Armstrong ABoxes for ALC TBoxes Henriette Harmse ICTAC 2018 - - PowerPoint PPT Presentation

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

• f 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 for n-ary relations 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?

Theoretical Basis

Armstrong ABoxes for ALC TBoxes Ontology Completion Partial Contexts Attribute Exploration Formal Concept Analysis Closed Sets in Lectic Order Description Logics

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 NC (concept names), NR (role names) and NI (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 Top ⊤I △I Bottom ⊥I ∅ Negation (¬C)I △I\C I Conjunction (C1 ⊓ C2)I C I

1 ∩ C I 2

Disjunction (C1 ⊔ C2)I C I

1 ∪ C I 2

Existential restriction (∃r.C)I {x ∈ △I|A y exists s.t. (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 ⊆ DI, I C(x) iff xI ∈ C I, and I r(x, y) iff (xI, yI) ∈ rI.

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

• ntology (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 = {σ0, . . . , σn}, we abbreviate O σ0, . . . , O σn 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?

Armstrong ABoxes for ALC TBoxes Ontology Completion Partial Contexts Attribute Exploration Formal Concept Analysis Closed Sets in Lectic Order Description Logics

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 m1 m2 m3 m4

• 1

× × ×

• 2

× ×

• 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. {m2} → {m1, m3}

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 : 2M → 2M 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.

1

Start with L = {∅}, find largest R such that L → R does not have a counterexample in the K′.

2

Find next L to consider using NextClosure and the implication closure operator.

Lectic order: Fixes some linear order on M and defines lectic

• rder 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 m1 m2 m3 m4

• 1

+ + ? −

• 2

? ? + −

• 3

+ ? + ? Table: A Partial Context for an ontology (T , A) KT ,A(M) C1 C2 C3 C4 x1 + + ? − x2 ? ? + − x3 + ? + ?

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?

Armstrong ABoxes for ALC TBoxes Ontology Completion Partial Contexts Attribute Exploration Formal Concept Analysis Closed Sets in Lectic Order Description Logics

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 (T0, A0) 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

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Armstrong ABoxes - Preliminaries

For convenience the notation Ci and Dj will respectively be used as shorthand for Ci0 ⊓ . . . ⊓ Cin and Dj0 ⊓ . . . ⊓ Djm. Armstrong ABoxes are restricted an interesting set M of concepts. M is said to be permissible if it is finite and no concept in M is equivalent to ⊤. We define M→ to be the set of GCIs representing the finite set of all the implications L → R over M.

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Armstrong ABoxes - Violating Exemplar

Let T be a consistent ALC TBox and let σ′ :=

• Ci ⊑
• Dj

for which T σ′ holds. An ABox A′ is a violating exemplar of the entailment T σ′ if

• (
• Ci)(x), (¬
• Dj)(x)
• ⊆ A′

holds for some named individual x that does not appear in any

• ther assertions of A′. This is denoted by A′ σ′.

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Armstrong ABoxes - Satisfying Exemplar

Let T be a consistent ALC TBox, and let σ :=

• Ci ⊑
• Dj

for which T σ and σ holds. An ABox A is a satisfying exemplar of the entailment T σ if

• (
• Ci)(x), (
• Dj)(x)
• ⊆ A

holds for some named individual x that does not appear in any

• ther assertions of A. This is denoted by A σ.

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Armstrong ABoxes - Candidate Axiom Set

Let T be a consistent ALC TBox and let M be permissible. Let Σ′ := {σ′ | T σ′ and σ′ ∈ M→}. Σ′ is called the candidate axiom set of T over M. Assume Σ′ = {σ′

0, . . . , σ′ n}. An ABox A′ is a violating exemplar of

T Σ′ if A′ σ′

0, . . . , A′ σ′ n holds. An ABox A′ is a minimal

violating exemplar of T Σ′ iff there is no ABox A′

0 ⊂ A′ that is

violating exemplar of T Σ′.

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Armstrong ABoxes - Entailment Set

Let T be a consistent ALC TBox and let M be permissible. Let Σ := {σ | T σ, σ and σ ∈ M→}. Σ is called the entailment set of T over M. Assume Σ = {σ0, . . . , σn}. An ABox A is a satisfying exemplar of T Σ if A σ0, . . . , A σn holds, which is denoted by A Σ. An ABox A is a minimal satisfying exemplar of T Σ iff there is no ABox A0 ⊂ A that is satisfying exemplar of T Σ.

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Armstrong ABoxes - Armstrong ABox Definition

Let T be a consistent ALC TBox with Σ and Σ′ respectively the entailment- and candidate axiom sets of T . A is said to be an Armstrong ABox for T if and only if:

1 for every σ ∈ Σ, A σ holds, 2 for every σ′ ∈ Σ′, A σ′ holds and 3 there is no proper subset of A such that properties (1) and

(2) hold. O = T ∪ A is called an Armstrong ontology.

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

An Example

T0 = {Course ⊑ ¬Person, Teacher ≡ Person ⊓ ∃teaches.Course, ∃teaches.⊤ ⊑ Person, Student ≡ Person ⊓ ∃attends.Course, ∃attends.⊤ ⊑ Person}

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

An Example (cont.)

Table: Armstrong ABox for T0 and M = {Person, Student ⊔ Teacher} Entailment Satisfying exemplar Student ⊔ Teacher ⊑ Person {(Student ⊔ Teacher)(x2), Person(x2)} Non-Entailment Violating exemplar Person ⊔ (Student ⊔ Teacher) ⊑ Person ⊓ (Student ⊔ Teacher) {(Person ⊔ (Student ⊔ Teacher))(x1), ¬(Person ⊓ (Student ⊔ Teacher))(x1)}

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

An Example (cont.)

T1 = T0 ∪ {Person ≡ Student ⊔ Teacher}

Table: Armstrong ABox for T1 and M = {Person, Student ⊔ Teacher} Entailment Satisfying exemplar Person ⊔ (Student ⊔ Teacher) ⊑ Person ⊓ (Student ⊔ Teacher) {(Person ⊔ (Student ⊔ Teacher))(x1), (Person ⊓ (Student ⊔ Teacher))(x1)} Student ⊔ Teacher ⊑ Person {(Student ⊔ Teacher)(x2), Person(x2)} Person ⊑ Student ⊔ Teacher {Person(x3), (Student ⊔ Teacher)(x3)}

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Potential Benefits

Can help stakeholders to understand the meaning of entailments and non-entailments, which may be helpful to identify missing axioms. Where stakeholders are reasonably sure T is representative of their application domain, Armstrong ABoxes can give quick feedback when compared to ontology completion. Does not replace ontology completion. Rather, Armstrong ABoxes complements ontology completion.

Henriette Harmse Armstrong ABoxes for ALC TBoxes

Overview Theoretical Basis Description Logics FCA, Partial Contexts, Ontology Completion Armstrong ABox Formal Definition An Example Questions?

Questions?

Henriette Harmse Armstrong ABoxes for ALC TBoxes