Christian Alrabbaa Stefan Borgwardt Patrick Koopmann Alisa Kovtunova
Finding Proofs for Description LogicEntailments in PracticeBased on �Finding Small Proofs for Description Logic Entailments�Theory and
Practice� (LPAR'20) // Explainable Logic-Based Knowledge Representation
(XLoKR 2020), September 14, 2020
Description Logics and Ontologies
Syntax of DL ALC
Concepts: C ::= A | ¬C | C u C | C t C | ∃r .C | ∀r .CAxioms: α ::= C v C | C ≡ C
Description Logics� Well-established formalism for specifying terminological knowledge inOntologies
� Used for many large-scale ontologies– SNOMED CT: over 300,000 concepts– BioPortal: repository of bio-medical ontologies, currently hosting 889
ontologies defining 12,084,317 terms– MOWLCorp: ontologies obtained by web-crawling, containing 21,000
ontologies
� With increasing complexity of the ontology, understanding entailmentsbecomes both crucial and difficult
� One typical reasoning task is classification– compute all entailed axioms of form A v B– obtain concept hierarchy
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 1 of 25
Description Logics and Ontologies
Syntax of DL ALC
Concepts: C ::= A | ¬C | C u C | C t C | ∃r .C | ∀r .CAxioms: α ::= C v C | C ≡ C
Description Logics� Well-established formalism for specifying terminological knowledge inOntologies
� Used for many large-scale ontologies� With increasing complexity of the ontology, understanding entailmentsbecomes both crucial and difficult
� One typical reasoning task is classification– compute all entailed axioms of form A v B– obtain concept hierarchy
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 1 of 25
Description Logics and Ontologies
Syntax of DL ALC
Concepts: C ::= A | ¬C | C u C | C t C | ∃r .C | ∀r .CAxioms: α ::= C v C | C ≡ C
Description Logics� Well-established formalism for specifying terminological knowledge inOntologies
� Used for many large-scale ontologies� With increasing complexity of the ontology, understanding entailmentsbecomes both crucial and difficult
� One typical reasoning task is classification– compute all entailed axioms of form A v B– obtain concept hierarchy
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 1 of 25
Current Tool of Choice: Justi�cations
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 2 of 25
Current Tool of Choice: Justi�cations
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 2 of 25
Current Tool of Choice: Justi�cations
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 2 of 25
Current Tool of Choice: Justi�cations
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 2 of 25
Justi�cations
Justifications: Minimal subsets entailing given subsumption
In practice often insufficient :� can be large� inferences often not obvious
Showing how to obtain the inference would be better� simple reasoning steps leading to conclusion� generally known as proof
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 3 of 25
Proofs for ELK in Protege
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 4 of 25
Proofs for ELK in Protege
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 4 of 25
Proofs for ELK in Protege
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 4 of 25
Proofs for ELK in Protege
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 4 of 25
Proofs for ELK in Protege
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 4 of 25
Proofs using Evonne (work in progress)
C. Alrabbaa, F. Baader, R. Dachselt, T. Flemisch, P. Koopmann: Visualizing Proofs andthe Modular Structure for Ontology Repair, DL 2020.
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 5 of 25
Proofs with ELK
� ELK using a consequence-based reasoning procedure⇒ inferences performed using a calculus
R0C v C
R>C v > R−u
C v D u E
C v D C v E
R+uC v D C v E
C v D u ER∃
C v ∃r .D D v E
C v ∃r .E
R⊥C v ∃r .D D v ⊥
C v ⊥ RvC v D
C v E: D v E ∈ O
R◦C0 v ∃r1.C1,C1 v ∃r2.C2 . . . Cn−1 v ∃rn.Cn
C0 v ∃r .Cn: r1 ◦ . . . ◦ rn v r ∈ O
⇒ proofs generated as part of reasoning process
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 6 of 25
Proofs For More Expressive DLs
� Currently, ELK is the only DL reasoner supporting proof generation
� ELK supports only a limited fragment of OWL, OWL EL
� More expressive reasoner often use other reasoning procedures– for a long time prominent: tableau reasoning– less convenient for understanding entailments
� Existing consequence-based reasoning for expressive DLs– often involved in complex systems– often combined with other reasoning paradigms– may use normal forms requiring different syntax⇒ generation of proofs not obvious
� Can we generate proofs without a calculus?
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 7 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
J
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Justi�cation Based Proofs
Justification-Based Proofs (Matthew Horridge 2011)� derive intermediate steps between conclusion and justification� consider all axioms of some predefined shapes� justification-relation between allows to construct a proof� involved ranking function allows to select axioms to be used in proof
Advantage of approach:� generates best proof according to ranking
Disadvantage of approach:� no clear inference principle� hard to implement� strongly depends on ranking function
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 8 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p
� Decide satisfiability by eliminating names one after the other:� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
b a ∨ b ¬b ∨ c ¬b ∨ ¬c ¬a ∨ c
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
b a ∨ b ¬b ∨ c ¬b ∨ ¬c ¬a ∨ c
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
b ¬b ∨ c ¬b ∨ ¬c b ∨ c
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
b ¬b ∨ c ¬b ∨ ¬c b ∨ c
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
c ¬c
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
c ¬c
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
A v C C v ∃r .D ∃r .> v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
A v C C v ∃r .D ∃r .> v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
A v C C v ∃r .> ∃r .> v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
A v C C v ∃r .> ∃r .> v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
A v C C v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
A v C C v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting-Based Proofs
Forgetting Based Proofs
� Idea from propositional resolution: q1 ∨ p q2 ∨ ¬pp1 ∨ p2
⇒ inference through elimination of p� Decide satisfiability by eliminating names one after the other:
⊥
� Idea: Use similar approach to prove axioms of form A v B
A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 9 of 25
Forgetting
DefinitionLet O be an ontology and X a predicate name. Then, O−X is a result offorgetting X iff� X does not occur in O−X� for every axiom α in which X does not occur, O |= α iff O−X |= α
⇒ strongest ontology without X entailed by O� which α to preserve also depends on underlying DL
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 10 of 25
Forgetting based proofs
� Use forgetting to produce sequence of ontologiesO0 O1 O2 O3 O4
A v B
−X0 −X1 −X2 −X3
� In each step, recompute justification for A v B
� Finally, reconstruct proof using justifications
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 11 of 25
Forgetting based proofs
� Use forgetting to produce sequence of ontologiesO0 O1 O2 O3 O4
A v B
−X0 −X1 −X2 −X3
� In each step, recompute justification for A v B
� Finally, reconstruct proof using justifications
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 11 of 25
Forgetting based proofs
� Use forgetting to produce sequence of ontologiesO0 O1 O2 O3 O4
A v B
� In each step, recompute justification for A v B
� Finally, reconstruct proof using justifications
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 11 of 25
Forgetting based proofs
� Use forgetting to produce sequence of ontologiesO0 O1 O2 O3 O4
A v B
� In each step, recompute justification for A v B� Finally, reconstruct proof using justifications, skipping steps if it makessense
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 11 of 25
Forgetting Based Proof
A v C
C v ∃r .D(D)
C v ∃r .>(C )
A v ∃r .> ∃r .> v B(r )A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 12 of 25
Forgetting Order
Forgetting order, as well as selection of justification, affects proof� Forgetting D first:
A v C
C v ∃r .D(D)
C v ∃r .>(C )
A v ∃r .> ∃r .> v B(r )A v B
� Forgetting r first:
A v C
C v ∃r .D ∃r .> v B(r )C v B
(C )A v B
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 13 of 25
Forgetting Order
To obtain nicer proofs practically, we process names using the followingheuristics:� role with non-trivial fillers last:– otherwise may hide most of inference:
A v ∃r .B A v ∀r .C t D B v ∃s.D C v ∀s.¬D(r )
A v D
� unnested names first– delay complex inferences
� less frequent names first– delay expensive forgetting operations– (also used by existing forgetting procedures)
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 14 of 25
Evaluation
� Implemented approach in modular fashion– easy exchange of different forgetting procedures, provided they produce OWL
ontologies– easy comparison with proofs generated by ELK
– Dijkstra-based search to extract shortest proof– use 2 forgetting tools in the evaluation
– ALCH variant of LETHE 0.6– ALCOI variant of FAME 1.01
1there is a much improved version FAME 2.0, but it often creates ontologies outside of theOWL-standard
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 15 of 25
Evaluation: Corpus
� Focus on proofs in ELH– to be able to compare with ELK– easier extraction of justification patterns (see below)
� Use ontologies from the OWL Reasoner Evaluation 2015, OWL ELClassification Track– well-balanced mix of ontologies from different repositories
� Extracted 1,573 justification patterns– all entailments of form A v B or A ≡ B– all justifications for these entailments– abstract away concept and role names– remove resulting duplicates
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 16 of 25
Evaluation: Metrics
� Hypergraph-size– number of distinct axioms used in the proof
� Tree-size– sub-proofs count as often as they are used
� Justification Complexity– Matthew et al. 2013: “Toward cognitive support for OWL justifications”– attempt to measure cognitive complexity of justification– provides value for each proof step– we measured maximum and sum for each proof
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 17 of 25
Evaluation: Proof size
0 30 60 90 1200
30
60
90
120
LETHE (114)
FAME(410)
Tree Size
0 30 60 90 1200
30
60
90
120
ELK (468)
FAME(87)
Tree Size
0 30 60 90 1200
30
60
90
120
ELK (1087)
LETH
E(142)
Tree Size
0 10 20 30 40 50 600
10
20
30
40
50
60
LETHE (100)
FAME(398)
Hypergraph Size
0 10 20 30 40 50 600
10
20
30
40
50
60
ELK (486)
FAME(48)
Hypergraph Size
0 10 20 30 40 50 600
10
20
30
40
50
60
ELK (1146)
LETH
E(54)
Hypergraph Size
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 18 of 25
Evaluation: ELK vs. Forgetting-Based Proofs
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 19 of 25
Evaluation: ELK Proof
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 20 of 25
Evaluation: LETHE Proof
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 21 of 25
Evaluation: FAME Proof
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 22 of 25
Evaluation: Justi�cation Complexity
0K 5K 10K0K
5K
10K
LETHE (101)
FAME(462)
Just. complexity (sum)
0K 5K 10K0K
5K
10K
ELK (411)FAME(214)
Just. complexity (sum)
0K 5K 10K0K
5K
10K
ELK (669)
LETH
E(603)
Just. complexity (sum)
0 250 500 7500
250
500
750
LETHE (37)
FAME(107)
Just. complexity (max)
0 250 500 7500
250
500
750
ELK (374)
FAME(189)
Just. complexity (max)
0 250 500 7500
250
500
750
ELK (493)
LETH
E(719)
Just. complexity (max)
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 23 of 25
Evaluation: Complexity of Inferences
� Both LETHE and FAME may use logical operators outside of EL
� Large number of distinct “inference rules” was used:– LETHE: 362 different rules– FAME: 281 different rules
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 24 of 25
Conclusion
� New proof generation procedure based on forgetting� Generate proof by repeated use of forgetting and justification� Proofs for expressive DLs without calculus� Can sometimes even compete with ELK� Several possibilities to improve:– better heuristics on forgetting order or when to skip steps– dynamic selection of forgetting order– use learned “rules” to shorten proof computation times– integrate newer version of FAME
Finding Proofs for Description Logic Entailments in PracticeTechnische Universität Dresden // Christian Alrabbaa, Stefan Borgwardt, Patrick Koopmann, Alisa KovtunovaExplainable Logic-Based Knowledge Representation (XLoKR 2020), September 14, 2020
Slide 25 of 25