Inconsistency Measurement based on Variables in Minimal - PowerPoint PPT Presentation

Inconsistency Measurement based on Variables in Minimal Unsatisfiable Subsets Guohui Xiao Yue Ma Institute of Informatics, Vienna University of Technology Theoretical Computer Science, Dresden University of Technology ECAI 2012 — August 30, 2012

Overview Motivation 1 Preliminaries 2 Inconsistency Measurement by Variables in MUSes 3 Computational Complexities 4 Experiments 5 Summary 6

Outline Motivation 1 Preliminaries 2 Inconsistency Measurement by Variables in MUSes 3 Computational Complexities 4 Experiments 5 Summary 6 G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 2 / 31

Background Consistent KBs are useful, but inconsistent KBs imply any conclusion (meaningless!) Inconsistency measurement: from “is inconsistent” to “how inconsistent” Ideas and approaches: ◮ based on different views of atomicity of inconsistency ◮ Semantics based approaches ◮ Syntax based approaches ◮ Semantics - syntax combined approaches (this paper) G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 3 / 31

Outline Motivation 1 Preliminaries 2 Inconsistency Measurement by Variables in MUSes 3 Computational Complexities 4 Experiments 5 Summary 6 G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 4 / 31

Inconsistency Measurement by Multi-valued Semantics G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 5 / 31

Inconsistency Measurement by Multi-valued Semantics Multi-Valued Semantics ◮ 4-valued, 3-valued, LP m , Quasi-Classical, . . . ◮ I : Var ( K ) → { t , f , Both , None } G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 5 / 31

Inconsistency Measurement by Multi-valued Semantics Multi-Valued Semantics ◮ 4-valued, 3-valued, LP m , Quasi-Classical, . . . ◮ I : Var ( K ) → { t , f , Both , None } ID of K respect to I under i -semantics ( i = 3 , 4 , LP m , Q ) ID i ( K , I ) = |{ p | p I = B , p ∈ Var ( K ) }| , if I | = i K | Var ( K ) | G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 5 / 31

Inconsistency Measurement by Multi-valued Semantics Multi-Valued Semantics ◮ 4-valued, 3-valued, LP m , Quasi-Classical, . . . ◮ I : Var ( K ) → { t , f , Both , None } ID of K respect to I under i -semantics ( i = 3 , 4 , LP m , Q ) ID i ( K , I ) = |{ p | p I = B , p ∈ Var ( K ) }| , if I | = i K | Var ( K ) | ID of K under under i -semantics ( i = 3 , 4 , LP m , Q ) ID i ( K ) = min = i K ID i ( K , I ) I | G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 5 / 31

Inconsistency Degree under 4-valued Semantics ID 4 ( K , I ) = |{ p | p I = B , p ∈ Var ( K ) }| Truth values: { t , f , B , N } | Var ( K ) | 4-model I : ID 4 ( K ) = min I | = 4 K ID 4 ( K ), K → { t , B } � K = { p , ¬ q , ¬ p ∨ q , r ∨ s } � I 1 : p I 1 = B , q I 1 = f , r I 1 = t , s I 1 = t , I 2 : p I 2 = B , q I 2 = B , r I 2 = t , s I 2 = t I 3 : p I 3 = B , q I 3 = B , r I 3 = t , s I 3 = N � ID 4 ( K , I 1 ) = 1 4 , ID 4 ( K , I 2 ) = 2 4 ID 4 ( K , I 3 ) = 2 4 Figure : Four-Valued Logic ID 4 ( K ) = 1 4 G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 6 / 31

Inconsistency Degree under Quasi-Classical Semantics ID Q ( K , I ) = |{ p | p I = B , p ∈ Var ( K ) }| | Var ( K ) | ID Q ( K ) = min I | = Q K ID Q ( K ), Quasi-Classical (Q) interpretation: � K = { p , ¬ q , ¬ p ∨ q , r ∨ s } 4-valued interpretation I 1 : p I 1 = B , q I 1 = f , r I 1 = t , s I 1 = t � ———————————————- Resolution laws are I 2 : p I 2 = B , q I 2 = B , r I 2 = t , s I 2 = t satisfied I 3 : p I 3 = B , q I 3 = B , r I 3 = t , s I 3 = N I | = Q α ∨ β, I | = Q ¬ β ∨ γ ID Q ( K , I 1 ) = 1 4 , ID Q ( K , I 2 ) = 2 � ——————– 4 ⇒ I | = Q α ∨ γ ID Q ( K , I 3 ) = 2 4 ID Q ( K ) = 2 4 Remark: ID 4 ( K ) = ID 3 ( K ) = ID LPm ( K ) ≤ ID Q ( K ) [Xiao et al., 2010] G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 7 / 31

MUS and MCS Definition A subset U ⊆ K is an Minimal Unsatisfiable Subset (MUS), if U is unsatisfiable and ∀ C i ∈ U , U \ { C i } is satisfiable. Definition A subset M ⊆ K is an Minimal Correction Subset (MCS), if K \ M is satisfiable and ∀ C i ∈ M , K \ ( M \ { C i } ) is unsatisfiable. Example Let K = { p , ¬ p , p ∨ q , ¬ q , ¬ p ∨ r } . Then MUSes ( K ) = {{ p , ¬ p } , {¬ p , p ∨ q , ¬ q }} and MCSes ( K ) = {{¬ p } , { p , p ∨ q } , { p , ¬ q }} . G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 8 / 31

Inconsistency Measurement by MUSes and MCSes [Hunter and Konieczny, 2008] The MI inconsistency measure is defined as the numbers of minimal inconsistent sets of K : I MI ( K ) = | MUSes ( K ) | . (minimal inconsistent sets = minimal unsatisfiable subsets) Example Let K = { p , ¬ p , p ∨ q , ¬ q , ¬ p ∨ r } . MUSes ( K ) = {{ p , ¬ p } , {¬ p , p ∨ q , ¬ q }} I MI ( K ) = 2 Note that I MI ( K ) can be exponentially large G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 9 / 31

Why another Inconsistency Measurement? Combination of Semantics and Syntax based IDs ◮ Shapley inconsistency measures [Hunter and Konieczny, 2006]: distribution of ID { 4 , Q ,... } among different formulas ◮ Ours: combination of semantics and syntax based IDs in the KB level Expected properties: ◮ Easier to compute than I MI : ⋆ I MI tends to be difficult to compute or approximate because of exponentially many MUSes ◮ More intuitive: ⋆ For K = { a ∧ ¬ a } and K ′ = { a ∧ ¬ a ∧ b ∧ ¬ b } , we have I MI ( K ) = I MI ( K ′ ) = 1, which is unintuitive ⋆ Later we see ID 4 tends to be “small”, while ID Q tends to be “large” G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 10 / 31

Outline Motivation 1 Preliminaries 2 Inconsistency Measurement by Variables in MUSes 3 Computational Complexities 4 Experiments 5 Summary 6 G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 11 / 31

Inconsistency Measurement by Variables in MUSes Definition For a given set of variables S and a given knowledge base K such that Var ( K ) ⊆ S , its MUS-variable based inconsistency degree, written ID MUS ( K ), is defined as: ID MUS ( K ) = | Var ( MUSes ( K )) | . | S | Example Let K = { p , ¬ p , p ∨ q , ¬ q , ¬ p ∨ r } and S = Var ( K ) = { p , q , r } , MUSes ( K ) = {{ p , ¬ p } , {¬ p , p ∨ q , ¬ q }} . Then ID MUS ( K ) = 2 / 3. Example For K = { a ∧ ¬ a } and K ′ = { a ∧ ¬ a ∧ b ∧ ¬ b } , let S = Var ( K ) ∪ Var ( K ′ ) = { a , b } . Then we have MUSes ( K ) = {{ a ∧ ¬ a }} and MUSes ( K ′ ) = {{ a ∧ ¬ a ∧ b ∧ ¬ b }} , ID MUS ( K ) = 1 / 2 and ID MUS ( K ′ ) = 1. So under ID MUS , K ′ is more inconsistent than K . G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 12 / 31

Inconsistency Measurement by Variables in MCSes Similarly to ID MUS ( K ), we can define another inconsistency degree through MCS as follows: Definition For a given set of variables S and a given knowledge base K such that Var ( K ) ⊆ S , its MCS-variable based inconsistency degree, written ID MCS ( K ), is defined as follows: ID MCS ( K ) = | Var ( MCSes ( K )) | . | S | Example Let K = { p , ¬ p , p ∨ q , ¬ q , ¬ p ∨ r } and S = Var ( K ), MCSes ( K ) = {{¬ p } , { p , p ∨ q } , { p , ¬ q }} , then ID MCS ( K ) = 2 / 3. G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 13 / 31

ID MUS = ID MCS MUSes ( K ) and MCSes ( K ) are hitting sets dual of each other [Liffiton and Sakallah, 2008] ⇒ � MUSes ( K ) = � MCSes ( K ) ⇒ Var ( � MUSes ( K )) = Var ( � MCSes ( K )) ⇒ ID MUS ( K ) = ID MCS ( K ) In the rest of the talk, the discussion is only about ID MUS ( K ), G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 14 / 31

ID 4 and ID MUS Corollary Let U be an MUS, then ID 4 ( U ) = 1 / | Var ( U ) | . The following theorem shows that ID 4 ( K ) can be determined by the cardinality minimal hitting sets of MUSes ( K ). Theorem For a given KB K, ID 4 ( K ) = min H {| H | | ∀ U ∈ MUSes ( K ) , Var ( U ) ∩ H � = ∅} . | Var ( K ) | Corollary ID MUS ( K ) ≥ ID 4 ( K ) . G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 15 / 31

ID Q and ID MUS Lemma Let U be an MUS, then U has only one Q-model which assigns B to all of its variables. Hence ID Q ( U ) = 1 . Proposition Let K be a KB and I ∈ PM Q ( K ) , then Conflict ( I , K ) ⊇ Var ( MUSes ( K )) . Corollary Let K be a KB, then ID Q ( K ) ≥ ID MUS ( K ) . G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 16 / 31

Outline Motivation 1 Preliminaries 2 Inconsistency Measurement by Variables in MUSes 3 Computational Complexities 4 Experiments 5 Summary 6 G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 17 / 31

Complexity Results ID-MUS ≥ k : Given a CNF KB, and a number k , deciding ID MUS ( K ) ≥ k . ID-MUS : Functional complexity of computing ID MUS Problem Complexity Σ p 2 -complete ID-MUS ≥ k Π p ID-MUS ≤ k 2 -complete D p 2 -complete ID-MUS = k FP Σ p 2 [log] ID-MUS Table : Complexity Results All the results are in the second layer of polynomial hierarchy Recall that ID 4 and ID Q are in first layer G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 18 / 31

Outline Motivation 1 Preliminaries 2 Inconsistency Measurement by Variables in MUSes 3 Computational Complexities 4 Experiments 5 Summary 6 G. Xiao & Y. Ma (TU Wien & TU Dresden) ECAI 2012 19 / 31