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 ELK (Java) Snorocket (Java) CEL (Common LISP) ℰℒ jcel (Java) Elephant (C) CB (OCaml) Horn- 𝒯ℋℐ𝒭 𝒝ℒ𝒟ℐ Horn- 𝒯ℛ𝒫ℐ𝒭 𝒝ℒ𝒟ℋ 𝒯ℛℐ𝒭 Condor (C++)

Key Facts ① Algorithm does not build models → Apply inference rules to derive local consequences of ontology ② Derived consequences not all stored together → Contexts store consequences corresponding to a conjunction of concepts and roles

Reasoning Stages 𝒯ℋℐ𝒭 DL Clauses FOL Clauses Taxonomy Calculus

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

𝒯ℋℐ𝒭 DL Clauses FOL Clauses Taxonomy Calculus

Structural transformation Translate into first-order clauses with equality 𝑜 𝑛 ሥ 𝑈 𝑗 → ሧ 𝑀 𝑘 𝑗=1 𝑘=1 Atoms or equations of the forms 𝑨 𝑗 ≈ 𝑨 𝑘 or 𝑨 𝑗 ≉ 𝑨 𝑘 Function-free atoms of the forms 𝐵 𝑦 , 𝑆 𝑦, 𝑨 𝑗 or 𝑆 𝑨 𝑗 , 𝑦

Vegetarian ⊑ Animal Animal ⊑ ≥ 5 eats Meat ⊓ SideDish ⊑ ⊥ Vegetarian ⊑ ∀ eats . SideDish ≥ 5 eats . ¬Meat ⊑ HealthyPerson HealthyPerson ⊑ Person 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 Meat 𝑦 ∧ SideDish 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 Meat 𝑦 ∧ SideDish 𝑦 → ⊥ Vegetarian 𝑦 ∧ eats 𝑦, 𝑨 1 → SideDish 𝑨 1

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 Meat 𝑦 ∧ SideDish 𝑦 → ⊥ Vegetarian 𝑦 ∧ eats 𝑦, 𝑨 1 → SideDish 𝑨 1 5 5 ሥ eats 𝑦, 𝑨 𝑗 → HealthyPerson 𝑦 ∨ ሧ 𝑨 𝑗 ≈ 𝑨 𝑘 ∨ ሧ Meat 𝑨 𝑗 𝑗=1 𝑗=1 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 Meat 𝑦 ∧ SideDish 𝑦 → ⊥ Vegetarian 𝑦 ∧ eats 𝑦, 𝑨 1 → SideDish 𝑨 1 5 5 ሥ eats 𝑦, 𝑨 𝑗 → HealthyPerson 𝑦 ∨ ሧ 𝑨 𝑗 ≈ 𝑨 𝑘 ∨ ሧ Meat 𝑨 𝑗 𝑗=1 𝑗=1 1≤𝑗<𝑘≤5 HealthyPerson 𝑦 → Person 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 Meat 𝑦 ∧ SideDish 𝑦 → ⊥ Vegetarian 𝑦 ∧ eats 𝑦, 𝑨 1 → SideDish 𝑨 1 5 5 ሥ eats 𝑦, 𝑨 𝑗 → HealthyPerson 𝑦 ∨ ሧ 𝑨 𝑗 ≈ 𝑨 𝑘 ∨ ሧ Meat 𝑨 𝑗 𝑗=1 𝑗=1 1≤𝑗<𝑘≤5 HealthyPerson 𝑦 → Person 𝑦

𝒯ℋℐ𝒭 DL Clauses FOL Clauses Taxonomy Calculus

Contexts Set 𝒲 of contexts Each context 𝑤 ∈ 𝒲 : core 𝑤 𝐵 𝑦 ∧ 𝐶(𝑦) ⊤core 𝑤 → ⋯ core 𝑤 ∧ 𝐷 𝑦 → ⋯ 𝒯 𝑤 ⋮ core 𝑤 ∧ 𝑆 𝑧, 𝑦 → ⋯ Edges between contexts labelled with functions Context structure 𝒠 is a the graph of labelled contexts and edges

Contexts Set 𝒲 of contexts Each context 𝑤 ∈ 𝒲 : core 𝑤 𝐵 𝑦 ∧ 𝐶(𝑦) |core 𝑤 ⊤ → ⋯ core 𝑤 ∧ 𝐷 𝑦 → ⋯ 𝒯 𝑤 ⋮ core 𝑤 ∧ 𝑆 𝑧, 𝑦 → ⋯ Edges between contexts labelled with functions Context structure 𝒠 is a the graph of labelled contexts and edges

Sound Context Structures core 𝑣 = 𝐷 𝑦 ∧ 𝐸(𝑦) core 𝑤 = 𝐵 𝑦 ∧ 𝐶(𝑦) 𝑔 𝑙 → ⋯ → ⋯ ⊤ ⊤ → ⋯ → ⋯ ⊤ ∧ 𝐷 𝑦 ⊤ ∧ 𝑇(𝑦, 𝑧) ⋮ ⋮ ⊤ ∧ 𝑆 𝑧, 𝑦 → ⋯ ⊤ → ⋯ ① 𝒫 ⊨ core 𝑤 ∧ Γ → Δ for each 𝑤 ∈ 𝑊 and each Γ → Δ ∈ 𝒯 𝑤 ② 𝒫 ⊨ core 𝑣 → core 𝑤 𝑦 ↦ 𝑔 𝑙 𝑦 , 𝑧 ↦ 𝑦 for each 𝑣, 𝑤, 𝑔 𝑙 ∈ ℰ

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 ሼ 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 ሽ ⊆ 𝒫 Vegetarian 𝑦 ∧ eats 𝑦, 𝑨 1 → SideDish 𝑨 1

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 5 ⊤ → HealthyPerson 𝑦 ∨ ሧ 𝑔 𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ Meat 𝑔 𝑗 𝑦 𝑗=1 1≤𝑗<𝑘≤5 5 5 ሼ ሽ ⊆ 𝒫 ሥ eats 𝑦, 𝑨 𝑗 → HealthyPerson 𝑦 ∨ ሧ 𝑨 𝑗 ≈ 𝑨 𝑘 ∨ ሧ Meat 𝑨 𝑗 𝑗=1 𝑗=1 1≤𝑗<𝑘≤5

Vegetarian 𝑦 ⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔 1 𝑦 𝑔 1 ⋮ ⊤ → eats 𝑦, 𝑔 5 𝑦 𝑔 2 , 𝑔 3 , 𝑔 4 , 𝑔 5 ⊤ → 𝑔 𝑗 𝑦 ≉ 𝑔 𝑘 𝑦 for 1 ≤ 𝑗 < 𝑘 ≤ 5 eats 𝑧, 𝑦 ∧ SideDish 𝑦 ⊤ → SideDish 𝑔 1 𝑦 ⋮ ⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑔 5 𝑦 ⊤ → SideDish 𝑦 5 ⊤ → HealthyPerson 𝑦 ∨ ሧ 𝑔 𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ Meat 𝑔 𝑗 𝑦 𝑗=1 1≤𝑗<𝑘≤5

Vegetarian 𝑦 ⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔 1 𝑦 𝑔 1 ⋮ ⊤ → eats 𝑦, 𝑔 5 𝑦 𝑔 2 , 𝑔 3 , 𝑔 4 , 𝑔 5 ⊤ → 𝑔 𝑗 𝑦 ≉ 𝑔 𝑘 𝑦 for 1 ≤ 𝑗 < 𝑘 ≤ 5 eats 𝑧, 𝑦 ∧ SideDish 𝑦 ⊤ → SideDish 𝑔 1 𝑦 ⋮ ⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑔 5 𝑦 ⊤ → SideDish 𝑦 5 ⊤ → HealthyPerson 𝑦 ∨ ሧ 𝑔 𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ Meat 𝑔 𝑗 𝑦 𝑗=1 1≤𝑗<𝑘≤5 5 5 ሼ ሽ ⊆ 𝒫 ሥ eats 𝑦, 𝑨 𝑗 → HealthyPerson 𝑦 ∨ ሧ 𝑨 𝑗 ≈ 𝑨 𝑘 ∨ ሧ Meat 𝑨 𝑗 𝑗=1 𝑗=1 1≤𝑗<𝑘≤5

Vegetarian 𝑦 ⊤ → Vegetarian 𝑦 ⊤ → Animal 𝑦 ⊤ → eats 𝑦, 𝑔 1 𝑦 𝑔 1 ⋮ ⊤ → eats 𝑦, 𝑔 5 𝑦 𝑔 2 , 𝑔 3 , 𝑔 4 , 𝑔 5 ⊤ → 𝑔 𝑗 𝑦 ≉ 𝑔 𝑘 𝑦 for 1 ≤ 𝑗 < 𝑘 ≤ 5 eats 𝑧, 𝑦 ∧ SideDish 𝑦 ⊤ → SideDish 𝑔 1 𝑦 ⋮ ⊤ → eats 𝑧, 𝑦 ⊤ → SideDish 𝑔 5 𝑦 ⊤ → SideDish 𝑦 5 Meat 𝑦 → Meat 𝑦 ⊤ → HealthyPerson 𝑦 ∨ ሧ 𝑔 𝑗 𝑦 ≈ 𝑔 𝑘 𝑦 ∨ ሧ Meat 𝑔 𝑗 𝑦 Meat 𝑦 → ⊥ 𝑗=1 1≤𝑗<𝑘≤5