Part C Approximate reasoning in general Holger Wache Overview of the Course A. Semantic Web in general and OWL syntax B. OWL Semantics (DLs) and tableaux reasoning C. Approximate reasoning in general D. Approximate reasoning on tableaux E. Approximate resolution for OWL 1
Knowledge-based Systems Knowledge Base Input Reasoning Output Motivation behind Approximation � Reducing complexity � Reasoning under time pressure � Reasoning with other limited resources � Reduce/increase number of solutions � Reasoning that is not “perfect” but “good enough” 2
Anytime Reasoning Quality output Computation time Reasoning Methods: Anytime Algorithms � Measurable Quality of the approximate result Quality of algorithm 1 � Recognizable Quality can be determined at run time 0.8 � Monotonicity over time and input quality 0.6 Quality � Consistency 0.4 � Diminishing returns with more improvements in the 0.2 beginning � Interuptibility at any time 0 0 1 2 3 4 5 6 7 � Preemptability ensures Algorithm step algorithm can be suspended and resumed 3
Types of Approximation � Numerical � Logical � Soundness � Completeness Approximation Approaches � Language � Language Weakening Weakening Knowledge Ontology Base � Theory + � Theory + Approximation Approximation � Approximate � Approximate Input � Approximate Input � Approximate DL- DL- Input Input Output Output Deduction Deduction Reasoning Reasoning 4
Approximation Approaches � Language � Language Weakening Weakening Knowledge Ontology Base � Theory + � Theory + Approximation Approximation � Approximate � Approximate Input � Approximate Input � Approximate DL- DL- Input Output Input Deduction Output Deduction Reasoning Reasoning Guidelines for Reasoning Methods � Semantically well-founded providing a clear answer to the problem � Computationally attractive Knowledge resulting in an easier Base computation of approximate answers Reasoning � Improvable approximate Input Output Method answers � Dual sound and complete � Flexible to apply to different problems 5
Logical Interpretation and Entailment � Interpretation : interpret predicate 0 1 as true or false � Entailment : x ¬x everything which is L interpreted as true Approximate Interpretation by Cadoli-Schaerf � S-1 -entailment: interpret everything 1/0 0/0 1/1 0/1 outside of S as S1 S3 false x ¬x � S-3 -entailment: L interpret everything S outside of S as true (or normal) 6
Approximate Entailment by Cadoli-Schaerf ² S ² ² S 1 3 Sound Complete But incomplete But unsound S1/S3-Entailment & Anytime S n = L L L L S 3 S 1 S 2 Quality of algorithm 1 0.8 � Anytime behavior when S i will be 0.6 Quality enlarged continuously 0.4 � Interesting Feature: 0.2 Reusing proof from previous level 0 0 1 2 3 4 5 6 7 Algorithm step 7
Approximate Entailment � Semantically well-founded � Computationally attractive � Improvable � Dual � Flexible Approximate Entailment � Unclear effect � Parameter S is crucial for approximate behaviour � Almost no quantitative analysis 8
Boolean Constraint Propagation (BCP) � Variant of unit resolution � Resolution only with at least one literal (P or ¬ P) � Example: Γ = { ¬ P, P ∨ ¬ Q, P ∨ Q ∨ ¬ R, P ∨ Q ∨ R} � ¬ Q from ¬ P, P ∨ ¬ Q � ¬ R from ¬ Q, ¬ P, P ∨ Q ∨ ¬ R � {} from ¬ P, ¬ Q, ¬ R, P ∨ Q ∨ R � Γ ² BCP φ iff {} ∈ Γ ∪ { ¬ φ } Variants of BCP � Clausal BCP (restricted to clauses) sound but incomplete � CNF-BCP � Prime-BCP (intractable) � Formula BCP (intractable) � Fact Propagation 9
Approximation through Abstraction � Process of mapping a representation of a Abstract Abstract problem onto a new Problem Solution representation � Helps deal with the problem in the original f search space by perserving certain desirable properties Problem Solution � Is simpler to handle � Abstraction: α → f( α ) Forms of Abstractions � TC(onstant)-Abstraction α ∈ Th( Σ 1 ) iff Th( Σ 1 ) Th( Σ 2 ) f( α ) ∈ Th( Σ 2 ) � TI(ncrease)-Abstraction if f( α ) ∈ Th( Σ 2 ) Th( Σ 1 ) then α ∈ Th( Σ 1 ) Th( Σ 2 ) � TD(ecrease)-Abstraction if α ∈ Th( Σ 1 ) Th( Σ 1 ) Th( Σ 2 ) then f( α ) ∈ Th( Σ 2 ) 10
Example: Predicate Abstraction � Base Theory: � Abstraction � JapaneseCar(X) => � ForeignCar (X) => Car(X) Car(X) � EuropeanCar(X) => � Toyota(X) => Car(X) ForeignCar (X) � Toyota(X) => � BMW(X) => JapaneseCar(X) ForeignCar (X) � BMW(X) => EuropeanCar(X) � Extension � Extension � EuropeanCar(X) => � ForeignCar (X) => Fast(X) Fast(X) � JapaneseCar(X) => � ForeignCar (X) => Reliable(X) Reliable(X) Problem with Syntactic Abstraction � Abstraction includes: � Captures the final result of abstraction � BMW(X) => Reliable(X) � BUT: � Toyota(X) => does not capture the Fast(X) underlying justification that leads to the abstraction � Stronger � ForeignCar(X) => � Which is the best (Fast(X) ∨ Reliable(X)) (TD)-Abstraction? 11
Semantic Abstraction � Interpretations I 1 , I 2 � I 1 ² Th( Σ 1 ) Abstract 0/1 � I 2 ² Th( Σ 2 ) Problem � Semantic Abstraction f(I 1 ) = I 2 f � Model Increasing Abstraction if I 1 is model of Th( Σ 1 ) 0/1 Problem then f(I 2 ) is model of Th( Σ 2 ) Example (cont.) � Predicate Abstraction: � I(ForeignCar) = I(EuropeanCar) ∪ I(JapaneseCar) 12
Best Semantic Abstraction � Strongest MI Abstraction Th( Σ 2 ) = { σ | all models I 1 of Th( Σ 1 ): f(I 1 ) ² σ } � Central question: How to find syntactic formulas for best Semantic Abstraction � ForeignCar(X) = EuropeanCar(X) ∪ JapaneseCar(X) � Obvious in Description Logics? Approximation Approaches � Language � Language Weakening Weakening Knowledge Ontology Base � Theory + � Theory + Approximation Approximation � Approximate � Approximate Input � Approximate Input � Approximate DL- DL- Input Input Output Output Deduction Deduction Reasoning Reasoning 13
Approximate Input � Approximating Terminological Queries � Top-k querying Conjunctive Queries Q ← q 1 ∧ … ∧ q n With q i : x:C or (x,y) : R C = concept name R = role name x,y = variables or constants � Developed for description logics � Most expressive query language 14
Example Approximating the Query � Hypothesis: Less complex queries can be answered in shorter time. � Query containment Q i w Q j : � results(Q j ) ⊆ results(Q i ) � Create a sequence of queries Q 1 , …, Q m with � i < j ⇒ Q i w Q j � Q m = Q 15
How to determine the query sequence � Observation: Q i contains less conjuncts than Q j ⇒ Q i w Q j � Q 0 = empty � Q i+1 = Q i ∧ q k ∧ … ∧ q l � How to determine which conjuncts q k ∧ … ∧ q l are included in Q i+1 Query Graph (Example) father-of Female degree works-at awarded-by {vu} 16
Strategy I: Node expansion father-of father-of father-of Female Female Female degree degree works-at works-at awarded-by {vu} Strategy II: Search father-of father-of father-of father-of Female Female Female Female degree degree degree works-at works-at awarded-by {vu} father-of father-of father-of father-of Female Female Female Female degree degree degree works-at awarded-by awarded-by {vu} {vu} 17
Approximate Input � Approximating Terminological Queries � Top-k querying Top-k queries � Well known in databases � Extends traditional database retrieval Top-k � Instead of returning an unordered set of results also rank the results � Top-K means return only the k best results � Query language SQL extended to facilitate rank and/or scoring algorithm 18
Ranking � Normally a (normalized) function F → [0..1] � Sources for ranking � Google’s Pagerank � User-preferences, e.g. user specify in which query predicates he is more interested in Naïve Algorithm for Top-k Naïve algorithm � Retrieval all answers 1. Order them according ranking function 2. Return the best k results 3. Problem � Too much unnecessary data accesses � 19
Optimal Solutions � Best solution: only k data accesses � With respect to the necessary data access are known for � Multimedia retrieval � Databases, and � Web searches Top-k in a peer-to-peer setting •0.7 •0.9 � More Web-alike: •0.6 assuming many peer sources for storing relevant answers 0.7 0.6 0.9 � Challenge: minimizing data transfer between 0.6 0.7 peers 0.9 � Each peer returns top- •0.8 0.8 k answers peer •0.9 peer •0.8 •0.7 20
Approximation Approaches � Language � Language Weakening Weakening Knowledge Ontology Base � Theory + � Theory + Approximation Approximation � Approximate � Approximate Input � Approximate Input � Approximate DL- DL- Input Output Input Deduction Output Deduction Reasoning Reasoning Knowledge Compilation � deals with translating the knowledgebase such that the computational complexity of reasoning decreases. � goal of knowledge compilation is to translate the knowledge into (or approximate it by) another knowledge base with better computational properties. 21
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