A Tableaux Decision Procedure for SHOIQ

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A Tableaux Decision Procedure for SHOIQ

• Ian Horrocks and Ulrike Sattler
• lthorrockssattler_at_cs.man.ac.ukgt
• University of Manchester
• Manchester, UK

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SHOIQ the Final Frontier

• Ian Horrocks and Ulrike Sattler
• lthorrockssattler_at_cs.man.ac.ukgt
• University of Manchester
• Manchester, UK

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Talk Outline

• Introduction to Description Logics
• Ontologies
• Ontology Reasoning
• Why do we want it?
• How do we do it?
• Tableaux Algorithms for Description Logic
  Reasoning
• Current Work and Research Challenges
• 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 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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Applications of DLs

• Databases
• Ontologies (knowledge bases)
• OWL Lite Web Ontology Language based on SHIF
• OWL DL Web Ontology Language based on SHOIN
• Motivation for OWL design was to exploit results
  of DL research
• Well defined semantics
• Formal properties well understood (complexity,
  decidability)
• Known tableaux decision procedures and
  implemented systems
• But not for SHOIN (up until now)

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DL Basics

• Concepts (unary predicates/formulae with one free
  variable)
• E.g., Person, Doctor, HappyParent, Doctor t
  Lawyer
• Roles (binary predicates/formulae with two free
  variables)
• E.g., hasChild, loves, (hasBrother hasDaughter)
• Individual names (constants)
• E.g., John, Mary, Italy
• Operators (for forming concepts and roles)
  restricted so that
• Language is decidable and, if possible, of low
  complexity
• No need for explicit use of variables
• Restricted form of 9 and 8 (direct correspondence
  with ? and )
• Features such as counting can be succinctly
  expressed

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The DL Family (1)

• Smallest propositionally closed DL is ALC (equiv
  modal 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 (2)

• S often used for ALC with transitive roles (R)
• Additional letters indicate other extension,
  e.g.
• H for role hierarchy (e.g., hasDaughter v
  hasChild)
• O for nominals/singleton classes (e.g., Italy)
• I for inverse roles (e.g., isChildOf
  hasChild)
• N for number restrictions (e.g., gt2hasChild,
  63hasChild)
• Q for qualified number restrictions (e.g.,
  gt2hasChild.Doctor)
• ALC R role hierarchy nominals inverse
  QNR SHOIQ

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Knowledge Bases (Ontologies)

• A TBox is a set of schema axioms (sentences),
  e.g.
• Doctor v Person,
• HappyParent Person u 8hasChild.(Doctor t
  9hasChild.Doctor)
• An ABox is a set of data axioms (ground facts),
  e.g.
• JohnHappyParent, John hasChild Mary
• A Knowledge Base (KB) is a TBox plus and ABox
• An ontology is usually taken to be equiv. to a
  TBox
• But in OWL, an ontology is an arbitrary set of
  axioms (i.e., equiv. to a KB)

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OWL Syntax (Abstract RDF)
E.g., Person u 8hasChild.(Doctor t
9hasChild.Doctor)

• intersectionOf(Person
• restriction(hasChild allValuesFrom(
• unionOf(Doctor
• restriction(hasChild someValuesFrom(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"hasCh
  ild"/gt
• ltowlsomeValuesFrom
  rdfresource"Doctor"/gt
• lt/owlRestrictiongt

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Description Logic Reasoning
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Tableaux Reasoning (1)

• 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 (2)

• 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 (3)

• 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 (4)

• 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 Procedure

• Algorithm is a decision procedure, 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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SHOIQ Why is it Hard?
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SHIQ is Already Tricky

• Does not have finite model property, e.g.
• ITN v 61 edge u 8 edge.ITN 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 8 edge.ITN u 9edge.ITN,
• RITN u 60 edge u 9edge.ITN
• Double blocking
• Block interpreted as infinite repetition
• Yo-yo 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 ExpTime ! NExpTime

• Interactions between O, I, and Q lead to
  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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SHOIQ Yo-Yo Problem is Back!

• E.g., KB
• VMP Person u 9loves.Mary u
  9hasFriend.VMP,
• John9hasFriend.VMP
• Mary62 loves
• Blocking prevented by N-nodes
• Repeated creation 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
  yo-yo 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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Summary

• DLs are a family of logic based KR formalisms
• Well known as basis of ontology languages such as
  OWL
• Key motivation for the design of OWL was the
  existence of DL tableaux decision procedures and
  implementations
• But, no procedure/implementation for OWL DL/SHOIN
  (up to now)
• SHOIQ algorithm solves this (very embarrassing)
  problem
• Ro?-rule introduces new source of non-determinism
• But good pay as you go characteristics
• Implementation already underway in FaCT and
  Pellet systems
• Should work well in realistic ontology
  applications

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Questions?