Extending Consequence-Based Reasoning to Andrew Bate, Boris Motik, - - PowerPoint PPT Presentation

Extending Consequence-Based Reasoning to 𝒯ℛℐ𝒭

Andrew Bate, Boris Motik, Bernardo Cuenca Grau, František Simančík, and Ian Horrocks

Department of Computer Science University of Oxford

Motivation

• Most reasoners based on (hyper)tableau

– FaCT++ – HermiT – Pellet – Konclude – Racer

• Work reasonably well in practice
• But building many counter models is expensive

– To prove 𝒫 ⊨ 𝐷 ⊑ 𝐸 show 𝐷 ⊓ ¬𝐸 is unsat – Bottleneck: large number of concepts – Rebuilds entire model for each test

Consequence-based Features

Optimal worse-case complexity One pass classification

No need for several counter models

Pay as you go Deterministic

State of the art

ℰℒ

Horn-𝒯ℋℐ𝒭

Horn-𝒯ℛ𝒫ℐ𝒭

𝒝ℒ𝒟ℋ

𝒝ℒ𝒟ℐ

ELK (Java)

CEL (Common LISP)

Snorocket (Java)

Elephant (C)

jcel (Java) CB (OCaml)

Condor (C++)

𝒯ℛℐ𝒭

Key Facts

① ② Algorithm does not build models

→ Apply inference rules to derive local consequences of ontology

Derived consequences not all stored together

→ Contexts store consequences corresponding to a conjunction of concepts and roles

DL Clauses 𝒯ℋℐ𝒭 Calculus FOL Clauses Taxonomy

Reasoning Stages

Example

Vegetarian ⊑ Person ?

Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person

DL Clauses 𝒯ℋℐ𝒭 Calculus FOL Clauses Taxonomy

Structural transformation

Translate into first-order clauses with equality

ሥ

𝑗=1 𝑜

𝑈𝑗 → ሧ

𝑘=1 𝑛

𝑀𝑘

Function-free atoms of the forms 𝐵 𝑦 , 𝑆 𝑦, 𝑨𝑗 or 𝑆 𝑨𝑗, 𝑦 Atoms or equations of the forms 𝑨𝑗 ≈ 𝑨𝑘 or 𝑨𝑗 ≉ 𝑨𝑘

Vegetarian 𝑦 → Animal 𝑦 Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person

Vegetarian 𝑦 → Animal 𝑦 Animal 𝑦 → eats 𝑦, 𝑔

1 𝑦

⋮ ⋮ ⋮ Animal 𝑦 → eats 𝑦, 𝑔

5 𝑦

Animal 𝑦 → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person

Vegetarian 𝑦 → Animal 𝑦 Animal 𝑦 → eats 𝑦, 𝑔

1 𝑦

⋮ ⋮ ⋮ Animal 𝑦 → eats 𝑦, 𝑔

5 𝑦

Animal 𝑦 → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person Meat 𝑦 ∧ SideDish 𝑦 → ⊥

Vegetarian 𝑦 → Animal 𝑦 Animal 𝑦 → eats 𝑦, 𝑔

1 𝑦

⋮ ⋮ ⋮ Animal 𝑦 → eats 𝑦, 𝑔

5 𝑦

Animal 𝑦 → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 Vegetarian 𝑦 ∧ eats 𝑦, 𝑨1 → SideDish 𝑨1 Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person Meat 𝑦 ∧ SideDish 𝑦 → ⊥

Vegetarian 𝑦 → Animal 𝑦 Animal 𝑦 → eats 𝑦, 𝑔

1 𝑦

⋮ ⋮ ⋮ Animal 𝑦 → eats 𝑦, 𝑔

5 𝑦

Animal 𝑦 → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 Vegetarian 𝑦 ∧ eats 𝑦, 𝑨1 → SideDish 𝑨1 ሥ

𝑗=1 5

eats 𝑦, 𝑨𝑗 → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑨𝑗 ≈ 𝑨𝑘 ∨ ሧ

𝑗=1 5

Meat 𝑨𝑗 Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person Meat 𝑦 ∧ SideDish 𝑦 → ⊥

Vegetarian 𝑦 → Animal 𝑦 Animal 𝑦 → eats 𝑦, 𝑔

1 𝑦

⋮ ⋮ ⋮ Animal 𝑦 → eats 𝑦, 𝑔

5 𝑦

Animal 𝑦 → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 Vegetarian 𝑦 ∧ eats 𝑦, 𝑨1 → SideDish 𝑨1 ሥ

𝑗=1 5

eats 𝑦, 𝑨𝑗 → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑨𝑗 ≈ 𝑨𝑘 ∨ ሧ

𝑗=1 5

Meat 𝑨𝑗 Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person Meat 𝑦 ∧ SideDish 𝑦 → ⊥ HealthyPerson 𝑦 → Person 𝑦

Vegetarian 𝑦 → Animal 𝑦 Animal 𝑦 → eats 𝑦, 𝑔

1 𝑦

⋮ ⋮ ⋮ Animal 𝑦 → eats 𝑦, 𝑔

5 𝑦

Animal 𝑦 → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 Vegetarian 𝑦 ∧ eats 𝑦, 𝑨1 → SideDish 𝑨1 ሥ

𝑗=1 5

eats 𝑦, 𝑨𝑗 → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑨𝑗 ≈ 𝑨𝑘 ∨ ሧ

𝑗=1 5

Meat 𝑨𝑗 Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person Meat 𝑦 ∧ SideDish 𝑦 → ⊥ HealthyPerson 𝑦 → Person 𝑦

DL Clauses 𝒯ℋℐ𝒭 Calculus FOL Clauses Taxonomy

Contexts

Set 𝒲 of contexts Each context 𝑤 ∈ 𝒲: Edges between contexts labelled with functions Context structure 𝒠 is a the graph of labelled contexts and edges

𝐵 𝑦 ∧ 𝐶(𝑦) ⊤core𝑤 → ⋯ core𝑤 ∧ 𝐷 𝑦 → ⋯ ⋮ core𝑤 ∧ 𝑆 𝑧, 𝑦 → ⋯ 𝒯𝑤 core𝑤

Contexts

Set 𝒲 of contexts Each context 𝑤 ∈ 𝒲: Edges between contexts labelled with functions Context structure 𝒠 is a the graph of labelled contexts and edges

𝐵 𝑦 ∧ 𝐶(𝑦) |core𝑤⊤ → ⋯ core𝑤 ∧ 𝐷 𝑦 → ⋯ ⋮ core𝑤 ∧ 𝑆 𝑧, 𝑦 → ⋯ 𝒯𝑤 core𝑤

Sound Context Structures

core𝑤 = 𝐵 𝑦 ∧ 𝐶(𝑦) core𝑣 = 𝐷 𝑦 ∧ 𝐸(𝑦)

𝑔

𝑙

𝒫 ⊨ core𝑤 ∧ Γ → Δ for each 𝑤 ∈ 𝑊 and each Γ → Δ ∈ 𝒯𝑤 𝒫 ⊨ core𝑣 → core𝑤 𝑦 ↦ 𝑔

𝑙 𝑦 , 𝑧 ↦ 𝑦 for each 𝑣, 𝑤, 𝑔 𝑙 ∈ ℰ

① ②

⊤

→

⋯

⊤ ∧ 𝐷 𝑦

→ ⋯ ⋮

⊤ ∧ 𝑆 𝑧, 𝑦

→ ⋯

⊤

→

⋯

⊤ ∧ 𝑇(𝑦, 𝑧)

→ ⋯ ⋮

⊤

→ ⋯

Vegetarian 𝑦

⊤ → Vegetarian 𝑦

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

Vegetarian 𝑦 → Animal 𝑦 Animal 𝑦 → eats 𝑦, 𝑔

1 𝑦

⋮ ⋮ ⋮ Animal 𝑦 → eats 𝑦, 𝑔

5 𝑦

Animal 𝑦 → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5

ሼ ሽ ⊆ 𝒫

Vegetarian 𝑦 ∧ eats 𝑦, 𝑨1 → SideDish 𝑨1

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ 𝑗=1 5

Meat 𝑔

𝑗 𝑦

ሥ

𝑗=1 5

eats 𝑦, 𝑨𝑗 → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑨𝑗 ≈ 𝑨

𝑘 ∨ ሧ 𝑗=1 5

Meat 𝑨𝑗

ሼ ሽ ⊆ 𝒫

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ 𝑗=1 5

Meat 𝑔

𝑗 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ 𝑗=1 5

Meat 𝑔

𝑗 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦

ሥ

𝑗=1 5

eats 𝑦, 𝑨𝑗 → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑨𝑗 ≈ 𝑨

𝑘 ∨ ሧ 𝑗=1 5

Meat 𝑨𝑗

ሼ ሽ ⊆ 𝒫

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ 𝑗=1 5

Meat 𝑔

𝑗 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ 𝑗=1 5

Meat 𝑔

𝑗 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

Meat 𝑦 ∧ SideDish 𝑦 → ⊥

ሼ ሽ ⊆ 𝒫

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ 𝑗=1 5

Meat 𝑔

𝑗 𝑦

… ⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗<𝑘≤5

𝑔

𝑗 𝑦 ≈ 𝑔 𝑘 𝑦

… ⊤ → HealthyPerson 𝑦 ∨ ሧ

1≤𝑗≤5

𝑔

𝑗 𝑦 ≉ 𝑔 𝑗 𝑦

… ⊤ → HealthyPerson 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

eats 𝑧, 𝑦 ∧ SideDish 𝑦

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ⊤ → Person 𝑦

eats 𝑧, 𝑦 ∧ SideDish 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

HealthyPerson 𝑦 → Person 𝑦

ሼ ሽ ⊆ 𝒫

Vegetarian 𝑦

⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔

1 𝑦

⋮ ⊤ → eats 𝑦, 𝑔

5 𝑦

⊤ → 𝑔

𝑗 𝑦 ≉ 𝑔 𝑘 𝑦

for 1 ≤ 𝑗 < 𝑘 ≤ 5 ⊤ → SideDish 𝑔

1 𝑦

⋮ ⊤ → SideDish 𝑔

5 𝑦

⊤ → HealthyPerson 𝑦 ⊤ → Person 𝑦

eats 𝑧, 𝑦 ∧ SideDish 𝑦

𝑔

1

𝑔

2, 𝑔 3, 𝑔 4, 𝑔 5

Vegetarian ⊑ Person

✔

⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑦 Meat 𝑦 → Meat 𝑦 Meat 𝑦 → ⊥

But that’s not all…

Strategies Context overloading Triggers to restrict rule applications

– PAYG behaviour on fragments of 𝒯ℛℐ𝒭

Ordering on atoms and Skolem functions

Evaluation

• Prototype implementation called Sequoia
• Evaluated using the Oxford Ontology Repository

– Nominal → fresh class – Datatype → fresh class – Data property → fresh object property – Removed ABox assertions

• 777 ontologies
• Timeout 5 minutes
• Average over 3 runs, reporting exception or timeout as failure

Classification Times

Sequoia FaCT++ HermiT Konclude Pellet

10 100 1000 10000 100000

1 101 201 301 401 501 601 701

Classification Time (milliseconds) 102 103 104 105 100 200 300 400 500 600 700

H P F K S Profile H P F K S Horn H P F K S No Equality H P F K S With Equality

0% 25% 50% 75% 100%

H P F K S Profile H P F K S Horn H P F K S No Equality H P F K S With Equality

H P F K S Profile EL / QL / RL H P F K S Horn No Equality H P F K S Non-Horn No Equality H P F K S Non-Horn Equality Hard (timeout or exception) Medium (10 seconds to 5 mins) Easy (< 10 seconds)

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Percentage of Easy, Medium & Hard Ontologies

Main result

Consequence-based classification for 𝒯ℛℐ𝒭 Optimal worst-case complexity Pay as you go One pass classification Competitive preliminary evaluation