Title: Hybrid Logics and Ontology Languages
1Hybrid Logics and Ontology Languages
- Ian Horrocks
- lthorrocks_at_cs.man.ac.ukgt
- Information Management Group
- School of Computer Science
- University of Manchester
2Talk 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
3Introduction to Description Logics
4What 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)
5DL Basics
- Concepts (formulae)
- 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
6The 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)
7The 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)
8The 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., hiimgt) - S role hierarchy (H) nominals (O) inverse
(I) NR (N) SHOIN - SHOIN is the basis for W3Cs OWL Web Ontology
Language
9DL 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. - JohnHappyParent,
- John hasChild Mary
- i.e., non-logical axioms including (restricted)
use of nominals
10DL 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
11Ontologies and OWL
12The 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 DAMLOIL
- OWL now a W3C recommendation (i.e., a standard)
- OIL, DAMLOIL and OWL based on Description Logics
- OWL effectively a Web-friendly syntax for SHOIN
13OWL RDF/XML Exchange Syntax
E.g., Person u 8hasChild.(Doctor t
9hasChild.Doctor)
- ltowlClassgt
- ltowlintersectionOf rdfparseType"
collection"gt - ltowlClass rdfabout"Person"/gt
- ltowlRestrictiongt
- ltowlonProperty rdfresource"hasChild"/gt
- ltowlallValuesFromgt
- ltowlunionOf rdfparseType" collection"gt
- ltowlClass rdfabout"Doctor"/gt
- ltowlRestrictiongt
- ltowlonProperty rdfresource"hasChil
d"/gt - ltowlsomeValuesFrom
rdfresource"Doctor"/gt - lt/owlRestrictiongt
- lt/owlunionOfgt
- lt/owlallValuesFromgt
- lt/owlRestrictiongt
- lt/owlintersectionOfgt
- lt/owlClassgt
14Class/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
15Ontology Axioms
- OWL ontology equivalent to DL KB (Tbox Abox)
16Why (Description) Logic?
- OWL exploits results of 15 years of DL research
- Well defined (model theoretic) semantics
17Why (Description) Logic?
- OWL exploits results of 15 years of DL research
- Well defined (model theoretic) semantics
- Formal properties well understood (complexity,
decidability)
I cant 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.
18Why (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
19Why (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
20Why (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
21Applications 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
22Applications of Ontologies
- Medicine
- Building/maintaining terminologies such as
Snomed, NCI Galen
23Applications of Ontologies
- Organising complex and semi-structured
information - UN-FAO, NASA, Ordnance Survey, General Motors,
Lockheed Martin,
24Nominals 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.
25Ontology ReasoningHow do we do it?
26Using 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
27Tableaux Reasoning (1)
- E.g., KB
- HappyParent Person u 8hasChild.(Doctor t
9hasChild.Doctor), - JohnHappyParent, John hasChild Mary, Mary
Doctor - Wendy hasChild Mary, Wendy marriedTo John
Person 8hasChild.(Doctor t 9hasChild.Doctor)
28Tableaux Reasoning (2)
- Tableau rules correspond to constructors in logic
(u, 9 etc) - E.g., John(Person u Doctor) --! JohnPerson
and JohnDoctor - 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,
- JohnPerson
29Tableaux Reasoning (3)
- In general, (representation of) model consists
of - Named individuals forming arbitrary directed
graph - Trees of anonymous individuals rooted in named
individuals
30Decision 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
31Reasoning with NominalsA Tableaux Algorithm for
SHOIQ
32Recall 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?
33SHIQ 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
34SHIQ 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 gt and 6, e.g.
- John9hasChild.Doctor u gt2 hasChild.Lawyer
- u 62 hasChild
- Add inequalities between nodes generated by gt
rule - Clash if 6 rule only applicable to ? nodes
35SHOIQ 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., Mary61 r
- Result is anonymous nodes that act like named
individual nodes - Blocking sequence cannot include such nodes
- Dont know how to build a model from a graph
including such a block
36Intuition 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
37Generate Merge Problem is Back!
- E.g., KB
- VMP Person u 9loves.Mary u
9hasFriend.VMP, - John9hasFriend.VMP
- Mary62 loves
- Blocking prevented by N-nodes
- Repeated generation and merging of nodes leads to
non-termination
38Intuition Guess Exact Cardinality
- New Ro?-rule guesses exact cardinality constraint
on N-nodes - VMP Person u 9loves.Mary u
9hasFriend.VMP, - John9hasFriend.VMP
- Mary62 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
39Conjunctive Query AnsweringUsing binders (maybe)
40Conjunctive 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 gt v Cq,Ai is not
satisfiable
41Rolling 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
42Cyclical 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
43Rolling 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
44Rolling 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
45Summary
- 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
46Acknowledgements
- Thanks to
- Birte Glimm
- Uli Sattler
47Resources
- Slides from this talk
- http//www.cs.man.ac.uk/horrocks/Slides/HyLo06.pp
t - FaCT system (open source)
- http//owl.man.ac.uk/factplusplus/
- Protégé
- http//protege.stanford.edu/plugins/owl/
- W3C Web-Ontology (WebOnt) working group (OWL)
- http//www.w3.org/2001/sw/WebOnt/
- DL Handbook, Cambridge University Press
- http//books.cambridge.org/0521781760.htm
48Thank you for listening
Any questions?