1. Hybrid Logics and Ontology Languages
Ian Horrocks
Information Management Group
School of Computer Science
University of Manchester

2. Talk Outline Introduction to Description Logics Ontologies and OWL
OWL ontology language
Ontology applications
Nominals in Ontology Languages
Ontology Reasoning
Tableaux algorithms
Reasoning with nominals
Conjunctive Query Answering
Using binders and state variables
Summary

3. Introduction to Description Logics

4. What Are Description Logics?
A family of logic based Knowledge Representation formalisms
Descendants of semantic networks and KL-ONE
Describe domain in terms of concepts (classes), roles (properties, relationships) and individuals
Distinguished by:
Formal semantics (typically model theoretic)
Decidable fragments of FOL (often contained in C2)
Closely related to Propositional Modal, Hybrid & Dynamic Logics
Closely related to Guarded Fragment
Provision of inference services
Decision procedures for key problems (satisfiability, subsumption, etc)
Implemented systems (highly optimised)

5. DL Basics Concepts (formulae) Roles (modalities)
E.g., Person, Doctor, HappyParent, (Doctor t Lawyer)
Roles (modalities)
E.g., hasChild, loves
Individuals (nominals)
E.g., John, Mary, Italy
Operators (for forming concepts and roles) restricted so that:
Satisfiability/subsumption is decidable and, if possible, of low complexity
No need for explicit use of variables
Restricted form of 9 and 8 (direct correspondence with hii and [i])
Features such as counting (graded modalities) succinctly expressed

6. The DL Family (1)
Smallest propositionally closed DL is ALC (equivalent to K(m))
Concepts constructed using booleans
u, t, :,
plus restricted quantifiers
9, 8
Only atomic roles
E.g., Person all of whose children are either Doctors or have a child who is a Doctor:
Person u 8hasChild.(Doctor t 9hasChild.Doctor)

7. The DL Family (1)
Smallest propositionally closed DL is ALC (equivalent to K(m))
Concepts constructed using booleans
u, t, :,
plus restricted quantifiers
9, 8
Only atomic roles
E.g., Person all of whose children are either Doctors or have a child who is a Doctor:
Person Æ [hasChild](Doctor Ç hhasChildiDoctor)

8. The DL Family (2) S often used for ALC extended with transitive roles
i.e., the union of K(m) and K4(m)
Additional letters indicate other extensions, e.g.:
H for role hierarchy (e.g., hasDaughter v hasChild)
O for nominals/singleton classes (e.g., {Italy})
I for inverse roles (converse modalities)
Q for qualified number restrictions (graded modalities, e.g., hiim)
N for number restrictions (graded modalities, e.g., hiim>)
S + role hierarchy (H) + nominals (O) + inverse (I) + NR (N) = SHOIN
SHOIN is the basis for W3C’s OWL Web Ontology Language

9. DL Knowledge Base
A TBox is a set of “schema” axioms (sentences), e.g.:
{Doctor v Person,
HappyParent ´ Person u 8hasChild.(Doctor t 9hasChild.Doctor)}
i.e., a background theory (a set of non-logical axioms)
An ABox is a set of “data” axioms (ground facts), e.g.:
{John:HappyParent,
John hasChild Mary}
i.e., non-logical axioms including (restricted) use of nominals

10. DL Knowledge Base
A TBox is a set of “schema” axioms (sentences), e.g.:
{Doctor ! Person,
HappyParent $ Person Æ [hasChild](Doctor Ç hhasChildiDoctor)}
i.e., a background theory (a set of non-logical axioms)
An ABox is a set of “data” axioms (ground facts), e.g.:
{John ! HappyParent,
John ! hhasChildiMary}
i.e., non-logical axioms including (restricted) use of nominals
A Knowledge Base (KB) is just a TBox plus an Abox

11. Ontologies and OWL

12. The Web Ontology Language OWL
Semantic Web led to requirement for a “web ontology language”
set up Web-Ontology (WebOnt) Working Group
WebOnt developed OWL language
OWL based on earlier languages OIL and DAML+OIL
OWL now a W3C recommendation (i.e., a standard)
OIL, DAML+OIL and OWL based on Description Logics
OWL effectively a “Web-friendly” syntax for SHOIN

13. OWL RDF/XML Exchange 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:allValuesFrom>
<owl:unionOf rdf:parseType=" collection">
<owl:Class rdf:about="#Doctor"/>
<owl:someValuesFrom rdf:resource="#Doctor"/>
</owl:Restriction>
</owl:unionOf>
</owl:allValuesFrom>
</owl:intersectionOf>
</owl:Class>

14. Class/Concept Constructors
C is a concept (class); P is a role (property); xi is an individual/nominal
XMLS datatypes as well as classes in 8P.C and 9P.C
Restricted form of DL concrete domains

15. Ontology Axioms
OWL ontology equivalent to DL KB (Tbox + Abox)

16. Why (Description) Logic?
OWL exploits results of 15+ years of DL research
Well defined (model theoretic) semantics

17. Why (Description) Logic?
OWL exploits results of 15+ years of DL research
Well defined (model theoretic) semantics
Formal properties well understood (complexity, decidability)
I can’t find an efficient algorithm, but neither can all these famous people.
[Garey & Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, 1979.]

18. Why (Description) Logic?
OWL exploits results of 15+ years of DL research
Well defined (model theoretic) semantics
Formal properties well understood (complexity, decidability)
Known reasoning algorithms

19. Why (Description) Logic?
OWL exploits results of 15+ years of DL research
Well defined (model theoretic) semantics
Formal properties well understood (complexity, decidability)
Known reasoning algorithms
Implemented systems (highly optimised)
Pellet

20. Why (Description) Logic?
Foundational research was crucial to design of OWL
Informed Working Group decisions at every stage, e.g.:
“Why not extend the language with feature x, which is clearly harmless?”
“Adding x would lead to undecidability - see proof in […]”
Heath-Robinson approach to language design; big tree is undecidability.

21. Applications of Ontologies
e-Science, e.g., Bioinformatics
Open Biomedical Ontologies Consortium (GO, MGED)
Used e.g., for “in silico” investigations relating theory and data
E.g., relating data on phosphatases to (model of) biological knowledge
Graphic showing protein functional domains (sequences of amino acids?); the identifying characteristics of different proteins.

22. Applications of Ontologies
Medicine
Building/maintaining terminologies such as Snomed, NCI & Galen
Frontal Lobe
Temporal Lobe
Parietal Lobe
Occipital
Lobe
Central Sulcus
Lateral Sulcus
- Example from project to (semi-) automate the annotation of MRI images of the brain
- FMA derived ontology used to capture knowledge of brain anatomy

23. Applications of Ontologies
Organising complex and semi-structured information
UN-FAO, NASA, Ordnance Survey, General Motors, Lockheed Martin, …

24. Nominals in Ontologies
Used in extensionally defined classes
e.g., class EU might be defined as {Austria, …, UnitedKingdom}
Written in OWL as oneOf(Austria … UnitedKingdom)
Equivalent to a disjunction of nominals: Austria Ç … Ç UnitedKingdom
Allows inferences such as:
EU contains 25 countries (assuming UNA/axioms)
If in the EU and not in oneOf(Austria … Sweden) ! in UnitedKingdom
Used in extended OWL Abox axioms
e.g., individual(Jim value(friend individual(value(friend Jane))))
Equivalent to {Jim} v 9 friend.(9 friend.{Jane})
i.e., Jim ! hfriendi(hfriendiJane)
Widely used in ontologies
e.g. in Wine ontology used for colours, grape types, regions, etc.

25. Ontology Reasoning: How do we do it?

26. Using Standard DL Techniques
Key reasoning tasks reducible to KB (un)satisfiability
E.g., C v D w.r.t. KB K iff K [ {x:(C u :D)} is not satisfiable
State of the art DL systems typically use (highly optimised) tableaux algorithms to decide satisfiability (consistency) of KB
Tableaux algorithms work by trying to construct a concrete example (model) consistent with KB axioms:
Start from ground facts (ABox axioms)
Explicate structure implied by complex concepts and TBox axioms
Syntactic decomposition using tableaux expansion rules
Infer constraints on (elements of) model

27. Tableaux Reasoning (1) E.g., KB:
{HappyParent ´ Person u 8hasChild.(Doctor t 9hasChild.Doctor),
John:HappyParent, John hasChild Mary, Mary:: Doctor
Wendy hasChild Mary, Wendy marriedTo John}
Person
8hasChild.(Doctor t 9hasChild.Doctor)

28. Tableaux Reasoning (2)
Tableau rules correspond to constructors in logic (u, 9 etc)
E.g., John:(Person u Doctor) --! John:Person and John:Doctor
Stop when no more rules applicable or clash occurs
Clash is an obvious contradiction, e.g., A(x), :A(x)
Some rules are nondeterministic (e.g., t, 6)
In practice, this means search
Cycle check (blocking) often needed to ensure termination
E.g., KB:
{Person v 9hasParent.Person,
John:Person}

29. Tableaux Reasoning (3)
In general, (representation of) model consists of:
Named individuals forming arbitrary directed graph
Trees of anonymous individuals rooted in named individuals

30. Decision Procedures
Algorithms are decision procedures, i.e., KB is satisfiable iff rules can be applied such that fully expanded clash free graph is constructed:
Sound
Given a fully expanded and clash-free graph, we can trivially construct a model
Complete
Given a model, we can use it to guide application of non-deterministic rules in such a way as to construct a clash-free graph
Terminating
Bounds on number of named individuals, out-degree of trees (rule applications per node), and depth of trees (blocking)
Crucially depends on (some form of) tree model property

31. Reasoning with Nominals: A Tableaux Algorithm for SHOIQ

32. Recall Motivation for OWL Design
Exploit results of DL research: …
Known tableaux decision procedures and implemented systems
But not for SHOIN (until recently)!
So why is/was SHOIN so hard?

33. SHIQ is Already Tricky Does not have finite model property, e.g.:
{ITN v 61 edge– u 9edge.ITN,
R:(ITN u 60 edge–)}
Double blocking
Block interpreted as infinite repetition

34. SHIQ is Already Tricky Does not have finite model property, e.g.:
{ITN v 61 edge– u 9edge.ITN,
R:(ITN u 60 edge–)}
Double blocking
Block interpreted as infinite repetition
Termination problem due to > and 6, e.g.:
{John:9hasChild.Doctor u >2 hasChild.Lawyer
u 62 hasChild}
Add inequalities between nodes generated by > rule
Clash if 6 rule only applicable to  nodes

35. SHOIQ: Loss (almost) of TMP
Interactions between O, I, and Q lead to new termination problems
Anonymous branches can loop back to named individuals (O)
E.g., 9r.{Mary}
Number restrictions (Q) on incoming edges (I) lead to non-tree structure
E.g., Mary:61 r–
Result is anonymous nodes that act like named individual nodes
Blocking sequence cannot include such nodes
Don’t know how to build a model from a graph including such a block

36. Intuition: Nominal Nodes
Nominal nodes (N-nodes) include:
Named individual nodes
Nodes affected by number restriction via outgoing edge to N-node
Blocking sequence cannot include N-nodes
Bound on number of N-nodes
Must initially have been on a path between named individual nodes
Length of such paths bounded by blocking
Number of incoming edges at an N-node is limited by number restrictions

37. Generate & Merge Problem is Back!
E.g., KB:
{VMP ´ Person u 9loves.{Mary} u hasFriend.VMP,
John:9hasFriend.VMP
Mary:62 loves–}
Blocking prevented by N-nodes
Repeated generation and merging of nodes leads to non-termination

38. Intuition: Guess Exact Cardinality
New Ro?-rule guesses exact cardinality constraint on N-nodes
{VMP ´ Person u 9loves.{Mary} u hasFriend.VMP,
John:9hasFriend.VMP
Mary:62 loves–}
Inequality between resulting N-nodes fixes generate & merge problem
Introduces new source of non-determinism
But only if nominals used in a “nasty” way
Usage in ontologies typically “harmless”
Otherwise behaves as for SHIQ

39. Conjunctive Query Answering: Using binders (maybe)

40. Conjunctive Queries
Want to query KB using DB style conjunctive query language
e.g., hx,zi à Winehxi Æ drunkWithhx,yi Æ Dishhyi Æ fromRegionhy,zi
How to answer such queries?
Reduce to boolean queries w.r.t. candidate answer tuples
e.g., hi à WinehChiantii Æ drunkWithhChianti,yi Æ Dishhyi Æ fromRegionhy,Venetoi
Transform query into concept Cq by “rolling up”
e.g., Cq = {Chianti} u 9 drunkWith.(Dish u 9 fromRegion.{Veneto})
such that query can be reduced to KB satisfiability test
hT,Ai ² q iff hT [ {> v :Cq},Ai is not satisfiable

41. Rolling Up (1)
View query as a labeled graph and “roll up” from leaves to root
e.g., hi à Ahwi Æ Rhw,xi Æ Bhxi Æ Phx,yi Æ Chyi Æ Shx,zi Æ Chyi
A u 9R.(B u 9P.C u 9S.D)
B u 9P.C u 9S.D
B u 9P.C

42. Cyclical Queries
Problems arise when trying to roll up cyclical queries
e.g., hi à Ahwi Æ Rhw,xi Æ Bhxi Æ Phx,yi Æ Chyi Æ Shx,zi Æ Chyi Æ Rhy,zi

43. Rolling Up with Binders (1)
Problem could be solved by extending DL with binder:
e.g., hi à Ahwi Æ Rhw,xi Æ Bhxi Æ Phx,yi Æ Chyi Æ Shx,zi Æ Chyi Æ Rhy,zi
A u 9R.(x.(B u 9S.(D u 9R-.(C u 9P-..x))))
D u 9R-.(C u 9P-..x)
x.B
C u 9P-..x
x.B
x.(B u 9S.(D u 9R-.(C u 9P-..x)))

44. Rolling Up with Binders (2)
Unfortunately, already known that ALC + binder is undecidable [Blackburn and Seligman]
But, when used in rolling up, only occurs in very restricted form:
Only intersection, existential and positive state variables
and when negated (in sat test), only union, universal and negated vars
in form 8R.:x
Now known that SHIQ conjunctive query answering is decidable
Binders would potentially lead to a more “practical” algorithm
But not trivial to extend tableaux algorithm to SHIQ + binder
Blocking is difficult because binder introduces new concepts
Decidability of SHOIQ conjunctive query answering still open
Although believe we now have a solution

45. Summary DLs are a family of logic based KR formalisms
Describe domain in terms of concepts, roles and individuals
Closely related to Modal & Hybrid Logics
DLs are the basis for ontology languages such as OWL
Nominals widely used in ontologies
Reasoning with SHOIQ is tricky, but now reasonalby well understood
Binders potentially useful for conjunctive query answering
Allow for rolling up of arbitrary queries
Required extensions known to be decidable
But reasoning with extended languages still an open problem

46. Acknowledgements
Thanks to:
Birte Glimm
Uli Sattler

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

48. Thank you for listening
Any questions?