Lethe: Saturation-Based Reasoning for Non-Standard Reasoning Tasks - PowerPoint PPT Presentation

φ ormal µ ethods γ roup Lethe: Saturation-Based Reasoning for Non-Standard Reasoning Tasks Patrick Koopmann, Renate A. Schmidt

φ ormal µ ethods γ roup Lethe • “River of Forgetfulness” • Usage from command line, as Java library, or via GUI • Non-standard reasoning services relative to signatures – Forgetting / Uniform Interpolation – TBox Abduction – Logical Difference • Support for expressive description logics (up to SHQ ) • Problems reduced to forgetting, uses saturation-based reasoning 2/16

φ ormal µ ethods γ roup Uniform Interpolation/Forgetting • Core Functionality of Lethe • Restrict vocabulary in set of axioms • Preserve entailments over that signature Input Ontology Uniform Interpolant Margherita ⊑ ∀ topping . ( Tomato ⊔ Mozarella ) American ⊑ ∃ topping . Tomato Margherita ⊑ ∀ topping . VegTopping American ⊑ ∃ topping . Mozarella American ⊑ ∃ topping . MeatTopping American ⊑ ∃ topping . Pepperoni Tomato ⊔ Mozarella ⊑ VegTopping Pepperoni ⊑ MeatTopping 3/16

φ ormal µ ethods γ roup Applications of Forgetting • Exhibit hidden concept relations • Information hiding • Ontology reuse • Ontology summary • Obfuscation • . . . 4/16

φ ormal µ ethods γ roup TBox Abduction • Given TBox T , axioms O , find axioms H with T ∪ H | = O • “Complete” ontology such that given set of axioms is entailed • Abducibles Σ specify concepts and roles allowed in solution • Reducible to uniform interpolation: – T ∪ ¬ O | = ¬ H – Express ¬ ( C ⊑ D ) as ∃ r ∗ . ( C ⊓ D ) – Interpolate to set of abducibles • Optimisations for large TBoxes and small inputs 5/16

φ ormal µ ethods γ roup Logical Difference • “Semantical Diff” • Analyse ontology changes, compare ontologies • Look for differing entailments in specified signature Σ • Compute new entailments in O 2 : – LD ( O 1 , O 2 , Σ) = { α | α ∈ O Σ 2 , O 1 �| = α } – O Σ 1 : Uniform interpolant of O 1 for Σ • Optimised for two use cases: 1. Bigger changes, computation in minutes acceptable 2. Small changes, computation in seconds required 6/16

φ ormal µ ethods γ roup Challenges Uniform Interpolation 1. Need for new reasoning methods 2. Cyclic TBoxes A ⊑ B , B ⊑ ∃ r . B S = { A , r } – Uniform Interpolant in ALC : – A ⊑ ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . ∃ r . . . . – Solutions: Fixpoints: A ⊑ ν X . ( ∃ r . X ) Approximate: A ⊑ ∃ r . ∃ r . ∃ r . ⊤ Helper concepts: A ⊑ ∃ r . D , D ⊑ ∃ r . D 3. High Complexity – ALC with fixpoints: 2 2 n , where n is size of input – Goal-oriented approach necessary 7/16

φ ormal µ ethods γ roup Normal form, ALC ALC -Clause ⊤ ⊑ L 1 ⊔ . . . ⊔ L n L i : ALC -literal ALC -Literal A | ¬ A | ∃ r . D | ∀ r . D A : any concept symbol, D : definer symbol • Definer symbols: Special concept symbols, not part of signature • Invariant: max 1 neg. definer symbol per clause ✭ ⇒ ¬ D 1 ⊔ ∃ r . D 2 ⊔ ¬ B , ✭✭✭✭✭✭ ¬ D 1 ⊔ ¬ D 2 ⊔ A 8/16

φ ormal µ ethods γ roup Definer symbols Invariant: max 1 neg. definer symbol per clause • Allows easy translation to clausal form and back: C 1 ⊔ Q r . C 2 ⇐ ⇒ C 1 ⊔ Q r . D 1 , ¬ D 1 ⊔ C 2 C 1 ⊔ ν X . C 2 [ X ] ⇐ ⇒ C 1 ⊔ Q r . D 1 , ¬ D 1 ⊔ C 2 [ D ] ⇒ Any set of clauses can be converted into an ALC µ -ontology ( ALC with fixpoints) • New definer symbols introduced by calculus – Number finitely bounded 9/16

φ ormal µ ethods γ roup Calculus Resolution + Combination rules • Resolution rule: – Direct inference on concept symbol to forget – Resolvent has to obey invariant C 1 ⊔ A C 2 ⊔ ¬ A C 1 ⊔ C 2 • Combination rules: – Combine context of nested definer symbols – Introduce new definer symbols – Representing conjunctions of definers – Max. 2 n new definer symbols – Make further inferences possible 10/16

φ ormal µ ethods γ roup Combination Rules ¬ D 1 ⊔ A ¬ D 2 ⊔ B ⊔ ¬ A C 1 ⊔ ∃ r . D 1 C 2 ⊔ ∀ r . D 2 11/16

φ ormal µ ethods γ roup Combination Rules Cannot resolve due invariant ¬ D 1 ⊔ A ¬ D 2 ⊔ B ⊔ ¬ A C 1 ⊔ ∃ r . D 1 C 2 ⊔ ∀ r . D 2 11/16

φ ormal µ ethods γ roup Combination Rules Cannot resolve due invariant ¬ D 1 ⊔ A ¬ D 2 ⊔ B ⊔ ¬ A C 1 ⊔ ∃ r . D 1 C 2 ⊔ ∀ r . D 2 combine C 1 ⊔ C 2 ⊔ ∃ r . D 12 ¬ D 12 ⊔ A ¬ D 12 ⊔ B ⊔ ¬ A 11/16

φ ormal µ ethods γ roup Combination Rules Cannot resolve due invariant ¬ D 1 ⊔ A ¬ D 2 ⊔ B ⊔ ¬ A C 1 ⊔ ∃ r . D 1 C 2 ⊔ ∀ r . D 2 combine C 1 ⊔ C 2 ⊔ ∃ r . D 12 ¬ D 12 ⊔ A Resolves to ¬ D 12 ⊔ B ¬ D 12 ⊔ B ⊔ ¬ A 11/16

φ ormal µ ethods γ roup Combination Rules ALC ∀∃ -Combination C 1 ⊔ ∀ r . D 1 C 2 ⊔ ∃ r . D 2 C 1 ⊔ C 2 ⊔ ∃ r . D 12 ∀∀ -Combination C 1 ⊔ ∀ r . D 1 C 2 ⊔ ∀ r . D 2 C 1 ⊔ C 2 ⊔ ∀ r . D 12 12/16

φ ormal µ ethods γ roup Combination Rules SHQ ≤≤ -Combination: ≥≤ -Combination: C 1 ⊔ ≤ n 1 r 1 . ¬ D 1 C 2 ⊔ ≤ n 2 r 2 . ¬ D 2 r ⊑ r 1 r ⊑ r 2 C 1 ⊔ ≥ n 1 r 1 . ( D 1 ⊔ . . . ⊔ D m ) C 2 ⊔ ≤ n 2 r 2 . ¬ D a r 1 ⊑ R r 2 C 1 ⊔ C 2 ⊔ ≤ ( n 1 + n 2 ) r . ¬ D 12 C 1 ⊔ C 2 ⊔ ≥ ( n 1 − n 2 ) r 1 . ( D 1 a ⊔ . . . ⊔ D ma ) ≤≥ -Combination: ≥≥ -Combination: C 1 ⊔ ≤ n 1 r 1 . ¬ D 1 C 2 ⊔ ≥ n 2 r 2 . D 2 r 2 ⊑ R r 1 n 1 ≥ n 2 C 1 ⊔ ≥ n 1 r 1 . D 1 C 2 ⊔ ≥ n 2 r 2 . D 2 r 1 ⊑ R r r 2 ⊑ R r C 1 ⊔ C 2 ⊔ ≤ ( n 1 − n 2 ) r 1 . ¬ ( D 1 ⊔ D 2 ) ⊔ ≥ 1 r 1 . D 12 C 1 ⊔ C 2 ⊔ ≥ ( n 1 + n 2 ) r . ( D 1 ⊔ D 2 ) ⊔ ≥ 1 r . D 12 . . . . . . C 1 ⊔ C 2 ⊔ ≤ ( n 1 − 1) r 1 . ¬ ( D 1 ⊔ D 2 ) ⊔ ≥ n 2 r 1 . D 12 C 1 ⊔ C 2 ⊔ ≥ ( n 1 + 1) r . ( D 1 ⊔ D 2 ) ⊔ ≥ n 2 r . D 12 Transitivity: C ⊔ ≤ 0 r 1 . ¬ D trans( r 2 ) ∈ R r 2 ⊑ R r 1 ¬ D ′ ⊔ D ¬ D ′ ⊔ ≤ 0 r 2 . ¬ D ′ C ⊔ ≤ 0 r 2 . ¬ D ′ 13/16

φ ormal µ ethods γ roup Algorithm • Compute all inferences on symbol to forget • Use resolvents breaking invariant to choose combination rules • Filter out all occurrences of symbol to forget • Eliminate introduced symbols 14/16

φ ormal µ ethods γ roup Evaluation of Uniform Interpolation ALCH , forget 50 symbols ALCH , forget 100 symbols Success Rate: 91.10% Success Rate: 88.10% Without Fixpoints: 95.29% Without Fixpoints: 93.27% Duration Mean: 7.68 sec. Duration Mean: 18.03 sec. Duration Median: 2.74 sec. Duration Median: 3.81 sec. Duration 90th percentile: 12.45 sec. Duration 90th percentile: 21.17 sec. ALC w. ABoxes, forget 50 symbols ALC w. ABoxes, forget 100 symbols Success Rate: 94.79% Success Rate: 91.37% Without Fixpoints: 92.91% Fixpoints: 92.48% Duration Mean: 23.94 sec. Duration Mean: 57.87 sec. Duration Median: 3.01 sec. Duration Median: 6.43 sec. Duration 90th percentile: 29.00 sec. Duration 90th percentile: 99.26 sec. SHQ , forget 50 concept symbols SHQ , forget 100 concept symbols Success Rate: 95.83% Timeouts: 90.77% Without Fixpoints: 93.40% Fixpoints: 91.99% Duration Mean: 7.62 sec. Duration Mean: 13.51 sec. Duration Median: 1.04 sec. Duration Median: 1.60 sec. Duration 90th percentile: 4.89 sec. Duration 90th percentile: 11.65 sec. Corpus Respective fragments of 306 ontologies from BioPortal having at most 100,000 axioms. Timeout 30 minutes 15/16

φ ormal µ ethods γ roup Conclusion • Lethe supports different non-classical reasoning methods via reduction to forgetting • Usage as library, command line tool or via simple front end • Available at http://cs.man.ac.uk/~koopmanp/lethe • Future work – Better evaluation on abduction and logical difference – Use saturation-based approach for other non-classical reasoning problems such as approximation and ABox abduction – Investigate more expressive description logics 16/16