On Consensus in Belief Merging

(1)

On Consensus in Belief Merging

(extended version including the proofs of all propositions)

Nicolas Schwind

National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

nicolas-schwind@aist.go.jp

Pierre Marquis

CRIL-CNRS, Universit´e d’Artois, Institut Universitaire de France

Lens, France marquis@cril.fr

Abstract

We define a consensus postulate in the propositional belief merging setting. In a nutshell, this postulate imposes the merged base to be consistent with the pieces of information provided by each agent involved in the merging process.

The interplay of this new postulate with the IC postulates for belief merging is studied, and an incompatibility result is proved. The maximal sets of IC postulates which are consistent with the consensus postulate are exhibited. When satisfying some of the remaining IC postulates, consensus operators are shown to suffer from a weak inferential power. We then introduce two families of consensus operators having a better inferential power by setting aside some of these postulates.

Introduction

In this paper, we are interested in defining consensus operators for aggregating pieces of propositional information.

We consider a set ofncommunicating agents, each of them wanting to refine her own propositional beliefsϕ_i by merging them with the beliefs of the other agents of the group.

Reaching this goal requires first to merge all the belief bases, then to determine how to refine eachϕiwith the result of the merging process. To achieve the first step, many belief merging (BM) operators can be exploited, see e.g., (Lin 1996;

Revesz 1997; Liberatore and Schaerf 1998; Konieczny and Pino P´erez 2002a; Konieczny and Pino P´erez 2011). The second step calls for revision policies, as considered in (Schwindet al.2015; 2016).

Here the focus is laid on agents which are reluctant to change: each agent is ready to accept the merged base, provided that it allows her to refine her prior beliefs ϕ_i but does not question them. Thus, the revision policy which is adopted by each such agent consists in expanding her belief baseϕiby the merged base if the conjunction is consistent, and to keepϕiunchanged otherwise. In order to avoid the latter case, we are interested in definingconsensus operators, i.e., merging operators such that the merged base C that is generated satisfies theconsensus condition:Cis consistent with every input baseϕi that is consistent with the integrity constraintµ. In such a case,Cis said to be a consensus forKunderµ.

Since consensus operators are BM operators, it is im- portant to situate them within the family of BM operators.

This family received much attention in AI and some rationality postulates (the so-called IC postulates) associated with representation theorems have been pointed out (Konieczny and Pino P´erez 2002a). As BM operators, consensus operators are thus expected to satisfy as many IC postulates as possible. Unfortunately, using fully rational merging operators does not lead to compute merged bases which are consensuses in the general case.

Example 1 Consider a set of three US citizens, Alice, Bob, Charlie traveling together to Paris, and planning a trip to Normandy. They all know that they have to take a train at station Gare St Lazare but have different beliefs about the location and availability of the station from CDG Airport.

Thus, Alice (agent1) believes that Gare St Lazare is located at the north of Paris midtown, and at the west. Bob (agent2) has been told that Gare St Lazare is located at the south of Paris midtown, and that it is not reachable from CDG airport in less than 30mn, and Charlie (agent3) believes that it is located at the north of Paris midtown, and is reachable from CDG airport in less than 30mn. Representing the information using three propositional symbols (a: Gare St Lazare is located at the south of Paris midtown,b: Gare St Lazare is located at the east of Paris midtown, andc: Gare St Lazare is reachable from CDG airport in less than 30mn), the belief bases of Alice, Bob, and Charlie are respectively ϕ1={¬a∧ ¬b},ϕ2={a∧ ¬c}, andϕ3={¬a∧c}. Here there are no integrity constraint (µ=>). Suppose one takes advantage of the distance-based merging operator ∆^d^H^,Σ to merge the belief bases of the profileK = {ϕ1, ϕ2, ϕ3} underµ.∆^d^H^,Σis based on the Hamming distance and uses sum as the aggregation function. It is known to satisfy all the IC postulates (Konieczny and Pino P´erez 2002a), thus it appears prima facie as a good candidate for this job. However,

∆^d_µ^H^,Σ({ϕ₁, ϕ₂, ϕ₃})is equivalent to the base{¬a∧ ¬b},¹ which is not a consensus for{ϕ1, ϕ2, ϕ3}underµsince it conflicts with Bob’s beliefs.

In order to satisfy the consensus condition for this example, a logically weaker merged base must be computed.

Thus, merging operators having quite a low inferential power look at a first glance as suitable consensus operators.

1Details of the computations are reported in Table 1.

(2)

Among them is thebasic merging operator∆^b (Konieczny and P´erez 1999):∆^b_µ({ϕ1, ϕ2, ϕ3})is equivalent to the disjunction of the three basesϕ₁, ϕ₂, ϕ₃, hence to{(¬a∨¬c)∧

(a∨¬b∨c)}, so that it is consistent with every of them. How- ever, this consensus is not convenient since it does not con- vey any new piece of information to any of the three agents (expandingϕ₁,ϕ₂, orϕ₃with it does not change anything:

there is no belief refinement at all). One consequence is that, for instance, it does not entail¬bwhich is not questioned by any of the three agents. Contrastingly, merged bases equivalent to{(¬a∨ ¬c)∧ ¬b}or to{(¬a∨ ¬c)∧(a∨c)∧ ¬b}

would be much better consensuses, since the first one would lead Bob and Charlie to improve their beliefs and the second one would lead each of the three agents to improve her beliefs. This example shows that one must take care of the inferential power of the consensus operators which are used.

The very objective of this paper is to determine the ex- tent to which the consensus condition is compatible with the IC postulates, to point out some consensus operators, and to delineate borders of the trade-off between the rationality conditions and the inferential power offered by such operators. The main contributions are as follows. First of all, a consensus postulate is formally defined. The interplay of this new postulate with the “standard” IC postulates for belief merging is studied. We show that the consensus postulate is incompatible with the conjunction of the IC postulates(IC2)and(IC6), while compatible with each of them taken separately. Since(IC2) is more central to BM than (IC6), the focus is laid on consensus operators satisfying (IC2). We also investigate how our consensus postulate interacts with the majority, arbitration and disjunction postulates. Then we show that the consensus operators satisfying (IC0)and(IC8)have a weak inferential power. Thus setting aside(IC8)is necessary for getting consensus operators satisfying(IC0)and(IC2), and which are not too weak from the inferential standpoint. On this ground, we introduce two families of consensus operators. The first one is composed of the distance-based operators relying on a Pareto aggregation.

Such operators satisfy all IC postulates but(IC6)and(IC8).

The second one gathers consensus operators∇induced by merging operators ∆, and can be viewed as a generaliza- tion of the family of arbitration operators from (Liberatore and Schaerf 1998). When the underlying BM operator∆is an IC one, the corresponding consensus operator ∇is en- sured to satisfy all IC postulates but(IC5),(IC6)and(IC8).

We show how the expected consensuses for Example 1 as given above can be computed using some of the introduced consensus operators, and provide comparative results on the behaviour of these operators from the inferential standpoint.

A Glimpse at Propositional Merging

LetLP be a propositional language built up from a finite set of propositional variablesP and the usual connectives.

⊥(resp.>) is the Boolean constant always false (resp. true).

An interpretation is a mapping fromPto{0,1}, denoted by a bit vector whenever a strict total order onP is specified.

The set of all interpretations is denotedW.[ϕ]denotes the

set of models of the formulaϕ, i.e.,[ϕ] ={ω ∈ W |ω |= ϕ}.|=denotes logical entailment and≡logical equivalence, i.e.,ϕ|=ψiff[ϕ]⊆[ψ]andϕ≡ψiff[ϕ] = [ψ].

An integrity constraint µ is a propositional formula. A belief base ϕ denotes the set of beliefs of an agent, it is a finite and consistent set of propositional formulae, inter- preted conjunctively, so that ϕis identified with the conjunction of its elements. AprofileK = {ϕ1, . . . , ϕn}is a finite, multi-set of belief bases.VK(resp.WK) denotes the conjunction (resp. the disjunction) of all bases from K.t refers to the union of multi-sets. Given a profileKand a for- mulaµ,M C(K, µ)denotes the set of maximal subsets of bases from K which are jointly consistent with µ, that is, M C(K, µ) = {K⁰ ⊆ K |VK⁰∧µ 6|=⊥,∀K⁰⁰ ⊆ K,K⁰ ( K⁰⁰ ⇒ V

K⁰⁰∧µ|=⊥}. Lastly,T

M C(K, µ)denotes the setT

K⁰∈M C(K,µ)K⁰.

A BM operator ∆ is a mapping associating with every integrity constraintµand every profileK ={ϕ1, . . . , ϕ_n} withn ≥ 1, a belief base ∆µ(K)called the merged base.

A set of “standard” properties denoted(IC0)-(IC8)(called IC postulates) expected for BM operators have been pointed out (Konieczny and Pino P´erez 2002a). Operators satisfying them are calledIC merging operators.

(IC0) ∆µ(K)|=µ;

(IC1) Ifµis consistent, then∆µ(K)is consistent;

(IC2) IfV

K ∧µis consistent, then∆_µ(K)≡VK ∧µ;

(IC3) IfK₁≡ K₂andµ₁≡µ₂, then∆µ₁(K1)≡∆µ₂(K2);

(IC4) Ifϕ₁ |=µ,ϕ₂ |=µand∆_µ({ϕ1, ϕ₂})∧ϕ₁is consistent, then∆µ({ϕ1, ϕ2})∧ϕ2is consistent;

(IC5) ∆µ(K1)∧∆µ(K2)|= ∆µ(K1t K2);

(IC6) If∆µ(K1)∧∆µ(K2)is consistent, then∆µ(K1t K2)|= ∆µ(K1)∧∆µ(K2);

(IC7) ∆µ₁(K)∧µ2|= ∆µ₁∧µ2(K);

(IC8) If∆µ₁(K)∧µ2is consistent, then∆_µ₁_∧µ₂(K)|= ∆_µ₁(K)∧µ₂.

In these postulates, when K1 = {ϕ1,1, . . . , ϕ1,n} and K₂ = {ϕ_2,1, . . . , ϕ_2,n},K₁ ≡ K₂ means that there exists a bijection π from {1, . . . , n} to {1, . . . , n} such that for eachi ∈ {1, . . . , n}, we haveϕ1,i ≡ϕ_2,π(i). We refer the reader to (Konieczny and Pino P´erez 2002a) for a detailed explanation about the rationale of these postulates.

The following additional postulates will also be considered in this paper (Konieczny and Pino P´erez 2002a; Everaere et al.2010):(Maj),(Arb)and(Disj), that respectively char- acterize the class of majority, arbitration (Konieczny and Pino P´erez 2002a) and disjunctive (Everaere et al. 2010) operators:

(Maj) ∃n≥1 ∆µ(K1t K2t. . .t K2

| {z }

n

)|= ∆µ(K2).

(Arb) If

∆µ₁({ϕ1})≡∆µ₂({ϕ2})

∆_µ₁_↔¬µ₂({ϕ₁, ϕ₂})≡(µ₁↔ ¬µ₂) µ16|=µ2

µ₂6|=µ₁

(3)

then∆µ1∨µ2({ϕ1, ϕ2})≡∆µ1({ϕ1}).

(Disj) IfW

K ∧µis consistent, then∆µ(K)|=W K.

Among IC merging operators are some distance-based operators, i.e., operators which are based on the selection of some models of the integrity constraint, the “closest”

ones to the given profile. Those operators are characte- rized by a distance d between interpretations and an aggregation function f (Konieczny et al. 2004). They asso- ciate with every integrity constraint µ and every profile K a belief base ∆^d,f_µ (K) which satisfies [∆^d,f_µ (K))] = min([µ],≤^d,f_K ), where ≤^d,f_K is the total preorder over interpretations induced by K defined by ω ≤^d,f_K ω⁰ if and only if d^f(ω,K) ≤ d^f(ω⁰,K), where d^f(ω,K) = f_ϕ_i_∈K{d(ω, ϕ_i)}andd(ω, ϕ_i) = min_ω0|=ϕ_id(ω, ω⁰). Usual distances aredD, the drastic distance anddH, the Hamming distance. Note that some distance-based operators are not IC merging ones (some conditions must be satisfied byf, see (Koniecznyet al.2004)) but taking advantage of usual aggregation functions asΣ,GMaxandGMin(Everaereet al.

2010) lead to IC merging operators.

Example 1 (continued) LetLP be built up from the set of propositional variablesP = {a, b, c},ϕ1 = {¬a∧ ¬b}, ϕ2 = {a∧ ¬c}, ϕ3 = {¬a∧c} and µ = >. We consider Σ, GMax and GMin operators based on the drastic distance on the one hand, the Hamming distance on the other hand. Table 1 gives for each interpretationω ∈ [µ]

the distancesd(ω, ϕ_i)ford∈ {d_D, d_H}andi∈ {1,2,3}, and shows for each interpretation whether it is selected in

∆^d,Σ_µ (K),∆^d,GMax_µ (K)and∆^d,GMin_µ (K)respectively (inter- pretationsωare denoted as binary sequences following the orderinga < b < c).

• ∆^d_µ^D^,Σ(K) ≡ ∆^d_µ^D^,GMax(K) ≡ ∆^d_µ^D^,GMin(K) ≡ ¬a∧

¬b∧c≡ϕ1∧ϕ3.

• ∆^d_µ^H^,Σ(K)≡ ¬a∧ ¬b≡ϕ₁;

• ∆^d_µ^H^,GMax(K)≡ ¬a∧ ¬b∧ ¬c;

• ∆^d_µ^H^,GMin(K)≡ ¬a∧ ¬b∧c≡ϕ1∧ϕ3.

Consensus and Propositional Merging

Aconsensus operatoris a BM operator satisfying the following consensus postulate(CO):

(CO) ∀ϕi∈ K, ifϕi∧µis consistent, thenϕi∧∆µ(K)is consistent.

When ∆ is a merging operator satisfying (CO), the merged base∆_µ(K)is said to be a consensus forKunderµ.

Let us investigate how the consensus postulate interacts with the IC postulates. First of all,(CO)can be viewed as a stronger version of the equity postulate(IC4):

Proposition 1 Every consensus operator satisfies(IC4).

Proof:Let∆be a consensus operator, i.e., it satisfies(CO).

Letµbe a formula, and let ϕ1, ϕ2 be two consistent bases such that ϕ₁ |= µ, ϕ₂ |= µ and∆_µ({ϕ₁, ϕ₂})∧ϕ₁ is consistent. Sinceϕ2∧µis consistent, by(CO)we get that

∆_µ({ϕ₁, ϕ₂})∧ϕ₂is consistent. Hence,∆satisfies(IC4).

Then a key issue is to determine whether(CO)is compatible or not with the IC postulates. It turns out that the answer to it is negative. More precisely:

Proposition 2 There is no BM operator jointly satisfying (IC2),(IC6), and(CO).

Proof: Let K = {ϕ₁, ϕ₂} such that [ϕ₁] = {ω₁} and [ϕ2] = {ω2}, and let µ such that [µ] = {ω1, ω2}.

Towards a contradiction, assume there exists a BM operator ∆ satisfying (IC2), (IC6), and (CO). By (IC2), ∆µ({ϕ2}) ≡ ϕ2, so [∆µ({ϕ2})] = {ω2}.

Furthermore, by (CO), {ω1, ω₂} ⊆ [∆_µ({ϕ1, ϕ₂})].

Hence, [∆µ({ϕ1, ϕ2}) ∧ ∆µ({ϕ2})] = {ω2}, so

∆_µ({ϕ1, ϕ₂}) ∧ ∆_µ({ϕ2}) 6|= ⊥. Then, by (IC6),

∆µ({ϕ1, ϕ2, ϕ2}) |= ∆µ({ϕ1, ϕ2}) ∧ ∆µ({ϕ2}). Thus [∆_µ({ϕ₁, ϕ₂, ϕ₂})] = {ω₂}. However, by(CO) we must also have{ω1, ω2} ⊆[∆µ({ϕ1, ϕ2, ϕ2})]. Contradiction.

The other IC postulates do not jointly conflict with(CO).

Indeed, one can find a BM operator∆satisfying(CO)and all the IC postulates but(IC2). Thetrivial merging operator

∆^tgiven by∆^t_µ(K) ={µ}does the job.

Proposition 3 ∆^t satisfies(CO)and all IC postulates but (IC2).

Proof:

• (CO),(IC0),(IC1),(IC3),(IC5),(IC6),(IC7), and(IC8) are obviously satisfied.

• (IC4)is satisfied:∆^t_µ({ϕ1, ϕ2})∧ϕ2is equivalent toµ∧ ϕ₂. Ifϕ₂|=µ, it is equivalent toϕ₂, which by assumption is consistent (as any belief base considered as input in a profile).

• (IC2)is not satisfied: as a counter-example, just consider the singleton profile K = {{a}} and the integrity constraint µ = b. Since VK ∧µis consistent, (IC2)asks

∆^t_µ(K)to be equivalent toa∧b. But∆^t_µ(K)is equivalent tob.

But∆^tis not an “interesting” operator as it gives up all information from any input profile, even when all belief bases are jointly consistent with the integrity constraint.

On the other hand, one can define consensus operators satisfying(IC2)and “almost” all other IC postulates. More precisely, there exist BM operators ∆satisfying(CO)and all the IC postulates but(IC6). This is the case of thedrastic merging operator∆^d (Konieczny and P´erez 1999) defined by:

∆^d_µ(K) = V

K ∧µ if consistent,

µ otherwise;

and of thebasic merging operator∆^b(Konieczny and P´erez 1999) defined by:

∆^b_µ(K) =







VK ∧µ if consistent,

WK ∧µ ifVK ∧µis inconsistent andW

K ∧µis consistent,

µ otherwise.

(4)

ω ϕ₁ ϕ₂ ϕ₃ Σ GMax GMin P ar ∇^d^D^,f ϕ₁ ϕ₂ ϕ₃ Σ GMax GMin P ar ∇^d^H^,GMin ∇^d^H^,Σ/GMax 000

d=dD

0 1 1

d=dH

0 1 1 • • • •

100 1 0 1 • • 1 0 2 • • •

010 1 1 1 1 1 1

001 0 1 0 • • • • • 0 2 0 • • • • •

110 1 0 1 • • 2 0 2

101 1 1 1 1 1 1

011 1 1 0 1 2 0

111 1 1 1 2 1 1

Table 1: The BM operators∆^d,Σ,∆^d,GMax,∆^d,GMin,∆^{d,P ar}and∇^d,f ford∈ {d_D, d_H}andf ∈ {Σ,GMax,GMin}. The left part (resp. right part) depicts the case whered=dD(resp.d=dH).

Proposition 4 ∆^d and ∆^b satisfy (CO)and all IC postulates but(IC6).

Proof:The fact that∆^dand∆^bsatisfies all IC postulates but (IC6)is shown in (Konieczny and P´erez 1999).

Let us first show that ∆^d satisfies(CO). IfV

K ∧µ is consistent, then∆^d_µ(K) =VK ∧µ. This implies that every base ϕfrom K is consistent with ∆^d_µ(K), hence(CO) is satisfied. IfV

K ∧µis inconsistent, then∆^d_µ(K) =µ, and (CO)is satisfied by definition.

In order to show that∆^b satisfies(CO), we consider the different cases:

• VK ∧µis consistent. Then the proof is similar to the case of∆^d, since∆^b_µ(K) =V

K ∧µ. Every baseϕfromKis consistent with∆^b_µ(K), hence(CO)is satisfied.

• V

K ∧µis inconsistent andW

K ∧µis consistent. In this case,∆^b_µ(K) =W

K ∧µ. Letϕ∈ Kbe such thatϕ∧µis consistent. Then∆^b_µ(K)∧ϕ≡W

K ∧µ∧ϕ≡µ∧ϕis consistent as well, and(CO)is satisfied.

• WK ∧µis inconsistent. This means that everyϕ∈ Kis such thatϕ∧µis inconsistent, and in this case(CO)is trivially satisfied.

The last two propositions can be used to strengthen Propo- sition 2, showing that(IC2),(IC6), and(CO)is aminimal set of incompatible postulates (i.e., while there is no merging operator satisfying the three postulates, one can find merging operators satisfying any proper subset of it). Yet(IC2)is es- sential to belief merging. Indeed, a fundamental expectation of BM is to ensure that the beliefs of a group of agents is at least as (logically) strong as the individual beliefs, when there is no conflict between them. That way, synergetic ef- fects are possible, i.e., some logical consequences of the beliefs of the group may not be among the consequences of any isolated agent. This is what(IC2)captures. On the other hand,(IC6)is sometimes considered as too “strong.”

Loosely speaking it is the counterpart of a very restrictive Pareto condition, considering the aggregation of sets of cri- teria, not only individual ones. This explains why many merging operators of interest proposed in the literature fail to satisfy(IC6), e.g., the formula-based operators (Konieczny 2000) and the quota operators (Everaereet al. 2010); and

this is also why weaker postulates (like (IC6b)introduced in (Everaereet al.2014)²) have been previously considered.

Therefore, as we are interested in consensus operators satisfying(IC2),(IC6)must be set aside.

Let us now take look at how consensus operators relate to(Maj),(Arb)and(Disj)together with the IC postulates (except(IC6)). It turns out that similarly to(IC6),(Maj)appears as antagonistic with(CO)under(IC2)(Proposition 5), and that(IC2),(Maj), and(CO)is a minimal set of incompatible postulates (Proposition 6):

Proposition 5 There is no BM operator jointly satisfying (IC2),(Maj), and(CO).

Proof: Let K = {ϕ₁, ϕ₂} such that [ϕ₁] = {ω₁} and [ϕ2] ={ω2}, and letµsuch that[µ] ={ω1, ω2}. Consider any BM operator ∆ satisfying (IC2)and (Maj). Since ∆ satisfies (IC2), one must have ∆µ({ϕ2}) ≡ ϕ2, so that [∆µ({ϕ2})] ={ω2}. Furthermore, by(CO), for anyn≥1 we get that{ω1, ω₂} ⊆ [∆_µ({ϕ1} t {ϕ2} t. . .t {ϕ2}

| {z }

n

)].

Hence,(Maj)is not satisfied.

Proposition 6 ∆^tsatisfies(Maj).

Proof:The proof is obvious by definition of∆^t.

On the other hand,(Arb)and(Disj)are compatible with (CO)and all IC postulates (except(IC6)):

Proposition 7 ∆^dand∆^bsatisfy(Arb)and(Disj).

Proof:By definition of∆^d and∆^b,(Disj)is obviously satisfied. Let∆^?∈ {∆^d,∆^b}and let us show that∆^?satisfies (Arb). Assume that ∆^? satisfies the four preconditions of (Arb), i.e.,

(1) ∆^?_µ₁({ϕ1})≡∆^?_µ₂({ϕ2});

(2) ∆^?_µ₁_↔¬µ₂({ϕ1, ϕ2})≡(µ1↔ ¬µ2);

(3) µ₁6|=µ₂; (4) µ₂6|=µ₁.

Let us first show that ϕ1 ∧ µ1 is consistent. Toward a contradiction, assume that ϕ₁ ∧ µ₁ is inconsistent. By

2(IC6b)requires that if∆µ(ϕ1)∧. . .∧∆µ(ϕn)is consistent, then∆µ(ϕ1, . . . , ϕn)|= ∆µ(ϕ1)∧. . .∧∆µ(ϕn).

(5)

definition of∆^?,∆^?_µ₁({ϕ1}) ≡ µ1. So by condition (1), µ1 ≡ ∆^?_µ₂({ϕ2}). Yet ∆^?_µ₂({ϕ2}) |= µ2, thusµ1 |= µ2, which contradicts condition (3). We can show thatϕ₂∧µ₂ is consistent using a similar reasoning. So we have that ϕ₁ ∧ µ₁ is consistent and ϕ₂ ∧ µ₂ is consistent. We need to show that ∆^?_µ₁_∨µ₂({ϕ1, ϕ2}) ≡ ∆^?_µ₁({ϕ1}). Yet

∆^?_µ

1({ϕ₁}) ≡ ϕ₁ ∧ µ₁ and ∆^?_µ

2({ϕ₂}) ≡ ϕ₂ ∧ µ₂. So by condition (1) we get that ϕ1 ∧ µ1 ≡ ϕ2 ∧ µ2. So (µ1 ∨ µ2) ∧ ϕ1 ∧ ϕ2 is consistent. Hence,

∆^?_µ₁_∨µ₂({ϕ1, ϕ2}) ≡ (µ1 ∨ µ2) ∧ ϕ1 ∧ ϕ2 ≡ (µ1∧ϕ1∧ϕ2)∨(µ2∧ϕ1∧ϕ2)≡(µ1∧ϕ1∧ϕ2)∨(µ1∧ϕ1)≡ µ₁∧ϕ₁ ≡ ∆^?_µ

1({ϕ1}). This shows that both∆^d and∆^b satisfy(Arb).

According to Proposition 4, the drastic and basic operators appear as relatively “well-behaved” in terms of IC properties. In addition (Proposition 7), they satisfy (Arb) and (Disj). However, wheneverVK ∧µis inconsistent nothing new can be inferred from the merged base∆^d_µ(K)or∆^b_µ(K) that cannot be inferred by at least one the bases from the profile. That is to say, for every baseϕ_i∈ K,ϕ_i|= ∆^d_µ(K)and ϕ_i|= ∆^b_µ(K). A natural question is then whether there exist some consensus operators which satisfy all (or a subset of) IC postulates apart from(IC6)and which offer a reasonable compromise from the inferential standpoint. It turns out that the consensus operators satisfying a few IC postulates do not preserve much information from the input profile:

Proposition 8 Let ∆ be a BM operator satisfying (CO), (IC0)and(IC8). For any profileKand each baseϕi ∈ K, if ϕ_i∧µ 6|= ∆_µ(K)then (i) ϕ_i ∈ T

M C(K, µ) and (ii)

∀ϕj ∈ K,ϕj ∈/T

M C(K, µ)⇒ϕj∧µ|=ϕi∧µ.

Proof:We first prove the following lemma:

Lemma 1 Let∆be an operator satisfying(CO),(IC0)and (IC8). For each baseϕi ∈ K, ifϕi∧µis consistent then ϕ_i∧µ|= ∆_µ(K)or∆_µ(K)|=ϕ_i∧µ.

Proof: Given a set of interpretations M, let φ(M) denote any formula satisfying [φ(M)] = M. Assume ϕi∧µ6|= ∆µ(K)and let us prove that∆µ(K)|=ϕi∧µ. Let ω|=ϕi∧µ∧ ¬∆µ(K). Assume towards a contradiction that

∆µ(K)6|=ϕi∧µ. Then letω⁰ |= ∆µ(K)∧ ¬(ϕi∧µ). By (IC0),ω⁰ |=µsoω⁰ |= ∆_µ(K)∧ ¬ϕi. Sinceω 6|= ∆_µ(K) and ω⁰ |= ∆µ(K), ∆µ(K)∧φ({ω, ω⁰}) ≡ φ({ω⁰}). So by (IC8), ∆_φ({ω,ω0})(K) |= φ({ω⁰}). On the one hand, ω⁰ 6|= ϕ_i so ∆_φ({ω,ω0})(K) ∧ ϕ_i |= ⊥. On the other hand, ω |= ϕ_i so φ({ω, ω⁰})∧ϕ_i 6|= ⊥, and by (CO)

∆_φ({ω,ω⁰_})(K)∧ϕi6|=⊥. Contradiction.

We now prove the proposition. Let ϕi ∈ K, ϕi ∧ µ 6|= ∆µ(K). We prove (i) ϕi ∈ TM C(K, µ).

Assume towards a contradiction that ϕi ∈/ T

M C(K, µ).

Sinceϕi∧µ6|=⊥,M C(K, µ)6=∅. So letS⊆M C(K, µ), ϕ_i ∈/ S. By Lemma 1,∆_µ(K) |= ϕ_i∧µ. Yetϕ_i ∈/ S, so VS ∧µ∧ϕi |= ⊥. So let ϕj ∈ S,∆µ(K) 6|= ϕj∧µ.

We haveϕ_j∧µ 6|=⊥, so by Lemma 1,ϕ_j∧µ|= ∆_µ(K).

We got that ∆µ(K) |= ϕi ∧µ and ϕj ∧µ |= ∆µ(K), therefore ϕ_j ∧ µ |= ϕ_i ∧ µ. This contradicts that VS ∧µ∧ϕi |= ⊥. Therefore, ϕi ∈ T

M C(K, µ). We

now prove (ii). We already got that∆µ(K) |= ϕi∧µ. Let ϕj ∈/ TM C(K, µ). By using the contrapositive of (i) on ϕ_j, we getϕ_j∧µ|= ∆_µ(K). Hence,ϕ_j∧µ|=ϕ_i∧µ. This concludes the proof.

Proposition 8 (i) tells us that using a consensus operator

∆satisfying(IC0)and(IC8), leads the merged base∆_µ(K) to be entailed by each base from the profile (in conjunction with µ) which does not belong to all maximal consistent subsets from M C(K, µ); and (ii) each one these bases (in conjunction with µ) necessarily entails each base that be- longs to all maximal consistent subsets from M C(K, µ).

This shows that consensus operators satisfying (IC0) and (IC8)have a weak inferential power.

The next proposition is a noticeable consequence of these results. We say that a profileKcontains a disagreement over µwheneverV

{ϕi∈ K |ϕi∧µ6|=⊥}is inconsistent:

Proposition 9 Let ∆ be a BM operator satisfying (CO), (IC0)and(IC8). For each profileKand each formulaµ, ifK contains a disagreement overµ, thenW

K⁰∈M C(K,µ)

VK⁰∧ µ|= ∆µ(K).

Proof:In this proof, given a finite setE we denote by|E|

the number of elements in E. Let ∆ be a BM operator satisfying (CO), (IC0) and (IC8), µ be a formula and K be a profile that contains a disagreement over µ. Let K⁰∈M C(K, µ). We need to prove thatVK⁰∧µ|= ∆_µ(K).

By Proposition 8 (i) we already have that for each base ϕi ∈ K⁰, if ϕi ∈/ TM C(K, µ)then ϕi∧µ |= ∆µ(K).

Then it is enough to prove that ifϕi ∈T

M C(K, µ), then eitherϕi∧µ |= ∆µ(K)orV{K⁰ \ϕi} ∧µ ≡VK⁰∧µ.

So letϕ_i ∈ T

M C(K, µ), assume that ϕ_i ∧µ 6|= ∆_µ(K) and let us prove that V

{K⁰ \ϕi} ∧µ ≡ V

K⁰ ∧µ. Yet by hypothesis, K contains a disagreement over µ, so V{ϕj ∈ K |ϕj∧µ6|=⊥}is inconsistent. This means that for each K⁰⁰ ∈ M C(K, µ), there exists a baseϕ_k ∈ K⁰⁰ such that ϕk ∈/ T

M C(K, µ). This is true in particular for K⁰⁰ = K⁰, i.e., there exists a base ϕk ∈ K⁰ such that ϕk ∈/ T

M C(K, µ). And by Proposition 8 (ii) we have that ϕk∧µ|=ϕi∧µ. Hence,V{K⁰\ϕi} ∧µ≡VK⁰∧µ. This shows thatW

K⁰∈M C(K,µ)

VK⁰∧µ|= ∆_µ(K).

Proposition 9 states that in presence of (IC0) and (IC8), the disjunction of all maximal consistent subsets of M C(K, µ)entails the merged base when the profileKcon- tains a disagreement over the integrity constraintsµ. Yet pro- files containing a disagreement over the integrity constraints are the most “interesting” ones in belief merging, as dealing with jointly inconsistent bases is precisely what makes the merging issue a non-trivial one.

Before closing this section, we show that the indecisive- ness of consensus operators gets more critical in presence of additional IC postulates. Let us consider a weakening of (IC6)introduced in (Konieczny and Pino P´erez 2002b):

(IC6’) If∆µ(K1)∧∆µ(K2)is consistent, then∆µ(K1t K2)|= ∆µ(K1)∨∆µ(K2).

Clearly, a BM operator satisfying(IC6)also satisfies(IC6’).

Merging operators satisfying all IC postulates yet repla- cing (IC6) by (IC6’) are called quasi-merging operators

(6)

(Konieczny and Pino P´erez 2002b). Among them are the Max operators (Konieczny and Pino P´erez 2002b), i.e., distance-based operators usingMaxas an aggregation function. Interestingly, although (IC6) is not compatible with (CO)and(IC2), one can find a quasi-merging operator that is a consensus one. Indeed, the drastic merging operator∆^d is a quasi-merging operator (cf. Theorem 87 in (Konieczny 1999)) and Proposition 4 shows that it satisfies(CO). How- ever, as we argued before this operator makes poor use of information conveyed by the input profile. Actually, in presence of(IC2), (IC7) and(IC6’) in addition to(IC0) and (IC8), the result given in Proposition 8 is strengthened:

when dealing with any profile whose belief bases are jointly inconsistent with the integrity constraints, every consensus quasi-merging operator returns a merged base which is entailed by the disjunction of all non-valid bases, once con- joined with the integrity constraints:

Proposition 10 Let ∆ be a merging operator satisfying (CO),(IC0),(IC2),(IC6’),(IC7)and(IC8). IfV

K ∧µis inconsistent, then for each baseϕi ∈ Ksuch thatϕiis not valid, we have thatϕ_i∧µ|= ∆_µ(K).

Proof: In this proof, given a set of interpretation M, let φ(M)denote any formula satisfying[φ(M)] = M. Let∆ be a merging operator satisfying(CO),(IC0),(IC2),(IC6’), (IC7) and (IC8). Let K be such that V

K ∧ µ is inconsistent. Let us first prove that for each baseϕi ∈ Ksuch thatµ 6|= ϕ_i,ϕ_i∧µ |= ∆_µ(K). Letϕ_i be a base fromK such thatµ 6|= ϕiand let us prove thatϕi∧µ |= ∆µ(K).

∆µ(K t {φ({ω⁰, ω⁰⁰})}). Assume thatϕi ∈/ S

M C(K t {φ({ω⁰, ω⁰⁰})}, µ). In this case, by point (i) of Proposition 8 we get thatϕ_i∧µ |= ∆_µ(K t {φ({ω⁰, ω⁰⁰})}), and since ω|=ϕi∧µ,ω |= ∆µ(K t {φ({ω⁰, ω⁰⁰})})holds. Then assume thatϕ_i∈SM C(K t {φ({ω⁰, ω⁰⁰})}, µ). Let us prove thatφ({ω⁰, ω⁰⁰})∈/S

M C(K t {φ({ω⁰, ω⁰⁰})}, µ). Toward a contradiction, assume that φ({ω⁰, ω⁰⁰}) ∈ SM C(K t {φ({ω⁰, ω⁰⁰})}, µ). Yetϕi∈S

M C(K t {φ({ω⁰, ω⁰⁰})}, µ) andϕi∧φ({ω⁰, ω⁰⁰}) ≡ φ({ω⁰⁰}), so this means that for each ϕ_j ∈ K, ω⁰⁰ |= ϕ_j, that is,V

K ∧ µis consistent, which contradicts our initial hypothesis. So now, we have ϕ_i ∈ S

M C(K t {φ({ω⁰, ω⁰⁰})}, µ) and φ({ω⁰, ω⁰⁰}) ∈/ SM C(K t{φ({ω⁰, ω⁰⁰})}, µ). By Proposition 8 we get that ϕ_i∧µ |= ∆_µ(K t {φ({ω⁰, ω⁰⁰}))or φ({ω⁰, ω⁰⁰})∧µ |= ϕi∧µ. In the first case, sinceω |=ϕi∧µ,ω |= ∆µ(K t {φ({ω⁰, ω⁰⁰})})holds. The second case leads to a contradiction, since ω⁰ |= φ({ω⁰, ω⁰⁰})∧µ andω⁰ 6|= ϕi ∧µ.

We have now proved that ω |= ∆µ(K t {φ({ω⁰, ω⁰⁰})}).

Yet we have defined ω such that ω 6|= ∆_µ(K). More- over, ω⁰, ω⁰⁰ |= µ so by (IC2) ∆µ(φ({ω⁰, ω⁰⁰})) ≡ φ({ω⁰, ω⁰⁰}); thusω 6|= ∆_µ(φ({ω⁰, ω⁰⁰})). Hence,∆_µ(K t {φ({ω⁰, ω⁰⁰})}) 6|= ∆µ(K)∨∆µ(φ({ω⁰, ω⁰⁰})). Yet ω⁰ |=

∆_µ(K)∧∆_µ(φ({ω⁰, ω⁰⁰})), so∆_µ(K)∧∆_µ(φ({ω⁰, ω⁰⁰})) is consistent. This contradicts (IC6’). Therefore, we have

proved that

∀ϕi∈ K, µ6|=ϕi⇒ϕi∧µ|= ∆µ(K). (1) We are now ready to prove the proposition. Letϕ_i ∈ K, ϕinot valid. We need to prove thatϕi∧µ|= ∆µ(K). This is trivially true if ϕ_i ∧µ is inconsistent, so assume that ϕi ∧µis consistent. Let ω 6|= ϕi (such an interpretation exists since ϕi is not valid.) Let µ⁰ = µ∨φ({ω}). Note thatV

K ∧µ⁰ is inconsistent sinceV

This concludes the proof.

Let us summarize: first, by Proposition 2, there is no consensus operator satisfying (IC2) and (IC6), so that (IC6) must be set aside. Second, by Proposition 8, we know that one cannot find a consensus operator with a reasonable inferential power which satisfies(IC0)and(IC8)(things getting worse in presence of (IC7),(IC2)and(IC6’), cf. Proposi- tion 10). Yet(IC0)captures a fundamental principle whereas (IC8)could be considered as too strong: it is the BM counterpart of the postulate(R6)in belief revision (Katsuno and Mendelzon 1991), and revision operators of interest yet not satisfying (R6) have been proposed in the literature (Kat- suno and Mendelzon 1991; Benferhatet al.2005) Therefore, in addition to(IC6),(IC8)appears as the best condition to be relaxed to get more interesting consensus operators. This is what we do in the following.

Pareto Consensus Operators

We now focus on a class of consensus operators, called Pareto operators. As distance-based operators, Pareto operators take advantage of a distancedbetween interpretations which defines a distance between an interpretationωand a belief base asd(ω, ϕ) = min_ω0|=ϕd(ω, ω⁰). Every interpre- tationωis then associated with a list of numbers(δ1, . . . , δn) where for eachϕi ∈ K,δi = d(ω, ϕi). Pairs of interpretations are then compared using the Pareto criterion on the list of numbers associated with them. The selected models of the merged base are the “closest” ones to the profile in terms of Pareto optimality:

Definition 1 (Pareto dominance) Given a profileK, a distance d between interpretations and two interpretations ω, ω⁰,ωis said to(weakly) Pareto dominateω⁰ w.r.t.dand K, noted ω ≤^{d,P ar}_K ω⁰, if for each ϕi ∈ K, d(ω, ϕi) ≤ d(ω⁰, ϕ_i).

Pareto operators are then formally defined as follows:

Definition 2 (Pareto operator) Given a distance d between interpretations, the Pareto merging operator based ond, denoted by ∆^{d,P ar}, associates with every formulaµ and every profileKa belief base∆^{d,P ar}_µ (K)which satisfies [∆^{d,P ar}_µ (K)] = min([µ],≤^{d,P ar}_K ).

Proposition 11 For any distancedbetween interpretations,

∆^{d,P ar} satisfies (CO), (IC0), (IC1), (IC2), (IC3), (IC4),

(7)

(IC5)and (IC7). It does not satisfy(IC6),(IC8),(Arb)or (Disj)in the general case.

Proof: (IC0),(IC1)and(IC3)are directly satisfied by definition of∆^{d,P ar}.

(CO): Letϕ_i ∈ Ksuch thatϕ_i∧µis consistent. We need to show thatϕi∧∆^{d,P ar}_µ (K)is consistent. Toward a contradiction, assume thatϕ_i∧∆_µ(K)is inconsistent, i.e., for everyω |= ϕ_i,ω /∈ min([µ],≤^{d,P ar}_K ), that is, there existsω⁰ |= µ∧ ¬ϕi such thatω⁰ <^{d,P ar}_K ω. We have that ω⁰_i ≤ ω_i, or equivalently, d(ω⁰, ϕ_i) ≤ d(ω, ϕ_i).

Yet ω |= ϕi and ω⁰ 6|= ϕi, thus d(ω, ϕi) = 0 and d(ω⁰, ϕi)>0. Contradiction. So∆^{d,P ar}satisfies(CO).

(IC2): Assume thatV

K ∧µis consistent. Letω|=V K ∧µ and ω⁰ |= µ, ω⁰ 6|= V

K. It is enough to show that ω <^{d,P ar}_K ω⁰. Yet for each ϕ_i ∈ K, ω |= ϕ_i, thus ωi =d(ω, ϕi) = 0. On the other hand, for eachϕi ∈ K, ω_i⁰ = d(ω⁰, ϕ_i) ≥ 0, and there existsϕ_j ∈ Ksuch that ω⁰6|=ϕj, thusω_j⁰ =d(ω⁰, ϕj)>0. Hence,ω <^{d,P ar}_K ω⁰. Therefore,∆^{d,P ar}_µ (K)≡V

K ∧µ, showing that∆^{d,P ar} satisfies(IC2).

(IC4): By Proposition 1 and since ∆^{d,P ar}_µ (K) satisfies (CO).

(IC5): Letω |= ∆^{d,P ar}(K₁)∧∆^{d,P ar}(K₂). Letω⁰ |= µ.

We have thatω⁰ 6<^{d,P ar}_K

1 ωandω⁰ 6<^{d,P ar}_K

2 ω. This means that for eachϕ_i ∈ K1,ω_i ≤ω⁰_ior there existsϕ_j ∈ K1

such thatωj < ω⁰_j. Similarly, for eachϕi ∈ K2,ωi ≤ω⁰_i or there existsϕj ∈ K2 such thatωj < ω_j⁰. So for each ϕ_i ∈ K₁t K₂,ω_i ≤ ω⁰_i or there exists ϕ_j ∈ K₂ such thatω_j < ω⁰_j. Hence,ω⁰ 6<^{d,P ar}_K₁_tK₂ ω. So we get thatω|=

∆^{d,P ar}_µ (K1tK2). This shows that∆^{d,P ar}satisfies(IC5).

(IC7): Letω |= ∆^{d,P ar}_µ

1 (K)∧µ₂. For everyω⁰ |=µ₁, we know thatω⁰ 6<^{d,P ar}_K ω. In particular, for everyω⁰|=µ1∧ µ₂, we know thatω⁰ 6<^{d,P ar}_K ω. Hence,ω|= ∆^{d,P ar}_µ₁_∧µ₂(K).

Therefore,∆^{d,P ar}satisfies(IC7).

The fact that∆^{d,P ar} does not satisfy(IC6)comes from Proposition 2 and since∆^{d,P ar}satisfies(CO)and(IC2).

To show that∆^{d,P ar}does not satisfy(IC8), consider the Pareto operator∆^d^D^{,P ar} based on the drastic distance, the profileK={ϕ1, ϕ2, ϕ3}such that[ϕ1] ={ω1, ω2},[ϕ2] = {ω1, ω3},[ϕ3] = {ω3},µ1 =>and[µ2] = {ω2, ω3}. We have that∆_µ₁(K)≡ϕ₂, so∆_µ₁(K)∧µ₂is consistent. Yet [∆µ₁(K)∧µ2] ={ω3}and[∆µ₁∧µ2(K)] ={ω2, ω3}, thus

∆_µ₁_∧µ₂(K)6|= ∆_µ₁(K)∧µ₂. Therefore(IC8)is not satisfied.

To show that∆^{d,P ar}does not satisfy(Arb)or(Disj), consider the Pareto operator∆^d^H^{,P ar} based on the Hamming distance,P = {a, b},K ={ϕ1, ϕ2}withϕ1 = a∧band ϕ₂=¬a∧ ¬b, andµ=>. We get that∆^d_µ^H^{,P ar}(K)≡ >, so∆^d_µ^H^{,P ar}(K)6|=ϕ₁∨ϕ₂, which shows that(Disj)is not satisfied. Furthermore, letµ1 =¬a∨ ¬bandµ2 = a∨b.

We can verify that all preconditions of (Arb) are satisfied: ∆^d_µ^H₁^{,P ar}({ϕ1}) ≡ ∆^d_µ^H₂^{,P ar}({ϕ2}) ≡ a ⇔ ¬b;

∆^d_µ^H₁_⇔¬µ^{,P ar}₂(K) ≡ µ1 ⇔ ¬µ2; µ1 6|= µ2; and

µ2 6|= µ1. However, we have that ∆^d_µ^H₁_∨µ^{,P ar}₂ (K) ≡ >, so ∆^d_µ^H₁_∨µ^{,P ar}₂ (K) 6≡ ∆^d_µ^H₁^{,P ar}({ϕ1}), which shows that (Arb)is not satisfied.

Example 1 (continued) We get that (see Table 1)

∆^d_µ^D^{,P ar}(K)≡(a∧ ¬c)∨(¬a∧ ¬b∧c)≡ϕ₂∨(ϕ₁∧ϕ₃), and ∆^d_µ^H^{,P ar}(K) ≡ (¬a∨ ¬c)∧ ¬b. Both ∆^d_µ^D^{,P ar}(K) and ∆^d_µ^H^{,P ar}(K) are consensuses for K under µ. Using

∆^d^D^{,P ar} leads agents 1 and 3 to refine their beliefs, and using∆^d^H^{,P ar}leads agents2and3to refine their beliefs.

From this example, one can observe that ∆^d^D^{,P ar} and

∆^d^H^{,P ar}are incomparable in terms of inferential power in the general case, since here ∆^d_µ^D^{,P ar}(K) 6|= ∆^d_µ^H^{,P ar}(K) and ∆^d_µ^H^{,P ar}(K) 6|= ∆^d_µ^D^{,P ar}(K). Table 1 also suggests that all three distance-based merging operators have an inferential power stronger than the Pareto operator based on the same distance, i.e., for d ∈ {dD, dH}, ∆^d,Σ_µ (K) |=

∆^{d,P ar}_µ (K)(and similarly for∆^d,GMax_µ (K)and∆^d,GMin_µ (K)).

This is actually always the case for any distancedand any aggregation functionf satisfying the (strict monotonicity) property (Everaereet al.2012), (which is offered by standard aggregation functions asGMax,GMinandΣ):

Definition 3 (strict monotonicity) An aggregation functionf satisfies(strict monotonicity)ifx < y ⇒f(x1, . . . , x, . . . , xn)< f(x1, . . . , y, . . . , xn).

Proposition 12 Letdbe any distance andf be an aggregation function satisfying (strict monotonicity). For every pro- fileKand formulaµ, we have∆^d,f_µ (K)|= ∆^{d,P ar}_µ (K).

Proof:Letdbe any distance,f be an aggregation function satisfying (strict monotonicity), K be a profile and µ a formula. If µis inconsistent, then ∆^d,f_µ (K)is inconsistent by(IC0)and the result trivially holds. So assumeµis consistent. Letωbe an interpretation such thatω6|= ∆^{d,P ar}_µ (K) and let us prove that ω 6|= ∆^d,f_µ (K). By(IC0) and(IC1) there existsω⁰ |=µsuch thatω⁰|= ∆^{d,P ar}_µ (K). This means that for each baseϕi ∈ K,d(ω⁰, ϕi)≤ d(ω, ϕi)and there existsϕ_j ∈ Ksuch thatd(ω⁰, ϕ_i)< d(ω, ϕ_i). Sincefsatis- fies (strict monotonicity), we get thatd^f(ω⁰,K)< d^f(ω,K).

Hence,ω6|= ∆^d,f_µ (K). So∆^d,f_µ (K)|= ∆^{d,P ar}_µ (K).

Example 1 also illustrates that the merged base obtained using Pareto operators (for any of the two distances) is not entailed by the disjunction of all three input bases, as it would be required for consensus operators satisfying(IC0), (IC2)and(IC8), since none of these three bases belong to all maximal consistent subsets ofM C(K, µ)(cf. point (i) of Proposition 8). This shows that while ensuring the consensus condition, Pareto operators have a reasonable inferential power compared to consensus merging operators satisfying (IC0)and(IC8). Especially, Corollary 1 below shows that the Pareto operator based on the drastic distance has an inferential power which is higher thananyconsensus operator satisfying (IC0) and (IC8), when dealing with profilesK containing a disagreement overµ: