1. Ontology Reasoning: the Why and the How
Ian Horrocks
University of Manchester
Manchester, UK

2. Talk Outline Ontologies: What are they?
Ontology Reasoning: Why do we need it?
Tools and services for ontology design and deployment
Importance of decidability, soundness and completeness
Ontology Reasoning: How do we do it?
Description Logics
Tableaux algorithms
Research Challenges
Expressive power, scalability etc.
Summary

3. Ontologies: What are they?

4. Ontology: Origins and History
a philosophical discipline—a branch of philosophy that
deals with the nature and the organisation of reality
Science of Being (Aristotle, Metaphysics, IV, 1)
Tries to answer the questions:
What characterizes being?
Eventually, what is being?
How should things be classified?

5. Classification: An Old Problem
Extract from Bills of Mortality, published weekly from s
The Diseases and Casualties this Week:
Aged 54
Apoplectic 1 ….
Fall down stairs 1
Gangrene
Grief
Griping in the Guts 74 …
Plague …
Suddenly 1
Surfeit
Teeth
Ulcer 2
Vomiting 7
Winde 8
Worms

6. Classification: An Old Problem
On those remote pages it is written that animals are divided into:
a. those that belong to the Emperor
b. embalmed ones
c. those that are trained
d. suckling pigs
e. mermaids
f. fabulous ones
g. stray dogs
h. those that are included in this classification
i. those that tremble as if they were mad
j. innumerable ones
k. those drawn with a very fine camel's hair brush
l. others
m. those that have just broken a flower vase
n. those that from a long way off look like flies
Attributed to “a certain Chinese encyclopaedia entitled Celestial Empire of benevolent Knowledge”. Jorge Luis Borges: The Analytical Language of John Wilkins

7. Ontology in Computer Science
An ontology is an engineering artifact consisting of:
A vocabulary used to describe (a particular view of) some domain
An explicit specification of the intended meaning of the vocabulary.
almost always includes how concepts should be classified
Constraints capturing additional knowledge about the domain
Ideally, an ontology should:
Capture a shared understanding of a domain of interest
Provide a formal and machine manipulable model of the domain

8. Example Ontology Vocabulary and meaning (“definitions”)
Elephant is a concept whose members are a kind of animal
Herbivore is a concept whose members are exactly those animals who eat only plants or parts of plants
Adult_Elephant is a concept whose members are exactly those elephants whose age is greater than 20 years
Background knowledge/constraints on the domain (“general axioms”)
Adult_Elephants weigh at least 2,000 kg
All Elephants are either African_Elephants or Indian_Elephants
No individual can be both a Herbivore and a Carnivore

9. Example Ontology (Protégé)

10. Example Ontology (OilEd)

11. Where are ontologies used?
e-Science, e.g., Bioinformatics
The Gene Ontology
The Protein Ontology (MGED)
“in silico” investigations relating theory and data
Medicine
Terminologies
Databases
Integration
Query answering
User interfaces
Linguistics
The Semantic Web

12. Ontology Driven User Interface
Structured Data Entry
File Edit Help
FRACTURE SURGERY
Reduction
Fixation
Fixation
Open
Closed
Open
Tibia
Fibula
Ankle
More...
Radius
Ulna
Wrist
Humerus
Femur
Femur
Left
Left
Right
More...
Gt Troch
Shaft
Neck
Neck
Fixation of open fracture of neck of left femur

13. Scientific American, May 2001:
Beware of the Hype!

14. Ontology Reasoning: Why do we need it?

15. Philosophical Reasons
Applications such as the Semantic Web aim at “machine understanding”
Understanding is closely related to reasoning
Recognising semantic similarity in spite of syntactic differences
Recognising implicit consequences given explicitly stated facts

16. Practical Reasons
Given key role of ontologies in many applications, it is essential to provide tools and services to help users:
Design and maintain high quality ontologies, e.g.:
Meaningful — all named classes can have instances
Correct — captured intuitions of domain experts
Minimally redundant — no unintended synonyms
Richly axiomatised — (sufficiently) detailed descriptions
Answer queries over ontology classes and instances, e.g.:
Find more general/specific classes
Retrieve individuals/tuples matching a given query
Integrate and align multiple ontologies

17. Why Decidable Reasoning?
OWL constructors/axioms have been restricted so reasoning is decidable
Consistent with Semantic Web's layered architecture
XML provides syntax transport layer
RDF(S) provides basic relational language and simple ontological primitives
OWL provides powerful but still decidable ontology language
Further layers (e.g. SWRL) will extend OWL
Will almost certainly be undecidable
W3C requirement for “implementation experience”
“Practical” algorithms for sound and complete reasoning
Several implemented systems
Evidence of empirical tractability

18. Why Sound & Complete Reasoning?
Important for ontology design
Ontologists need to have complete confidence in reasoner
Otherwise they will cease to trust results
Doubting unexpected results makes reasoner useless
Important for ontology deployment
Many realistic web applications will be agent ↔ agent
No human intervention to spot glitches in reasoning
Incomplete reasoning might be OK in 3-valued system
But “don’t know” typically treated as “no”

19. System Demonstration (OilEd)

20. Ontology Reasoning: How do we do it?

21. Use a (Description) Logic
OWL DL based on SHIQ Description Logic
In fact it is equivalent to SHOIN(Dn) DL
OWL DL Benefits from many years of DL research
Well defined semantics
Formal properties well understood (complexity, decidability)
Known reasoning algorithms
Implemented systems (highly optimised)
In fact there are three “species” of OWL (!)
OWL full is union of OWL syntax and RDF
OWL DL restricted to FOL fragment (¼ DAML+OIL)
OWL Lite is “simpler” subset of OWL DL

22. OWL Class Constructors
XMLS datatypes as well as classes in 8P.C and 9P.C
Restricted form of DL concrete domains

23. RDFS Syntax E.g., Person u 8hasChild.(Doctor t 9hasChild.Doctor):
<owl:Class>
<owl:intersectionOf rdf:parseType=" collection">
<owl:Class rdf:about="#Person"/>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasChild"/>
<owl:toClass>
<owl:unionOf rdf:parseType=" collection">
<owl:Class rdf:about="#Doctor"/>
<owl:hasClass rdf:resource="#Doctor"/>
</owl:Restriction>
</owl:unionOf>
</owl:toClass>
</owl:intersectionOf>
</owl:Class>

24. OWL Axioms Axioms (mostly) reducible to inclusion (v)
C ´ D iff both C v D and D v C
Obvious FOL equivalences
E.g., C ´ D , x.C(x) $ D(x), C v D , x.C(x) !D(x)

25. Basic Inference Tasks Knowledge is correct (captures intuitions)
Does C subsume D w.r.t. ontology O? (CI µ DI in every model I of O)
Knowledge is minimally redundant (no unintended synonyms)
Is C equivallent to D w.r.t. O? (CI = DI in every model I of O)
Knowledge is meaningful (classes can have instances)
Is C is satisfiable w.r.t. O? (CI  ; in some model I of O)
Querying knowledge
Is x an instance of C w.r.t. O? (xI 2 CI in every model I of O)
Is hx,yi an instance of R w.r.t. O? ((xI,yI) 2 RI in every model I of O)
Above problems can be solved using highly optimised DL reasoners

26. DL Reasoning: Basics
Tableau algorithms used to test satisfiability (consistency)
Try to build a tree-like model of the input concept C
Decompose C syntactically
Apply tableau expansion rules
Infer constraints on elements of model
Tableau rules correspond to constructors in logic (u, t etc)
Some rules are nondeterministic (e.g., t, 6)
In practice, this means search
Stop when no more rules applicable or clash occurs
Clash is an obvious contradiction, e.g., A(x), :A(x)
Cycle check (blocking) may be needed for termination
C satisfiable iff rules can be applied such that a fully expanded clash free tree is constructed

27. DL Reasoning: Advanced Techniques
Satisfiability w.r.t. an Ontology O
For each axiom C v D 2 O , add :C t D to every node label
More expressive DLs
Basic technique can be extended to deal with
Role inclusion axioms (role hierarchy)
Number restrictions
Inverse roles
Concrete domains/datatypes
Aboxes
etc.
Extend expansion rules and use more sophisticated blocking strategy
Forest instead of Tree (for Aboxes)
Root nodes correspond to individuals in Abox

28. DL Reasoning: Decision Procedures
Theorem: Tableaux algorithms are decision procedures for concept satisfiability (& subsumption & w.r.t. an ontology)
i.e., algorithms return “SAT” iff input concept is satisfiable
Terminating
Bounds on out-degree (rule applications per node) and depth (blocking) of tree
Sound
Can construct a tableau, and hence a model, from a fully expanded and clash-free tree
Complete
Can use a model to guide application of non-deterministic rules and so construct a clash-free tree

29. DL Reasoning: Optimised Implementations
Naive implementation can lead to effective non-termination
10 GCIs £ 10 nodes → 2100 different possible expansions
Modern systems include MANY optimisations
Optimised classification (compute partial ordering)
Enhanced traversal (exploits information from previous tests)
Use structural information to select classification order
Optimised satisfiability/subsumption testing
Normalisation and simplification of concepts
Absorption (simplification) of axioms
Dependency directed backtracking
Caching of satisfiability results and (partial) models
Heuristic ordering of propositional and modal expansion …

30. Research Challenges Increased expressive power Scalability
Existing DL systems implement (at most) SHIQ
OWL extends SHIQ with datatypes and nominals (SHOIN(Dn))
Future (undecidable) extensions such as SWRL
Scalability
Very large ontologies
Reasoning with (very large numbers of) individuals
Other reasoning tasks (non-standard inferences)
Querying
Matching
Least common subsumer
...
Tools and Infrastructure
Support for large scale ontological engineering and deployment

31. Summary An Ontology is an engineering artifact consisting of:
A vocabulary of terms
An explicit specification their intended meaning
Ontologies are set to play a key role in many applications
e-Science, Medicine, Databases, Semantic Web, etc.
Reasoning is important because
Understanding is closely related to reasoning
Essential for design, maintenance and deployment of ontologies
Reasoning support based on DL systems
Sound and complete reasoning
Highly optimised implementations
Challenges remain
Expressive power; scalability; new reasoning tasks; tools and infrastructure

32. Acknowledgements
Thanks to the many people who inspired me and with whom I have had the privilege of collaborating, in particular:
Alan Rector
Franz Baader
Uli Sattler

33. Resources Slides from this talk FaCT system (open source)
FaCT system (open source)
OilEd (open source)
Protégé
W3C Web-Ontology (WebOnt) working group (OWL)
DL Handbook, Cambridge University Press

34. Select Bibliography
Ian Horrocks, Peter F. Patel-Schneider, and Frank van Harmelen. From SHIQ and RDF to OWL: The making of a web ontology language. Journal of Web Semantics, 2003.
Franz Baader, Ian Horrocks, and Ulrike Sattler. Description logics as ontology languages for the semantic web. In Festschrift in honor of Jörg Siekmann, LNAI. Springer, 2003.
I. Horrocks and U. Sattler. Ontology reasoning in the SHOQ(D) description logic. In Proc. of IJCAI 2001.
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