on the logic of merging
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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. - PowerPoint PPT PresentationTRANSCRIPT
On the logic of merging
Sebasien Konieczy and et el
Muhammed Al-Muhammed
What you should get out of this paper
• Three major themes1- 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 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
2- Let E be a knowledge set and I an
interpretation we define:
3-
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”ForMod(2)={(0,1,0),(0,0,1)}ForMod(3)={(1,1,1)}
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.
• 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.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.
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)