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Hybrid Logics and Ontology Languages

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Title: Hybrid Logics and Ontology Languages


1
Hybrid Logics and Ontology Languages
  • Ian Horrocks
  • lthorrocks_at_cs.man.ac.ukgt
  • 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)
  • 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., hiimgt)
  • S role hierarchy (H) nominals (O) inverse
    (I) NR (N) SHOIN
  • SHOIN is the basis for W3Cs 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.
  • JohnHappyParent,
  • 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 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

13
OWL 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

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 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.
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

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

22
Applications of Ontologies
  • Medicine
  • Building/maintaining terminologies such as
    Snomed, NCI Galen

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 ReasoningHow 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),
  • JohnHappyParent, 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) --! 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

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 NominalsA 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 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

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., 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

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
    9hasFriend.VMP,
  • John9hasFriend.VMP
  • Mary62 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
    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

39
Conjunctive Query AnsweringUsing 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 gt 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

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

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
  • 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

48
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