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 be shared; A Context is a model of some domain which is supposed to encode a view of a party A context is built to be kept 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): 1. Share as much as possible (extended OWL import construct) 2. Keep it local whenever sharing does not work (C-OWL context mappings) Notes: 1. In many (most in the Web?) cases sharing does not work and produces undesired results (semantic heterogeneity) 2. 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 Index OWL Ontologies: and their languages (e.g., i:C, j:E, i:  r.C) (Local language). A local concept (role, individual), C i ( R i, O i ) is an element of C that appears in O i 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 { } such that the language of every O i contains all the other foreign languages

A Global Semantics for OWL (cont’ed) (OWL interpretation). An OWL interpretation for the OWL space { } is a pair I =, such that  I(i, C)  ∆ I for any i  I and C  C i ;  I(i, r)  ∆ I x ∆ I for any i  I and r  R i ;  I(i, o)  ∆ I for any i  I and o  O i ; 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 O i 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 An OWL interpretation I satisfies an OWL space { }, if I satisfies each axiom and fact of O i, for any i

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

Example 1: directionality Need to keep track of source and target ontology Example: Construct O 2 by importing O 1 and adding it some new axiom Want that axioms added to O 2 do not affect O 1 O 1 contains axioms A B and C D O 2 contains also axiom 1: B 1:C In new semantics, we want 1: A 1:D in O 2, but not in O 1.

Example 1 (cont’ed): directionality We want to avoid propagation of inconsistency Example: O 1 contains axioms A B and C D O 2 contains also axiom 1 :B 1:C We want to derive 1: A 1:D in O 2 but not in O 1 … O2 contains also 1: A(a) and 1: not D(a) O 2 is inconsistent In new semantics, we want to keep O 1 consistent

Example 2: local domains Need to give up hypothesis that of single global domain of interpretation Example: Car manufacturing ontology O WCM 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, O Ferrari describing Ferrari’s production Imports O WCM 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) IWCM since 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: O FIAT describes cars from manufacturer point of view O Sale describes cars from car vendor point of view O FIAT and O Sale are largely independent and different Two concepts of car defined in O FIAT and O Sale, (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

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

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 { } is a family I = {I i }, where each I i =, called the local interpretation of O i, is either an interpretation of L i on ∆ Ii, or a hole

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

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 L i and L j (Context mapping). Given a OWL space { } a mapping M ij from O i to O j is a set of bridge rules from O i to O j.

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 1. OWL space { } (of local ontologies) 2. family {M ij } of (context) mappings from i to j, for any pair i,j (Interpretation for context spaces). It is the pair 1. I, where I is an OWL interpretation with holes and local domains and 2. r ij, 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 (**): r ij (Car I Sale )= Car I FIAT From example 2: (*) Domain relation satisfying (*): r WCM, Ferrari (Petrol) I WCM  {F23 I Ferrari, F34i I Ferrari }

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: Bouquet, F. Giunchiglia, F. van Harmelen, L. Serafini, H. Stuckenschmidt: C-OWL: Contextualizing Ontologies // In Proceedings of ISWC'03. C-OWL: Contextualizing OntologiesISWC'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): , Short versions in Proceedings of SI workshop at 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): , 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