On the belief merging by negotiation (2)

The contribution of this paper is two-fold: i we develop a negotiation-based model for belief merging, and ii we investigate the computational complexity of the belief merging problem wi

Procedia Computer Science 35 ( 2014 ) 147 – 155

1877-0509 © 2014 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of KES International.

doi: 10.1016/j.procs.2014.08.094

ScienceDirect

18th International Conference on Knowledge-Based and Intelligent

Information & Engineering Systems - KES2014

On the belief merging by negotiation Trong Hieu Trana,∗, Quoc Bao Vob, Thi Hong Khanh Nguyenc

a VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

b Swinburne University of Technology, PO Box 218 Hawthorn VIC 3122 Victoria, Australia

c Electric Power University, 235 Hoang Quoc Viet, Tu Liem, Hanoi, Vietnam

Abstract

Belief merging is an active research field with many important applications Most existing work addresses the belief merging issue using a centralised approach In this paper, we investigate a distributed approach to the problem of belief merging The contribution

of this paper is two-fold: (i) we develop a negotiation-based model for belief merging, and (ii) we investigate the computational complexity of the belief merging problem within the proposed framework Through the proposed model of negotiation-based belief merging, we will present and discuss several significant logical properties and computational complexity results

c

 2014 The Authors Published by Elsevier B.V

Peer-review under responsibility of KES International

Keywords: Computational complexity; Belief merging; Knowledge integration; Negotiation

1 Introduction

Belief merging has emerged as an active research topic with important applications in many fields of Computer Science The main goal of belief merging problem is to obtain the common beliefs from several belief bases It is applied in database integration1,2, information retrieval3,4,5,6, sensor data fusion7, coordination in multi-agent8,9,10, and multimedia systems11,12

Several approaches have been proposed for addressing the belief merging problem In general, they can be clas-sified as either centralized approaches or distributed ones The centralized belief merging approaches constitute the major direction in the belief merging literature in which the merging process is considered an arbitration The typical approaches in this group include belief merging with arbitration operators proposed by Revesz13, belief merging with weighted belief bases by Lin14, belief merging with integrity constraints by Konieczny and P´erez15, belief merging in

a possibilistic logic framework proposed by Benferhat et al16, and belief merging with stratified bases by Qi et al17 The solutions induced in these approaches satisfy a number of rational axioms for belief merging However, these approaches are not without some shortcomings In particular, they require a mediator without taking into account the roles of the agents who provide the source of the belief bases to be merged and they assume that all belief bases are

∗Corresponding author Tel.:+84-945-89-3663 ; fax: +84-4-3858-8817.

E-mail address: hieutt@vnu.edu.vn

© 2014 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

Peer-review under responsibility of KES International

completely provided up front These requirements are generally inapplicable in many multi-agent systems.

The distributed belief merging approaches aim to overcome the above mentioned shortcoming In these approaches, belief merging is considered as a game in which agents who possess the source belief bases are self-interested and may act strategically The agents merge their belief bases by following a pre-defined protocol in order to reach the consensuses among themselves18,19,20,21,22,23,24 An approach inspired by two-stage belief revision operators are pro-posed by Booth18,19 It has been subsequently enriched by identifying a family of merging operators by Konieczny (see21) Another important approach is proposed by Zhang20 in which a negotiation model is built for the set of agents’ demands represented by logical formulas The negotiation is carried out by first aligning all the belief bases

in their lowest priority layers and then iteratively removing the lowest layers of belief bases until the remaining layers are jointly consistent or a disagreement situation arises However, this approach suffers the drowning effect and is syntax sensitive In22,24, we propose a solution for belief merging by negotiation, which can overcome the drowning

effect, but it is still syntax sensitive Moreover, our more recent work23can overcome both issues

While the literature of centralised belief merging has been quite comprehensive addressing both the rationality of the merging operators and their computational complexity, there is a noticeable lack of results regarding the compu-tational complexity of the distributed merging problems It is another aim of this work to address this issue In this paper, we propose a model for negotiation-based belief merging in which the merging process is organised into two stages In the first stage, the preferences over the set of all possible worlds are constructed from the stratified belief bases, based on several ordering strategies The second stage is a negotiation process which works on the preferences constructed in the first stage Then, several logical properties and complexity results for the proposed belief merging operator will be presented and discussed

The rest of this paper is organized as follows Section 2 provides some formal preliminaries A model for belief merging by negotiation is introduced in Section 3, in which we present a model for belief merging and a set of axioms

to characterize the negotiation solutions Several computational complexity results are presented and discussed in Section 4 Finally, Section 5 concludes the paper with a discussion on the future work

2 Formal preliminaries

2.1 Stratified knowledge base

In this paper, we consider a propositional languageL over a finite alphabet P and the constants {, ⊥} Symbol

W is used to denote the set of possible worlds, where each possible world is a function from P to {, ⊥}.

A model of a formula φ is a possible world ω which makes φ true, written as ω  φ With Φ being a set of

formulas, Mod( Φ) denotes the set of models of Φ, i.e Mod(Φ) = {ω ∈ W |∀φ ∈ Φ(ω  φ)} By abuse of notation, we use Mod(φ) instead of Mod({φ}) for some φ ∈ L We also use the symbol  to denote the consequence relation, for

example{φ, ψ}  θ means θ is a logical consequence of {φ, ψ}

Let be a binary relation on a non-empty set X ⊆ L The relation  is a total pre-order on X if it satisfies the

following properties:

-∀α ∈ X, α  α;(Reflexivity)

-∀α, β, γ ∈ X, if α  β and β  γ then α  γ;(Transitivity)

-∀α, β ∈ X, α  β or β  α (Totality)

A stratified belief base, sometimes also called ranked knowledge base or prioritized knowledge base, is a belief base K together with a total pre-order  on K Stratified belief base (K, ) can be equivalently defined as a sequence (K , ) = (S1, , S n ), where each S i (i = 1, , n) is a non-empty set and for φ ∈ S i , ψ ∈ S j , φ  ψ iff i ≤ j, i.e when

i ≤ j each formula in S i is more reliable than the formulas of the stratum S j Each subset S iis called a stratum of

K, and index i indicates the level of each formula of S i It is clear that each formula in S iis more reliable than any

formula of the stratum S j for j ≥ i.

Given stratified belief bases (K, ) = (S1, , S n ) and (K, ) = (S

1, , S

m), we say that these belief bases are

equivalent, denoted by (K , ) ≡ (K, ), if m = n and S i ≡ S

i for i = 1, , n Further, a belief set E= {(K

1,  1 ), , (K

n, 

n)} is logically equivalent to a belief set E = {(K1, 1), , (Kn, n)}, denoted E ≡ E, if and only if there exists a permutationπ on the set {1, , n} such that (K i, i)≡ (K

π(i), 

π(i) ) for all i = 1, , n.

Additionally, given any set S , we use S  to denote the cardinality of S

2.2 Computational complexity

In this section we briefly recall some computational complexity classes, including the complexity classΘp

2 (see e.g.,25) We assume the reader familiarity with the classes P, NP and coNP Some important classes in the polynomial

hierarchy are defined in the following

The complexity classesΔp

k,Σp

k, andΠp

k (k is a positive integer) are defined recursively as follows:

- Δp

0 = Σp

0= Πp

0 = P;

- Δp

k+1 = PΣp

k;

- Σp

k+1 = NPΣp

k;

- Πp

k+1 = coNPΣp

k

In addition, the class Θp

2 = Δp

2[O(log n)] is the class of problems ΔP

2 = P NP which can be determined via a

logarithmic number of calls to an NP oracle.

We can now define the hardness and the completeness of different complexity classes as follows:

Definition 1 Let C be a complexity class, and Q be a decision problem We say that Q is C-hard i ff for any decision

problem P belonging to the class C, there is a reduction from P to Q.

For instance, a decision problem can be shown to be NP-hard by identifying a reduction of the SAT (Boolean satisfiability) problem to this problem In general, in order to prove that a problem P is C-hard, we just need to prove that there exists a reduction of a known C-hard problem to P.

Definition 2 Let C be a complexity class, and P be a decision problem We say that P is C-complete i ff P ∈ C and P

is C-hard.

C-complete problems are the most “di fficult” problems in the complexity class C.

3 A model of negotiation for belief merging

This section presents an axiomatic model for merging stratified belief bases by negotiation We introduce the concept of mapping solution, which maps the preferences of agents into the layers, as a vehicle to represent the belief states of agents and their attitudes towards negotiation situations The belief merging process in our model is divided into two stages In the first stage, the stratified belief bases of agents are mapped to their preferences In the second stage, a negotiation between the agents is carried out based on these preferences To this end, a set of rational axioms for negotiation-based belief merging is proposed and a negotiation solution which satisfies the proposed axioms is introduced Finally, the logical properties of a family of merging-by-negotiation operators are discussed

We start the work in this section by considering a set of agentsA = {a1, , a n }, where each agent a i has a stratified

belief base (X i, i ) in which X i⊆ LV, and relationi ⊆ X i × X iis a total pre-order

A negotiation game is a sequence of stratified belief bases together with the integrity constraints presented logically

equivalent to a formula The set of all negotiation games from the set of agentsA in language LVis denoted by gA,LV The negotiation solution is defined as follows:

Definition 3 A negotiation solution is a function f : gA,LV → 2W\{∅} which maps each negotiation game to a

non-empty subset of all possible worlds.

Note that we consider the negotiation solution of any negotiation game as a set of possible worlds instead of a single one

3.1 From stratified belief base to preferences

In this section, we consider several ordering strategies from a given stratified belief base (K, ) = (S1, , S n) where{S1, , S n } is a partition of K w.r.t the total preorder  such that ∀φ ∈ S i , ∀ψ ∈ S j , φ  ψ iff i < j as follows:

−maxsat ordering26: let r MO(ω) =

 +∞ i f ∀S i(ω |= Si),

min {i : ω |= S i } otherwise. whereω ∈ W Then the maxsat ordering

maxsatonW is defined as: ω maxsatωiff r MO(ω) ≤ r MO(ω)

−leximin ordering27: let K i(ω) = {φ ∈ Si : ω |= φ} Then the leximin ordering leximin onW is defined as:

ω leximin ωiff K i(ω) = K i(ω) for all i = 1, , n or there exists j ≤ n such that K j(ω) < K j(ω) and

K i(ω) = Ki(ω) for all i < j

−vector ordering : let v i(ω) =



1 i f ω |= S i,

0 otherwise..

Then the vector orderingvectoronW is defined as: ω vectorωiff v i(ω) = v i(ω) for all i = 1, , n or there exists

j ≤ n such that v j(ω)< v j(ω) and vi(ω) = vi(ω) for all i < j.

Given a preorder on W, the associated strict partial order ≺ is defined by ω ≺ ωiff ω  ωbut notω ω

An orderingYis more specific than anotherXiff ω ≺Xωimpliesω ≺Yω We have the relation among the above ordering strategies as follows:

Proposition 1 Letω, ω∈ W, K be a stratified belief base The following relationships hold:

1)ω ≺maxsatωimpliesω ≺vectorω,

2)ω ≺maxsatωimpliesω ≺leximinω.

3.2 Negotiation on the preferences

Clearly, given a stratified belief base and an ordering strategy, one can easily partitionW into the classes of possible

worlds (W1, , W k ) where W i  ∅, i = 1, , n Therefore, for each possible world we can determine the unique class which contains this possible world We define the index function as follows:

Definition 4 Given a total preorder  on W and X ⊆ W, the index function l of  over X is defined as: lX :W →

N+, where for anyω, ω∈ X:

1) lX(ω) = 1 if ω ∈ min(X, ),

2) lX(ω) = l

X(ω) iff ω  ωandω ω,

3) lX(ω) ≤ lX(ω) iff ω  ω,

4) Ifω ≺ ωthen there existsω ∈ X such that lX(ω )= lX(ω) + 1 and if ω≺ ω then there exists ω ∈ X such that

lX(ω )= l(ω) − 1

5) ∀ω ∈ W, ω  X, lX(ω )= max{lX(ω)|ω∈ X} + 1

We use the index function lX(ω) to indicate the index of class that ω belongs to w.r.t the constraint X and relation

, i.e lX(ω) = i indicates ω ∈ X i Note that the indexes are consecutive integers up from 1, and the lower the index a possible world has, the more preferred it is, i.e formally, givenω, ω∈ W, lX(ω) ≤ lX(ω) iff ω  ω

Here, we define the solution mapping of a negotiation problem built from the set of preferences{1, , n}

achieved by the stratified belief bases and the ordering strategies, and a set C of models of the integrity constraintμ,

i.e C = Mod(μ) and C is called the feasible set of the negotiation problem, as follows:

Definition 5 Given a negotiation problem G = (C, 1, , n ) where C ⊆ W and 1, , n are the preferences

of agents a1, , a n respectively, a solution mapping of G is a function defined as: m G :W → Nn where m G(ω) =

(l1

C (ω), , ln

C (ω)) for any ω ∈ W

Because the index of each possible world in a preference is unique, we have the following proposition:

Proposition 2 For each negotiation problem G, the solution mapping m G is unique.

Now, we present a set of axioms to characterize the negotiation solutions Firstly, the Pareto E fficiency axiom can

be formulated in our model as follows:

PE If G = (C, 1, , n) is a negotiation problem withω ∈ C, ω∈ W and m G(ω) < mG(ω) thenω f (G).

Note that the Pareto efficiency we mention here is the Strong Pareto Efficiency It states that a solution is Pareto

efficient if no one can improve its utility without causing another utility to be worse off

Next, the Independence of Irrelevant Alternatives axiom can be formulated in our model as follows:

IIA If G1 = (C1, 1, , n ) and G2= (C2, 1, , n ) are negotiation problems with C2 ⊆ C1and f (G1)⊆ C2

then f (G1)= f (G2)

The Symmetry axiom can be formulated as follows:

SYM If G = (C, 1, , n ) and Gπ= (C, π(1), , π(n)) are negotiation problems withπ being any permutation

on{1, n} then m G(ω) = (m Gπ(ω))π

Obviously, the Invariant to equivalent utility representations axiom is applied to Affine spaces, while in this paper, we

are working on ordinal spaces, thus it is omitted

The Upper bound axiom can be formulated as follows:

UB Given a negotiation problem G = (C, 1, , n) and two possible outcomesω1, ω2∈ C If max m G(ω1)<

max m G(ω2) thenω2 f (G).

We say thatω1, ω2∈ W is upper bound equal iff max m G(ω1)= max m G(ω2)

The Upper bound axiom ensures that the negotiation process will be terminated immediately when an agreement

is reached

The Majority axiom can be formulated as follows:

MA Given a negotiation problem G = (C, 1, , n) and outcomesω1, ω2 ∈ C that are upper bound equal, if

{i : ω1iω2} < {i : ω2iω1} then ω1 f (G).

We also say thatω1, ω2 ∈ W are majority equal iff ω1, ω2are upper bound equal and{i : ω1 i ω2} = {i :

ω2iω1}

The Majority axiom states that if two feasible worldsω and ωare upper bound equal, whichever one is voted by the larger number of participants, is preferred to be the solution Although the majority property is studied in a wide range of works in Social Choice as well as Decision-making, it is usually criticized by being affected by the voting paradox However, it is not a serious problem in our work because if the paradox happens, we can take all the feasible

worlds as the outcomes Lastly, the Lower bound axiom can be formulated as follows:

LB Given a negotiation problem G = (C, 1, , n) and two possible outcomesω1, ω2 ∈ C If ω1andω2 are

majority equal and min m G(ω1)< min m G(ω2) thenω1 f (G).

The Lower bound axiom ensures the solution is fair in the sense that the difference between the best and the worst

is minimal

Given a set of possible outcomes S , we use ma x(S , #) to denote the subset of possible outcomes of S which is most

supported by agents w.r.t cardinality Formally, we have:

max(S , #) = {ω ∈ S : ω∈ S ({i : ωi ω} < {i : ω  iω})}

We also denoteG as the set of all negotiation problems

Now, we show the possibility of the set of the above axioms by pointing out a solution based on the idea of the well-known egalitarian solution as follows:

Theorem 1 Let f G:G → 2W\{∅} be a negotiation solution, where

- f G ((C, 1, , n))= arg max ω∈LS min(m G(ω)), where

- LS = max(BS, #), where

- BS = arg min ω∈C (max(m G(ω))).

A negotiation solution f :G → 2W\{∅} satisfies UB, MA and LB iff f = f G

We also see the relation between the negotiation solution f G and the axioms IIA, PE, S Y M as follows:

Proposition 3 The negotiation solution f G satisfies IIA, PE, and SYM.

3.3 Logical properties

Given a negotiation game G = ({(K i, i)|ai ∈ A}, μ) ∈ gA,L V,X i

i is the preference of agent a ionW according to

an ordering strategy X i∈ {maxsat, vector, leximin }, and X = {X1, , X n} Let ΔX

μ(G) be a belief merging operator such that Mod(Δ X

μ(G)) = f G ((Mod(μ),  X1

1 , , X n

n )) We call such operators the Negotiation-based Merging operators.

We need to modify some postulates for belief merging with integrity constraints (28) to accommodate the merging

on the stratified knowledge bases In particular, postulates (IC2) and (IC3) should be modified as follows:

(IC2’) Let∧G = ∧ a i∈A∧φ∈K i φ, if ∧G ∧ μ is consistent then Δ X

μ(G) ≡ ∧G ∧ μ.

(IC3’) Given two negotiation games G = ({(K i, i)|ai ∈ A}, μ) and G = ({(K

i, 

i)|ai∈ A}, μ), (G, G∈ gA,L V), if

μ ≡ μand there exists a permutation π on {1, , n} such that (K i, i) ≡ (K

π(i), 

π(i) ) and X i = X

π(i)for all

i ∈ {1, , n} then Δ X

μ(G)≡ ΔX

μ (G)

Proposition 4 IfΔX

μ(G) is a Negotiation-based Merging operator, thenΔX

μ(G) satisfies (IC0), (IC1), (IC2’), (IC5),

(IC7), (IC8).

If X i∈ {maxsat, vector } for all i then Δ X

μ(G) also satisfies (IC3’).

We also have the relation between negotiation solutions according to the ordering strategies as follows:

Proposition 5 Given a negotiation game G = ({(K i, i)|a i ∈ A}, μ) ∈ gA,L V, if X i , X

i ∈ {maxsat, vector, leximin } and

X i is more specific than X ifor all i = 1, , n, then

f G ((Mod(μ), X1

1 , , Xn

n ))⊆ f G ((Mod(μ), X1

1 , , X n

n )).

4 Computational complexity of belief merging by negotiation

In this sub-section, we discuss the computational complexity of the family of belief merging operators by negotia-tion Firstly, we consider the computational complexity of stratifyingW from a stratified belief base and an ordering strategy According to29, the problems for logical preference representation languages need to be taken into account

as follows:

Definition 6 (29) Given a stratified belief base (K , ), an ordering strategy X, a formula φ and two interpretations ω

andω.

• The COMPARISON problem determines whether ω  X ω.

• The NON-DOMINANCE problem determines whether ω is non-dominated by  X , i.e there is notωsuch that

ω≺X ω.

We have a proposition for the ordering strategies presented in Subsection 3.1 as follows:

Proposition 6 Let (K , ) be a stratified belief base and X be an ordering strategy For X ∈ {maxsat, leximin, vector}

we have:

- COMPARISON is in P;

- NON-DOMINANCE is coNP − complete.

In order to stratifyW, we need to take into account the problem determining all non-dominated interpretations.

This problem is computationally much harder than the NON-DOMINANCE problem To simplify the computation

of our merging operators, we assume thatW is stratified from each stratified knowledge base during an off-line preprocessing stage

Let f be a negotiation solution, and we define the decision problem ME R NEGO( f ) as follows:

Input: a tupleE, μ, φ, X where E = {(K1, 1), , (K n, n)} is a belief set of stratified belief bases, μ and φ are

formulas, X = {X , , X } is set of ordering strategies (X is attached to K, respectively)

Question: Does f (E , μ, X) |= φ hold?

Theorem 2 Let X = {X1, , X n }, where X i ∈ {maxsat, leximin, vector} (i=1, ,n) We have MER NEGO( f ) is

Θp

2− complete.

Proof:

• Membership of MER NEGO( f ) in Θ p

2is demonstrated by the following algorithm:

1 Determine the smallest k such that for all ω |= μ, li(ω) ≤ k by using binary search, resulting in O(log n) calls to a S AT − oracle.

2 Determine the largest m such that for all ω |= μ, max(mod(ω), #) ≤ m, resulting in O(log n) calls to a

S AT − oracle.

3 Determine the largest n such that for all ω |= μ, li(ω) ≥ n by using binary search, resulting in O(log n) calls to a S AT − oracle.

4 Determine whether an interpretation obtained by applying in sequence the above steps is a model ofφ by

one call to a S AT − oracle.

• Hardness is proved by using the Θp

2− complete problem UOCS AT30stated as follows:

Given a set of clausesΦ, decide if all truth-assignments that satisfy a maximum number of clauses in Φ always satisfy the same set of clauses inΦ

We will prove that UOCS AT can be reducible to MER NEGO in polynomial time as follows Suppose we have an instance of UOCS ATΦ = {φ1, , φm } over the variables p1, , p n Let c1, , c m , c m+1 , c m+2be new variables that have not occurred inΦ We define:

- K = {{c1}, , {c m }, {c m+1 }, {¬c m+2}},

- μ = (φ1∨ ¬c1)∧ ∧ (φm ∨ ¬c m)∧ c m+1 ∧ c m+2

It is easy to see that by any ordering strategy in{maxsat, leximin, vector} each belief base K iofK classifies W

into two layers, one being the set of models of K iand the other the set of the rest Moreover, for allω |= μ,

we also haveω |= c m+1andω |= ¬c m+2 Hence, for allω ∈ W we always have max(li(ω)) = 2 = lm+2(ω)

and min(li(ω)) = 1 = lm+1(ω) Therefore, for all ω |= μ, ω ∈ f ((Mod(μ), 1, , m+2)) if and only if

ω ∈ max(Mod(μ), #) if and only if ω satisfies a subset with maximal number of elements of Φ.

Similarly, we consider c1, , c

mbe new variables that have not occurred inΦ and

- K= {{c

1}, , {c

m }, {c m+1 }, {¬c m+2}},

- μ= (φ1∨ ¬c

1)∧ ∧ (φm ∨ ¬c

m)∧ c m+1 ∧ c m+2

We also have: for allω |= μ,ω ∈ f ((Mod(μ), 

1, , 

m, m+1, m+2)) if and only ifω ∈ max(Mod(μ), #) if and only ifω satisfies a subset with maximal number of elements of Φ

Now we define:

- K∗= KK,

- μ∗= μ ∧ μ,

- φ = (c1≡ c

1)∧ ∧ (c m ≡ c

m)

Clearly, f ((Mod(μ∗), ∗

1, ∗

m+2 )) “merges” each model on f ((Mod(μ), 1, m+2 ) on Var(K)

Var(μ)

with a model on f ((Mo d(μ), 

1, 

m, m+1, m+2 )) on Var(K)

Var(μ) Therefore f (K∗, μ∗, X) |= φ if

and only if all truth assignments that satisfy a maximum number of clauses inΦ always satisfy the same set of clauses inΦ

Thus, MEG NEGO isθp

2− hard.

Therefore, MEG NEGO hasθp

2membership and MEG NEGO isθp

2− hard It concludes that MEG NEGO is

θp − complete.

5 Conclusion

This paper proposed a two-stage model-based approach for belief merging by negotiation The first stage lets each agent build its own preference on the set of possible outcomes from its stratified belief base and an ordering strategy The second stage allows the agents to negotiate with each other based on the constructed preferences to reach agree-ment as the result of merging A set of rational axioms for merging by negotiation is proposed and analyzed and a negotiation solution that satisfies these axioms is identified Especially, several significant computational complexity results are also presented, evaluated, and discussed Although this paper has addressed the problem of belief merg-ing by negotiation in both axiomatic and constructive models, it is necessary to consider the strategic model in the negotiation stage Moreover, the complexity of the merging process in the presence of strategic behaviours is also an interesting topic to be investigated We will explore these open issues in future works

Acknowledgements

This study was fully supported by Science and Technology Development Fund (B) from Vietnam National Univer-sity, Hanoi under grant number QG.14.13 (2014-2015)

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3 A model of negotiation for belief merging

This section presents an axiomatic model for merging stratified belief bases by negotiation We introduce the. .. between the agents is carried out based on these preferences To this end, a set of rational axioms for negotiation- based belief merging is proposed and a negotiation solution which satisfies the proposed... the concept of mapping solution, which maps the preferences of agents into the layers, as a vehicle to represent the belief states of agents and their attitudes towards negotiation situations The