base Completion [Baader,Ganter,Sattler,Sertkaya’07]Role assertion analysis [Coulet, Smail-Tabbone, Napoli, Devignes’08] Exploring finite models [Baader,Distel’08,09]Slide14
Extended Subsumption Hierarchy of DL Concepts
traditional
TBox
classification: subsumption hierarchy of concepts not sufficient in some settings: interaction between defined concepts not visible consider the concepts , , and no subsumption relation between these three concepts but, subsumed by not visible from the subsumption hierarchy!
hierarchy of conjunctions of defined concepts enables
faster inferences
.
precompute
and store it.
how?
Using attribute exploration
define a formal context whose concept lattice represents this hierarchySlide15
Extended Subsumption Hierarchy of DL Concepts
Formal context
s.t
. the concept lattice is isomorphic to the hierarchy of conjunctions of DL concepts [Baader’95]: … X X …
…
…
…
and , but , which is not visible in the usual hierarchy
implication questions are
subsumption
tests
a DL
reasoner
can act as an expert
a modified DL
reasoner
is needed for providing
countexamplesSlide16
Contributions to bridging the gap:1) supporting bottom-up construction of KBs traditional way of creating
ontologies
: (top-down manner)
define concepts specify properties of individuals using them not always adequate which concepts are relevant? how to define them correctly? alternative: bottom-up construction of ontologies
ABox
User selects similar
ABox
individuals
Individuals automatically generalized into concept descriptions (MSC computation)
Commonalities automatically extracted (LCS computation)
The LCS inspected/modified by the ontology engineer and added to the ontology
Slide17
Supporting bottom-up construction of KBs subsumption hierarchy of conjunctions of concept names and their negations needed for computing LCS
requires
subsumption
tests for a TBox containing concept names each subsumption test computationally expensive computing the hierarchy smartly without checking all pairs? using attribute explorationAgain define an appropriate formal contextDL reasoner can answer implication questionsUse background knowledge implies implies on the FCA sideSlide18
Bridging the gap:
2) Ontology completion
Existing
ontologytools support:Detecting inconsistenciesInferring consequencesFinding reasons for them
Quality dimesion of soundnessWhat about completeness?
are there
missing relations between classes? missing individuals?
if so how to
extend
the ontology appropriately?Slide19
Ontology Completion
ABox
Asian
EUmember
European
Mediterranean
Russia
+
?
?
?
China
+
-
-
?
Montenegro
?
?
+
?
Germany
-
+
+
-
Italy
-
+
+
+
TBox
All European countries EU members?
All EU members that have a border to Mediterranean have territories in Europe?Slide20
The Phosphatese OntologyOWL Ontology for human protein
phosphatese
family
[Wolstencroft, Brass, Horrocks, Lord, Sattler, Turi, & Stevens (2005)] developed based on peer-reviewed publications detailed knowledge about different classes of such proteins TBox: classes of proteins, relations among these classes ABox: large set of human phospthateses identified and documented by expert biologistsGiven this ontology, the biologist wants to know: Are there relations that hold in the real world, but that do not follow from the TBox? Are there phospthateses that are not represented in the ABox, or even that have not yet been identified? Slide21
When is an ontology (formally) complete?
is complete
w.r.t
. the intended application domain if these are equivalent:
( and are sets of concept names)
is satisfied by
follows from
does not contain a counterexample to
Cannot be achieved by an automated tool alone, a domain expert needed!
questions ( the number of concept names)
Many of them redundant
Do not bother the expert unnecessarily
A smart way to get answers to these questions:
attribute exploration!Slide22
Attribute Exploration for DL OntologiesExtension for open-world semantics of DL
ABoxes
Attribute exploration for
partial/incomplete formal contextsAlready existing approaches [Burmeister & Holzer 2005]the resulting knowledge is incomplete (certain implications, uncertain implications)In contrast we want to have complete knowledge at the endOur expert has / can access to complete knowledgeBut he should be able to give partial descriptions of objects during explorationProved termination, correctness, minimum number of questionsAn ABox is a partial context Integrated a DL reasoner for avoiding questionsImproved usability. The expert can:Skip questionsStop exploration, see previous answers, undo previous actions,See why an implication automatically was accepted Slide23
Ontology Completion
When a question is asked:
first check if it follows from the ontology
if not ask the expert if the expert confirms, add a new axiomto the TBox if the expert rejects, get a new ABox assertion as counterexampleSlide24
Summary:How DLs benefit from FCA?
Mainly 2 categories:
using concept lattice to detect implicit relations between classes
Extended subsumption hierarchy (of conjunctions of concepts)Subsumption hierarchy of least common subsumersSupporting bottom-up construction using attribute exploration to complete knowledgeKnowledge base completionSlide25
FCA at SAP ResearchThe
Aletheia
Project
Obtaining product information through the use of semantic technologiesFCA used for requirement analysissponsored by the Federal Ministry of Education and Research (BMBF)Partners: SAP AG, ABB, BMW Group, Deutsche Post, OntoPrise, Otto, TU Dresden, FU Berlin, HU Berlin, Frauenhofer IIS, TecO, Giesecke & Devrient, Eurolog, http://www.aletheia-projekt.de New project CUBIST (Combining and Uniting Business Intelligence with Semantic Technologies) FCA used for visual analytics on top of business intelligencePartners: SAP AG, Sheffield Halam University, Heriot-Watt University, Innovantage, Ontotext Lab, Centrale Rechereche S.A. (CRSA) – Laboratoire MAS, Space Applications Services NV Academic articles at ICCS, ICFCA on Role Based Access Control for Ontologies, …Slide26
Thank youSlide27
Early Days of KR
PieceOfLand
OceanOceanCountryCountry
BodyOfWaterIslandCountry
is a
is a
is a
is a
hasBorderTo
hasBorderTo
Semantic Networks
[
Quilian
1967]
nodes represent classes links represent relations hasBorderTo: does it mean there existsa border,
or for all borders? ambigious semantics!KL-ONE [Brachman & Levesque 1985] logic-based semanticsSlide28