Download
1 / 24
240 likes | 358 Views
On the logic of merging. Sebasien Konieczy and et el Muhammed Al-Muhammed. What you should get out of this paper. Three major themes 1- what characterizations merging operators must have. 2- the difference between the majority operators and the arbitration operators. 3- their usefulness.
E N D
On the logic of merging Sebasien Konieczy and et el Muhammed Al-Muhammed
What you should get out of this paper • Three major themes 1- what characterizations merging operators must have. 2- the difference between the majority operators and the arbitration operators. 3- their usefulness
Concepts to be covered • Key definitions – revision theorem, merging operators • Some theorems • Example • Conclusions and future work
Revision theorem • Revision basic assumption “new information is more reliable than the knowledge base”. • However, this assumption does not hold always - three cases can be distinguished 1- the new piece of info. Is more reliable; 2- the new piece of info. Is less reliable and 3- the new piece of info. as reliable as the knowledge base.
Merging operators • Two merging operators of special interest - majority operators – satisfy the majority - arbitration operators- satisfy all individuals
Where they are useful They are useful in * finding a coherent information in distributed data base systems * Solving a conflict between several people or agents * Finding answer in a decision-making committee. Etc.
Key Definitions • Interpretation: let L be a language over a finite alphabet P of prepositional letters, we say that the function I: P {0,1} is interpretation if it maps each pP to true or false. • Formula Model: we call any interpretation I a formula model iff it makes a formula true. A set of models of formula represented by Mod().
A knowledge Base : if K is a finite set of prepositional formulae, then conjunction of of K’s formulae is a knowledge base. -Key Point: Knowledge base is consistent • Knowledge set: is the set in which each element is knowledge base. I.e. E={K1,..,Kn}. We define the conjunction as E=K1 … Kn. -Key point: a knowledge set is consistent E is consistent.
Two knowledge bases E1 and E2 are equivalent iff bijection f :E1={K11, …,k1n}E2={K21,…,K2n} such that f(K)K • Key definition: a function from set of knowledge to knowledge base called merging operator if and only if the following is met:
(A1)(E) is consistent. (A2) if E is consistent, then (E) =E (A3) if E1E2, then (E1) (E2) (A4)if K K’ is not consistent, then (KUK’) K (A5) (E1) (E2) (E1U E2) (A6) if (E1) (E2) is consistent, then (E1U E2) (E1) (E2)
Points to ponder carefully first point: Look at this postulate: if a merging operator satisfies (M7), we call it majority operator. Second point: consider this postulate: (A7’) K n such that (E U Kn) = (E UK) there is problem with this: what if E has conflict knowledge bases {K , ¬K}?
Point three: we call any merging operator satisfies (A7) an arbitration operator. • Key point: a merging operator cannot be arbitration and majority operator.
Some Merging operators Fundamental definitions: • Distance between two interpretations: let I and J be interpretations then we define the distance between them as: dis(I,J)=the number prepositional letters in which they differ. example: let I(0,1,0) and J(1,1,0) then dis(I,J)=1
The distance between an interpretation and knowledge base: is the minimum between the interpretation and the model(s) of the knowledge base, formally: Recall: Model() is all interpretations that makes true. Example: let Model() ={(1,1,1) ,(0,0,0)} and I=(0,1,1) then dis(I, )=min(1,2)=1
The distance between two knowledge bases we define such distance as: Example: let Model()={(1,1),(0,1)} and Model() ={(0,0), (1,1)} Then dis(, )=min(2,0,1,1)=0
Three operatorsdefinition: syncretic assignment is function between k.set and pre-order ETeorem : an operator is M.operator iff syncrtic Ass. That maps each knowledge set E to E such that Mod(E)=min(E ) 1- Let be a knowledge base and E a knowledge set, then we define
2- Let E be a knowledge set and I an interpretation we define:
Basic example • Suppose we have a database class with 3 students : the teacher can teach SQL,Database and O2. he asks his student to choose what courses they want to learn. This their responses:
Building the interpretations For Mod(1)={(1,0,0),(0,0,1),(1,0,1)} “assume that letter S, D and O in this order” For Mod(2)={(0,1,0),(0,0,1)} For Mod(3)={(1,1,1)}
the following table shows the results: For example let compute the dis. Between 1 and the interpr. I=(0,0,0). Recall And Mod(1)={(1,0,0),(0,0,1),(1,0,1)} so dis(1,I)=min(1,1,2)=1. The same for others. All possible interpretation
Mod(max(E)={(0,1,1),(1,0,1),(1,1,0)} note:Mod(max(E) = all interpretations with minimum value in dismax column • Mod((E)={(0,0,1),(1,0,1)} • Mod(GMax(E)={(1,0,1)} • It is obvious that max is arbitration operator and is majority operator. maxis arbitration operator?. Recall Let compute satisfaction of1=2(from(0,1,1))+0(from(1,0,1))+0(from(1,1,0))=2,2=3 and 3=3. So all of them satisfied. While majority merging operator. With the same logic we can prove that 1=4, 2=4, 3=0(not satisfied) but that is ok since the majority satisfied.
Conclusions and future work • Building postulates that all rational merging operators have to satisfy. • Distinguishing between majority and arbitration operators. • Proposing new merging operator Gmax • (Future work) finding the minimum conditions that a distance must meet to ensure that the operators defined using such distance satisfy the axiomatic characterization (A1– A6)
