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On the logic of merging Sebasien Konieczy and et el Muhammed Al-Muhammed

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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

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Concepts to be covered Key definitions – revision theorem, merging operators Some theorems Example Conclusions and future work

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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.

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Merging operators Two merging operators of special interest - majority operators – satisfy the majority - arbitration operators- satisfy all individuals

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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.

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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 p P 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( ).

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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.

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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:

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(A1) (E) is consistent. (A2) if E is consistent, then (E) = E (A3) if E1 E2, 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)

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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 K n ) = (E UK) there is problem with this: what if E has conflict knowledge bases {K, ¬K}?

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Point three: we call any merging operator satisfies (A7) an arbitration operator. Key point: a merging operator cannot be arbitration and majority operator.

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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

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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

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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

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Three operators definition: syncretic assignment is function between k.set and pre- order E Teorem : 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

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2- Let E be a knowledge set and I an interpretation we define:

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3-

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Basic example Suppose we have a database class with 3 students : the teacher can teach SQL,Database and O 2. he asks his student to choose what courses they want to learn. This their responses:

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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)}

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the following table shows the results: All possible interpretation 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.

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Mod( max (E)={(0,1,1),(1,0,1),(1,1,0)} note:Mod( max(E) = all interpretations with minimum value in dis max 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. max is arbitration operator?. Recall Let compute satisfaction of 1=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.

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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)

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1. The set N = {1,2,3,4,……..} is known as natural numbers or the set of positive integers The natural numbers are used mainly for : counting ordering.

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