Transcript Ontologe Reasoning: the Why and the How
A Tableaux Decision Procedure for
SHOIQ
Ian Horrocks and Ulrike Sattler
SHOIQ
: the Final Frontier
Ian Horrocks and Ulrike Sattler
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 relationships) and individuals (properties, Distinguished by: – Formal semantics (typically model theoretic) • Decidable fragments of FOL (often contained in C 2 ) • 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) Combining the strengths of UMIST and The Victoria University of Manchester
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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 Combining the strengths of UMIST and The Victoria University of Manchester
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The DL Family (1)
Smallest propositionally closed DL is ALC – Concepts constructed using booleans (equiv modal K (m) ) 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 8 hasChild.(Doctor t 9 hasChild.Doctor) Combining the strengths of UMIST and The Victoria University of Manchester
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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 O I N Q hasChild) for nominals/singleton classes (e.g., {Italy}) for inverse roles (e.g., isChildOf ´ hasChild
–
) for number restrictions (e.g., > 2 hasChild, 6 3 hasChild) for qualified number restrictions (e.g., > 2 hasChild.Doctor) ALC + R + + role hierarchy + nominals + inverse + QNR = SHOIQ Combining the strengths of UMIST and The Victoria University of Manchester
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Knowledge Bases (Ontologies)
A TBox is a set of “schema” axioms (sentences), e.g.: {Doctor v Person, HappyParent ´ Person u 8 hasChild.(Doctor t 9 hasChild.Doctor)} An ABox is a set of “data” axioms (ground facts), e.g.: {John:HappyParent, 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) Combining the strengths of UMIST and The Victoria University of Manchester
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 Combining the strengths of UMIST and The Victoria University of Manchester
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Tableaux Reasoning (2)
E.g., KB: { HappyParent ´ Person u 8 hasChild.(Doctor t 9 hasChild.Doctor), John:HappyParent, John hasChild Mary, Mary: : Doctor Wendy hasChild Mary, Wendy marriedTo John} Person 8 hasChild.(Doctor t 9 hasChild.Doctor) Combining the strengths of UMIST and The Victoria University of Manchester
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Tableaux Reasoning (3)
Tableau rules correspond to constructors in logic ( u , 9 – E.g., John: ( Person u Doctor) - !
John:Person and etc) John: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: { Person v 9 hasParent.Person, John:Person} Combining the strengths of UMIST and The Victoria University of Manchester
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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 Combining the strengths of UMIST and The Victoria University of Manchester
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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 Combining the strengths of UMIST and The Victoria University of Manchester
SHOIQ
: Why is it Hard?
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SHIQ
is Already Tricky
Does not have finite model property , e.g.: { ITN v 6 1 edge
–
u 8 edge.ITN u 9 edge.ITN, R:(ITN u 6 0 edge
–
)} – Double blocking – Block interpreted as infinite repetition Combining the strengths of UMIST and The Victoria University of Manchester
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SHIQ
is Already Tricky
Does not have finite model property , e.g.: { ITN v 6 1 edge
–
u 8 edge.ITN u 9 edge.ITN, R:ITN u 6 0 edge
–
u 9 edge.ITN
} – Double blocking – Block interpreted as infinite repetition Yo-yo problem due to > and 6 , e.g.: { John: 9 hasChild.Doctor u u 6 2 hasChild } > 2 hasChild.Lawyer
– Add inequalities between nodes generated by > rule – Clash if 6 rule only applicable to nodes Combining the strengths of UMIST and The Victoria University of Manchester
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SHOIQ
: ExpTime
!
NExpTime
Interactions between O , I , and Q termination problems lead to – Anonymous branches can loop back to named individuals ( O ) • E.g., 9 r.{Mary} – Number restrictions edges ( Q ) on incoming ( I ) lead to non-tree structure • E.g., Mary: 6 1 r
–
– 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 Combining the strengths of UMIST and The Victoria University of Manchester
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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 Combining the strengths of UMIST and The Victoria University of Manchester
SHOIQ
: Yo-Yo Problem is Back!
• • E.g., KB: { VMP ´ Person u 9 loves.{Mary} u 9 hasFriend.VMP, John: 9 hasFriend.VMP
Mary: 6 2 loves
–
} Blocking prevented by N-nodes Repeated creation and merging of nodes leads to non-termination Combining the strengths of UMIST and The Victoria University of Manchester
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Intuition: Guess Exact Cardinality
New Ro?-rule guesses exact cardinality constraint on N-nodes { VMP ´ Person u 9 loves.{Mary} u 9 hasFriend.VMP, John: 9 hasFriend.VMP
Mary: 6 2 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 Combining the strengths of UMIST and The Victoria University of Manchester
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 Combining the strengths of UMIST and The Victoria University of Manchester
Combining the strengths of UMIST and The Victoria University of Manchester