Hybrid Logics and Ontology Languages

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Hybrid Logics and Ontology Languages

• Ian Horrocks
• lthorrocks_at_cs.man.ac.ukgt
• Information Management Group
• School of Computer Science
• University of Manchester

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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

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Introduction to Description Logics
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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)

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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

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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)

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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)

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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

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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

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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

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Ontologies and OWL
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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

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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

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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

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Ontology Axioms

• OWL ontology equivalent to DL KB (Tbox Abox)

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Why (Description) Logic?

• OWL exploits results of 15 years of DL research
• Well defined (model theoretic) semantics

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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.
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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

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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
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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

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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

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Applications of Ontologies

• Medicine
• Building/maintaining terminologies such as
  Snomed, NCI Galen

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Applications of Ontologies

• Organising complex and semi-structured
  information
• UN-FAO, NASA, Ordnance Survey, General Motors,
  Lockheed Martin,

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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.

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Ontology ReasoningHow do we do it?
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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

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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)
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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

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Tableaux Reasoning (3)

• In general, (representation of) model consists
  of
• Named individuals forming arbitrary directed
  graph
• Trees of anonymous individuals rooted in named
  individuals

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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

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Reasoning with NominalsA Tableaux Algorithm for
SHOIQ
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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?

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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

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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

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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

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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

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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

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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

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Conjunctive Query AnsweringUsing binders (maybe)
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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

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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

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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

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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

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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

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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

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Acknowledgements

• Thanks to
• Birte Glimm
• Uli Sattler

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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

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Thank you for listening
Any questions?