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 K

n) = (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”

For

Mod(2)={(0,1,0),(0,0,1)}

For

Mod(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.

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)