Reasoning with context in the Semantic Web … or contextualizing ontologies

Reasoning with context in the Semantic Web… or contextualizing ontologies Fausto Giunchiglia July 23, 2004

The Key Idea: Scaling up to the (Semantic) Web • Diversity is a feature and not a defect! Only way to scale up • Keep diversity (local autonomy) in ontology (schema, ….) definition (at design and run time) • Coordinate at run time (do not integrate! … at design time) to find common understanding • Underlying theory: reasoning in context and Local Models Semantics

The Talk • Ontologies vs. Contexts • A (restated) global semantics for OWL – Intuitions • Three motivating examples • A (new) local models semantics for OWL – Intuitions • C-OWL: extending OWL with (context) mappings

Ontologies vs. Contexts • An Ontology is a model of some domain which is supposed to encode a view common to a set of different parties An ontology is built to beshared; • A Context is a model of some domain which is supposed to encode a view of a party A context is built to bekept local(where local implies not shared) • A context and an ontology of the same domain are likely to be very different (different goals, different approach, …)

Pro’s and Contra’s • Ontologies • Strengths • “easy” exchange of information • Weaknesses • consensus must be reached about their contents • maintenance may become arbitrarily hard • Contexts • Strengths • “easy” to define and to maintain • can be constructed with no consensus with the other parties • Weaknesses • Exchange of information by constructing explicit mappings among the elements of the contexts of the involved parties

Contextual Ontologies Contextual ontology = Ontology + Context mappings Key idea (in two steps): • Share as much as possible (extended OWL import construct) • Keep it local whenever sharing does not work (C-OWL context mappings) Notes: • In many (most in the Web?) cases sharing does not work and produces undesired results (semantic heterogeneity) • Using context allows for incremental, piece-wise construction of the Semantic Web (bottom up vs. top down approach).

The Talk • Contexts vs. Ontologies • A (restated) global semantics for OWL – Intuitions • Three motivating examples • A (new) local models semantics for OWL – Intuitions • C-OWL: extending OWL with (context) mappings

A Global Semantics for OWL IndexOWL Ontologies: <i, Oi> and their languages (e.g., i:C, j:E, i:r.C) (Local language). A local concept (role, individual), Ci (Ri,Oi) is an element of C that appears in Oi either without indexes or with index equal to i. (Foreign language): … Anything (concept, role, individual) which is not local (OWL space). An OWL space is a family of ontologies {<i, Oi>} such that the language of every Oi contains all the other foreign languages

A Global Semantics for OWL (cont’ed) (OWL interpretation). An OWL interpretation for the OWL space {<i, Oi>}is a pair I = <∆I, (.)I>, such that • I(i, C)∆I for any i I and CCi; • I(i, r)∆I x ∆I for any i  I and rRi; • I(i, o) ∆I for any i  I and oOi; With ∆I domain of interpretation and (.)I interpretation function Note: a global interpretation! (new local interpretation being defined within KWeb)

A Global Semantics for OWL (cont’ed) (OWL axiom and fact satisfiability). I satisfies a fact or an axiom ø of Oi according to the rules defined in [*] P.F. Patel-Schneider, P. Hayes, and I. Horrocks. Web Ontology Language (OWL) Abstract Syntax and Semantics. Technical report, W3C, February 2003. An OWL interpretation I satisfies an OWL space {<i, Oi>}, if I satisfies each axiom and fact of Oi, for any i

Example 1: directionality Need to keep track of source and target ontology Example: • Construct O2 by importing O1 and adding it some new axiom • Want that axioms added to O2 do not affect O1 • O1 contains axioms A Band C D • O2 contains also axiom 1:B 1:C In new semantics, we want 1:A 1:Din O2, but not in O1.

Example 1 (cont’ed): directionality We want to avoid propagation of inconsistency Example: • O1 contains axioms A Band C D • O2 contains also axiom 1:B 1:C • We want to derive 1:A 1:Din O2 but not in O1 • … • O2 contains also 1:A(a) and 1: not D(a) • O2 is inconsistent In new semantics, we want to keep O1 consistent

Example 2: local domains Need to give up hypothesis that of single global domain of interpretation Example: Car manufacturing ontology OWCM with domain of interpretation the totality of cars • individual constants Diesel and Petrol for Diesel engine and petrol engine • Axiom: a car has only one engine which is either Diesel or petrol Car (1) hasEngine.{Diesel, Petrol} Diesel  Petrol Ferrari ontology, OFerrari describing Ferrari’s production • Imports OWCM standard • Axiom: engine of a Ferrari is either an F23 or and F34i Ferrari (WCM:car (1) (WCM:hasEngine).{F23, F34i} F23F34i In new semantics, we want to avoid (F23)IFerrari = (Diesel)IWCMsince Ferrari produces only petrol engines

Example 3: context mappings Need to state that two elements of two ontologies, though being extensionally different, are contextually related Example: • OFIATdescribes cars from manufacturer point of view • OSale describes cars from car vendor point of view • OFIAT and OSale are largely independent and different • Two concepts of car defined in OFIAT and OSale, (i.e. Sale:Car and FIAT:Car) may be very different, still describing same real world object (different viewpoints) Not possible to state relation between two concepts with OWL syntax

Exampe 1: Directionality • Consider all (local) ontologies as part of a OWL space • Split global interpretation into a family of local interpretations, one for each ontology • Allow for an ontology to be locally inconsistent (i.e., not to have a local interpretation) • Technically: Associate inconsistent ontologies to a special “interpretation”, called a hole, that verifies any set of axioms

Example 2: Local Domains • Associate to each ontology a local domain • Local domains may overlap (two ontologies may refer to the same object) • Technically:An OWL interpretation with local domains for the OWL space {<i, Oi>} is a family I = {Ii}, where each Ii = <∆Ii, (.)Ii>, called the local interpretation of Oi, is either an interpretation of Li on ∆Ii, or a hole

Example 3: adding context mappings to syntax (Bridge rules). A bridge rule from i to j is a statement of one of the four following forms, where x and y are concepts, or individuals, or roles of the languages Li and Lj (Context mapping). Given a OWL space {<i, Oi>} a mapping Mij from Oi to Oj is a set of bridge rules from Oi to Oj.

Context mappings (cont’ed) (Contextual ontology): It is a local ontology plus a set of bridge rules (context mappings). We sometimes write context meaning contextual ontology. (Context space). A context space is the pair • OWL space {<i, Oi>} (of local ontologies) • family {Mij} of (context) mappings from i to j, for any pair i,j (Interpretation for context spaces). It is the pair • I, where I is an OWL interpretation with holes and local domains and • rij, the domain relation from i to j, is a subset of ∆Ii x ∆Ii

Examples: Context mappings From example 3: Sale:Car and FIAT:car describe the same set of objects from two different viewpoints: (**) • Domain relation satisfying (**): rij(CarISale)= CarIFIAT From example 2: (*) • Domain relation satisfying (*): rWCM, Ferrari(Petrol)IWCM {F23IFerrari , F34iIFerrari}

Context OWL (C-OWL) A contextual ontology is a pair: • OWL ontology • a set of context mappings where a mapping is a set of bridge rules with the same target ontology A context mapping is a 4-tuple: • A mapping identifier (URI) • A source context containing an OWL ontology • A target context containing an OWL ontology • A set of bridge rules from the local language of the source ontology to the local language of the target ontology NOTE: mappings are objects (!!)

The Key Idea (continued): using C-OWL to scale up to the (semantic) Web • How often in the Web we will import ontologies and how often we will define context mappings (diversity as a defect, or diversity as a feature)? • Shouldn’t the Semantic Web be a Web of Semantic links (e.g., context mappings)? Context mappings useful for: maintaining alignment, propagating info, (semantics driven) navigation, … • Shouldn’t discovering context mappings (e.g., Semantic matching) be one of the core issues in building the Semantic Web?

Conclusions • Ontologies: share knowledge • Contexts: keep knowledge local (not shared) • Contextual ontologies: share as much as possible, keep local whenever necessary • C-OWL (Context OWL): • OWL + • Local models semantics + • context mappings (limited, explicitly defined, visibility from outside)

References • Project website - ACCORD: http://www.dit.unitn.it/~accord/ • Bouquet, F. Giunchiglia, F. van Harmelen, L. Serafini, H. Stuckenschmidt: C-OWL: Contextualizing Ontologies // In Proceedings of ISWC'03. • F. Giunchiglia, P.Shvaiko, M. Yatskevich: S-Match: an algorithm and an implementation of semantic matching. In Proceedings of ESWS’04. • F. Giunchiglia, P.Shvaiko: Semantic matching. In The Knowledge Engineering Review journal, 18(3):265-280, 2004. Short versions in Proceedings of SI workshopat ISWC’03 and ODS workshop at IJCAI’03. • F. Giunchiglia, I. Zaihrayeu: Making peer databases interact – a vision for an architecture supporting data coordination. In Proceedings of CIA’02. • C. Ghidini, F. Giunchiglia: Local models semantics, or contextual reasoning = locality + compatibility. Artificial Intelligence journal, 127(3):221-259, 2001.

Context mappings (cont’ed) (Satisfiability of bridge rules) A interpretation for a context space is a model for it if all the bridge rules are satisfied

Reasoning with context in the Semantic Web … or contextualizing ontologies