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Hybrid Logics and Ontology Languages. Ian Horrocks <[email protected]> Information Management Group School of Computer Science University of Manchester. Talk Outline. Introduction to Description Logics Ontologies and OWL OWL ontology language Ontology applications

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Hybrid logics and ontology languages l.jpg

Hybrid Logics and Ontology Languages

Ian Horrocks

<[email protected]>

Information Management Group

School of Computer Science

University of Manchester


Talk outline l.jpg
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


Introduction to description logics l.jpg

Introduction to Description Logics


What are description logics l.jpg
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)


Dl basics l.jpg
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


The dl family 1 l.jpg
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 u8hasChild.(Doctor t 9hasChild.Doctor)


The dl family 17 l.jpg
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 ÇhhasChildiDoctor)


The dl family 2 l.jpg
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


Dl knowledge base l.jpg
DL Knowledge Base

  • A TBox is a set of “schema” axioms (sentences), e.g.:

    {Doctor v Person,

    HappyParent´Person u8hasChild.(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


Dl knowledge base10 l.jpg
DL Knowledge Base

  • A TBox is a set of “schema” axioms (sentences), e.g.:

    {Doctor ! Person,

    HappyParent$Person Æ [hasChild](Doctor ÇhhasChildiDoctor)}

    • 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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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


Owl rdf xml exchange syntax l.jpg
OWL RDF/XML Exchange Syntax

<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:Restriction>

<owl:onProperty rdf:resource="#hasChild"/>

<owl:someValuesFrom rdf:resource="#Doctor"/>

</owl:Restriction>

</owl:unionOf>

</owl:allValuesFrom>

</owl:Restriction>

</owl:intersectionOf>

</owl:Class>

E.g., Person u8hasChild.(Doctor t 9hasChild.Doctor):


Class concept constructors l.jpg
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


Ontology axioms l.jpg
Ontology Axioms

  • OWL ontology equivalent to DL KB (Tbox + Abox)


Why description logic l.jpg
Why (Description) Logic?

  • OWL exploits results of 15+ years of DL research

    • Well defined (model theoretic) semantics


Why description logic17 l.jpg
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.]


Why description logic18 l.jpg
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


Why description logic19 l.jpg
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


Why description logic20 l.jpg
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 […]”


Applications of ontologies l.jpg
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


Applications of ontologies22 l.jpg

CentralSulcus

Frontal Lobe

Parietal Lobe

Occipital

Lobe

Temporal Lobe

Lateral Sulcus

Applications of Ontologies

  • Medicine

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


Applications of ontologies23 l.jpg
Applications of Ontologies

  • Organising complex and semi-structured information

    • UN-FAO, NASA, Ordnance Survey, General Motors, Lockheed Martin, …


Nominals in ontologies l.jpg
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} v9 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.


Ontology reasoning how do we do it l.jpg

Ontology Reasoning:How do we do it?


Using standard dl techniques l.jpg
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


Tableaux reasoning 1 l.jpg
Tableaux Reasoning (1)

  • E.g., KB:

    {HappyParent´Person u8hasChild.(Doctor t 9hasChild.Doctor),

    John:HappyParent, John hasChild Mary, Mary:: Doctor

    Wendy hasChild Mary, Wendy marriedTo John}

Person

8hasChild.(Doctor t 9hasChild.Doctor)


Tableaux reasoning 2 l.jpg
Tableaux Reasoning (2)

  • Tableau rules correspond to constructors in logic (u, 9 etc)

    • E.g., John:(Person u Doctor) --!John:Person andJohn: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:

      {Personv9hasParent.Person,

      John:Person}


Tableaux reasoning 3 l.jpg
Tableaux Reasoning (3)

  • In general, (representation of) model consists of:

    • Named individuals forming arbitrary directed graph

    • Trees of anonymous individuals rooted in named individuals


Decision procedures l.jpg
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


Reasoning with nominals a tableaux algorithm for shoiq l.jpg

Reasoning with Nominals:A Tableaux Algorithm for SHOIQ


Recall motivation for owl design l.jpg
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?


Shiq is already tricky l.jpg
SHIQ is Already Tricky

  • Does not have finite model property, e.g.:

    {ITN v61edge–u9edge.ITN,

    R:(ITN u60edge–)}

    • Double blocking

    • Block interpreted as infinite repetition


Shiq is already tricky34 l.jpg
SHIQ is Already Tricky

  • Does not have finite model property, e.g.:

    {ITN v61edge–u9edge.ITN,

    R:(ITN u60edge–)}

    • Double blocking

    • Block interpreted as infinite repetition

  • Termination problem due to > and 6, e.g.:

    {John:9hasChild.Doctor u>2 hasChild.Lawyer

    u62hasChild}

    • Add inequalities between nodes generated by > rule

    • Clash if 6 rule only applicable to  nodes


Shoiq loss almost of tmp l.jpg
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:61r–

    • 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


Intuition nominal nodes l.jpg
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


Generate merge problem is back l.jpg
Generate & Merge Problem is Back!

E.g., KB:

{VMP´Person u9loves.{Mary} u9hasFriend.VMP,

John:9hasFriend.VMP

Mary:62loves–}

  • Blocking prevented by N-nodes

  • Repeated generation and merging of nodes leads to non-termination


Intuition guess exact cardinality l.jpg
Intuition: Guess Exact Cardinality

  • New Ro?-rule guesses exact cardinality constraint on N-nodes

    {VMP´Person u9loves.{Mary} u9hasFriend.VMP,

    John:9hasFriend.VMP

    Mary:62loves–}

  • 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 forSHIQ


Conjunctive query answering using binders maybe l.jpg

Conjunctive Query Answering:Using binders (maybe)


Conjunctive queries l.jpg
Conjunctive Queries

  • Want to query KB using DB style conjunctive query language

    • e.g., hx,zià WinehxiÆ drunkWithhx,yiÆ DishhyiÆ fromRegionhy,zi

  • How to answer such queries?

    • Reduce to boolean queries w.r.t. candidate answer tuples

      • e.g., hià WinehChiantiiÆ drunkWithhChianti,yiÆ DishhyiÆ fromRegionhy,Venetoi

    • Transform query into concept Cq by “rolling up”

      • e.g., Cq = {Chianti} u9 drunkWith.(Dish u9 fromRegion.{Veneto})

    • such that query can be reduced to KB satisfiability test

      • hT,Ai² q iff hT[ {>v:Cq},Ai is not satisfiable


Rolling up 1 l.jpg

B u9P.C u9S.D

A u9R.(B u9P.C u9S.D)

B u9P.C

Rolling Up (1)

  • View query as a labeled graph and “roll up” from leaves to root

    • e.g., hià AhwiÆ Rhw,xiÆ BhxiÆ Phx,yiÆ ChyiÆ Shx,ziÆ Chyi


Cyclical queries l.jpg
Cyclical Queries

  • Problems arise when trying to roll up cyclical queries

    • e.g., hià AhwiÆ Rhw,xiÆ BhxiÆ Phx,yiÆ ChyiÆ Shx,ziÆ ChyiÆ Rhy,zi


Rolling up with binders 1 l.jpg

C u9P-..x

A u9R.(x.(B u9S.(D u9R-.(C u9P-..x))))

x.B

x.B

x.(B u9S.(D u9R-.(C u9P-..x)))

D u9R-.(C u9P-..x)

Rolling Up with Binders (1)

  • Problem could be solved by extending DL with binder:

    • e.g., hià AhwiÆ Rhw,xiÆ BhxiÆ Phx,yiÆ ChyiÆ Shx,ziÆ ChyiÆ Rhy,zi


Rolling up with binders 2 l.jpg
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


Summary l.jpg
Summary

  • DLs are a family of logic basedKR 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


Acknowledgements l.jpg
Acknowledgements

Thanks to:

  • Birte Glimm

  • Uli Sattler


Resources l.jpg
Resources

  • Slides from this talk

    • http://www.cs.man.ac.uk/~horrocks/Slides/HyLo06.ppt

  • 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?


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