Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Armstrong ABoxes for ALC TBoxes Henriette Harmse ICTAC 2018 Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Outline 1 Overview 2 Theoretical Basis 3 Description Logics 4 FCA, Partial Contexts, Ontology Completion 5 Armstrong ABox Formal Definition 6 An Example 7 Questions? Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Overview Armstrong ABoxes are inspired by Armstrong relations of relational database theory. Armstrong ABoxes are the Description Logic counterpart of Armstrong relations. A DL ontology or knowledge base consists of a TBox and an ABox. An Armstrong ABox is an ABox that for a specific class of constraints, satisfies all constraints that hold, and violates all constraints that do not necessarily hold. Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Overview (cont.) Armstrong ABoxes are formalized relative to particular classes of constraints, with each class of constraints resulting in a different Armstrong ABox formalization. For arbitrary classes of constraints Armstrong ABoxes are undecidable. Armstrong ABoxes … Armstrong ABoxes Armstrong ABoxes ? for n-ary relations for ALC TBoxes Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Theoretical Basis Closed Sets in Formal Concept Lectic Order Analysis Attribute Exploration Partial Contexts Ontology Description Logics Completion Armstrong ABoxes for ALC TBoxes Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Description Logics - Axioms, ALC Concept Constructors The syntactic building blocks for a DL are based on the disjoint sets N C (concept names), N R (role names) and N I (individual names). TBox axioms: C ⊑ D , C ≡ D . ABox assertions: C ( x ), r ( x , y ). ALC concept descriptions (referred to as concepts) are constructed using the following concept constructors C := ⊤ | A | ¬ C | C ⊓ D | C ⊔ D | ∃ r . C where A is an atomic concept, C and D are (possibly complex) concepts and r is a role. Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Description Logics - Semantics of ALC For ALC for an given interpretation I = ( △ I , · I ), the interpretation function · I is extended to interpret complex concepts in the following way: Name Constructor Semantics ⊤ I △ I Top ⊥ I Bottom ∅ ( ¬ C ) I △ I \ C I Negation ( C 1 ⊓ C 2 ) I C I 1 ∩ C I Conjunction 2 ( C 1 ⊔ C 2 ) I C I 1 ∪ C I Disjunction 2 ( ∃ r . C ) I { x ∈ △ I | A y exists s.t. Existential restriction ( x , y ) ∈ r I and y ∈ C I } Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Description Logics - Satisfaction I � α indicates that an interpretation I satisfies an axiom α . Satisfaction of α is defined as I � C ⊑ D iff C I ⊆ D I , I � C ( x ) iff x I ∈ C I , and I � r ( x , y ) iff ( x I , y I ) ∈ r I . Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Description Logics - Models, Entailment I is a model of a TBox T (ABox A ) if it satisfies all its axioms (assertions). If I is a model of T and A , it is called a model of the ontology ( T , A ) and ( T , A ) is said to be consistent if such a model exists. An axiom α is said to be entailed by an ontology O , written as O � α , if every model of O is also a model of α . For a set of axioms Σ = { σ 0 , . . . , σ n } , we abbreviate O � σ 0 , . . . , O � σ n with O � Σ. If O is empty, we abbreviate O � α as � α . Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Why is All This Necessary? Closed Sets in Formal Concept Lectic Order Analysis Attribute Exploration Partial Contexts Ontology Description Logics Completion Armstrong ABoxes for ALC TBoxes Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Formal Concept Analysis - Basics K m 1 m 2 m 3 m 4 × × × o 1 × × o 2 o 3 × × × Formal context: K := ( G , M , I ) where G (objects), M (attributes), and I ⊆ G × M (relation). Implications between attributes can be used to analyse K , i.e. { m 2 } → { m 1 , m 3 } Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Formal Concept Analysis - Implications L → R holds in K if every object that has all the attributes in L also has all the attributes in R . X ⊆ M respects an implication L → R if L � X or R ⊆ X . X ⊆ M respects a set L of implications if X respects every implication in L . L → R follows from L if every X ⊆ M that respects all implications in L , also respects L → R . Mod( L ) := { X ⊆ M | X respects L} is a closure system on M , for which a closure operator L : 2 M → 2 M can be defined. Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Formal Concept Analysis - Attribute Exploration Attribute Exploration used to complete a subcontext K ′ . Iterates through implications from {∅} → M to M → {∅} . L is an implication base of K if every implication from L holds in K, every implication that holds in K follows from L , and no implication in L follows from other implications in L . Minimal implication base, in particular a Duquenne Guigues base. Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Formal Concept Analysis - Traversing Implications Steps to traverse Duquenne Guigues base. Start with L = {∅} , find largest R such that L → R does not 1 have a counterexample in the K ′ . Find next L to consider using NextClosure and the 2 implication closure operator. Lectic order: Fixes some linear order on M and defines lectic order such that for A , B ⊆ M we can answer whether A < B ? NextClosure : Given A ⊆ M and some closure operator ϕ it determines the next closed set B in the lectic order such that A < B . Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? From FCA to Ontology completion Table: Partial Context K m 1 m 2 m 3 m 4 − o 1 + + ? o 2 ? ? + − + ? + ? o 3 Table: A Partial Context for an ontology ( T , A ) K T , A ( M ) C 1 C 2 C 3 C 4 + + ? − x 1 x 2 ? ? + − + ? + ? x 3 Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? What are we doing again? Closed Sets in Formal Concept Lectic Order Analysis Attribute Exploration Partial Contexts Ontology Description Logics Completion Armstrong ABoxes for ALC TBoxes Henriette Harmse Armstrong ABoxes for ALC TBoxes
Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions? Ontology Completion vs Armstrong ABoxes Ontology Completion assumes we start with an ontology ( T 0 , A 0 ) and we want to determine an ontology ( T , A ) that is representative of the application domain. Armstrong ABoxes assumes we start with an ontology ( T , ∅ ) that is representative of the application. We want to determine an ontology ( T , A ) where A represents perfect synthetic test/example data. Henriette Harmse Armstrong ABoxes for ALC TBoxes
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