Majority Merging: from Boolean Spaces to Affine Spaces Jean-Franc¸ois Condotta

Majority Merging: from Boolean Spaces to Affine Spaces

Jean-Franc¸ois Condotta

and Souhila Kaci

and Pierre Marquis

and Nicolas Schwind

Abstract. This paper is centered on the problem of merging (possi- bly conflicting) information coming from different sources. Though this problem has attracted much attention in propositional settings, propositional languages remain typically not expressive enough for a number of applications, especially when spatial information must be dealt with. In order to fill the gap, we consider a (limited) first- order logical setting, expressive enough for representing and rea- soning about information modeled as half-spaces from metric affine spaces. In this setting, we define a family of distance-based majority merging operators which includes the propositional majority opera- tor∆^dH,P. We identify a subclass of interpretations of our repre- sentation language for which the result of the merging process can be computed and expressed as a formula.

1 INTRODUCTION

The problem of merging information coming from different sources arises in many applications, e.g. distributed knowledge systems. Due to the multiplicity of the sources providing information, combining them may lead to conflicts. However, one would need to get from a set of (possibly conflicting) belief/goal bases a single consistent belief/goal base representing a global view of the input set, taking into account every source as much as possible.

In the particular framework of propositional logic (PL), merging multiple belief/goal bases has been the point of interest of many works the last two decades [10, 11, 3, 7, 9, 8], a prominent reason being that many problems can be conveniently expressed using PL [4]. However, as PL only deals with true/false statements as units we often need a more expressive language to express information, in particular when the considered variables represent spatial entities, or when they range over an infinite, unbounded, or continuous domain.

For instance, consider the following goal merging problem about a family wanting to purchase a new house. Suppose that the members of family compare accommodations on the basis of their type (flat or house), price and location. As one would expect, there is generally no type, price and location completely satisfying every person, though they need to find a compromise. PL is not sufficient to represent the available information, since two features in question (price and loca- tion) range over continuous domains, the price in a one-dimensional domain and the location in a two-dimensional domain. Yet each one of these domains is naturally associated with a propermetric, ordis- tance. If half of the group would like to pay more than a certain price

1Universit´e Lille-Nord de France, CRIL, CNRS UMR 8188, IUT de Lens F-62307, France, email: [email protected]

2Universit´e Lille-Nord de France, CRIL, CNRS UMR 8188, IUT de Lens F-62307, France, email: [email protected]

3CLLE-LTC, CNRS UMR 5263, 5 All´ees Machado, 31058 Toulouse Cedex 9, France

4Universit´e Lille-Nord de France, Artois, CRIL, CNRS UMR 8188, Facult´e Jean Perrin, F-62307 Lens, France, email: [email protected]

5Universit´e Lille-Nord de France, Artois, CRIL, CNRS UMR 8188, Facult´e Jean Perrin, F-62307 Lens, France, email: [email protected]

Pwhile the other half would prefer a cheap house less than a price pwithp < P, then it is natural to search for a house which price would range betweenpandP.

In this paper, we deal with the representation of such kind of in- formation for which PL is not adequate. We define a class of merging operators that return a consistent set of information from a set of pos- sibly conflicting sources. We consider quite a general setting, allow- ing each variable considered in the merging process to take its value out of a proper domain defined as unions and intersections of half- spaces from finite-dimensional metric affine spaces. The essence of our merging method then exploits the metrics associated to these do- mains. For this purpose, we take our inspiration from distance-based merging operators proposed in the PL framework [10, 8]. We pro- pose a similar procedure for merging belief/goal bases expressed in a more general setting, which can be viewed as a fragment of First- Order Logic (FOL), monadic, typed, without symbol of quantifica- tion or function; furthermore we consider a particular class of inter- pretations. Since a common and natural choice to deal with conflicts among a group is to let the majority decide [10], we focus in this pa- per on the class ofmajority merging operators, which aims at com- puting a consistent result minimizing the global dissatisfaction of the input sources. The solid theoretical background on propositional ma- jority merging operators [10, 8] and their aptitude to satisfy a signif- icant set of rationality postulates motivates our choice. In particular we show that the class of merging operators we define includes the propositional majority merging operator∆^dH,P. We identify a sub- class of interpretations for which the result of the merging can be computed and expressed as a formula.

The rest of the paper is organized as follows: in the next section we provide some necessary mathematical background and we define the syntax and the semantics of our representation language. In Section 3 we define a class of merging operators. In Section 4 we point out how to compute the result of the merging under some restrictions. We conclude and present some perspectives for further work in Section 5.

The results obtained are illustrated through the motivating example sketched above.

2 PRELIMINARIES

2.1 Background on linear spaces

For a given positive integern,Rⁿdenotes the n-dimensional real affine space. Ahyperplane of Rⁿ is a (n−1)-dimensional affine subspace of Rⁿ. For instance, a line (resp. a plane) is a hyper- plane of R² (resp. of R³). Formally, a hyperplane h of Rⁿ is characterized by a vector(h0, . . . , hn)ofRⁿ⁺¹and is defined by the set{(x1, . . . , xn) ∈ Rⁿ |h1x1 +. . .+hnxn = h0}. Let k ∈ {1, . . . , n}. A hyperplanehofRⁿ is said to berectilinearif it is parallel ton−1axes ofRⁿ, formallyhis defined by the set {(x1, . . . , xn) ∈ Rⁿ|xk =h0}for somek ∈ {1, . . . , n}. A hy- perplanehof Rⁿis associated to two closed half-spacesh^≤ and

h^≥, namely two subsets ofRⁿdefined ash^≤ = {(x1, . . . , xn) ∈ Rⁿ | h1x1 +. . .+hnxn ≤ h0} and h^≥ = {(x1, . . . , xn) ∈ Rⁿ|h1x1 +. . .+hnxn ≥ h0}. Thus a closed half-space asso- ciated to a hyperplanehofRⁿis one of the two parts into whichh dividesRⁿ. Anopen half-spaceofRⁿis the complement of a closed half-space inRⁿ. A half-space isrectilinearif it is associated to a rectilinear hyperplane. Aconvex polyhedronofRⁿis the finite in- tersection of closed or open half-spaces ofRⁿ. A rectilinear convex polyhedron ofRⁿ, also called acuboidofRⁿ, is the finite intersec- tion of closed or open rectilinear half-spaces ofRⁿ.

AnormN onRⁿis a mapping fromRⁿtoRsatisfying the fol- lowing properties, for everye1, e2∈Rⁿ, for everyλ∈R:

8<

:

N(e1) = 0iffe1is the zero vector (positive definiteness), N(λ·e1) =|λ|.N(e1)(homogeneity),

N(e1+e2)≤N(e1) +N(e2)(subadditivity)

Given a normNonRⁿ, ametric(or distance)donRⁿis typically derived fromN, namely,dis a mapping fromRⁿ×RⁿtoRdefined for everye1, e2∈Rⁿasd(e1, e2) =N(e1+ (−e2)). LetE⊆Rⁿ, S⊆Eanddbe the metric induced by a norm onRⁿ.Sis called an open setofEiff∀e ∈ S,∃r > 0such that{e⁰ ∈ E|d(e, e⁰) <

r} ⊂S.Sis aclosed setif its complement inEis an open set. The closureofS, denoted◦S, is the smallest closed set containingS.S isboundedif∃λ >0such that∀c1, c2∈S,d(c1, c2)≤λ.

2.2 Syntax and semantics of L

Given the above preliminaries, we now define the syntax and seman- tics of our representation languageL. The alphabet ofLconsists of a finite set of variablesV={x¹, x², . . .}, a finite set of unary pred- icate symbolsP, notedX, Y, . . ., orXjⁱ,iandjbeing two positive integers, the usual logical connectives¬(not),∧(and),∨(or), the usual constant symbols>(true) and⊥(false) and the punctuation symbols⁰(⁰and⁰)⁰.

LetT ={t1, t2, . . .}be a finite set oftypes. We assume that we are given a mappingτ fromV ∪ P toT, namely every symbol of variable or predicate has a type. Anatomis of the form(xⁱ ∈ X), withxⁱ∈ V,X ∈ Pandτ(xⁱ) =τ(X). Aliteralis an atom or its negation. The language Lis inductively defined as follows: every atom is a formula,>,⊥are formulas and given two formulasαand β,(¬α),(α∧β),(α∨β)are also formulas.

We consider a particular classHof interpretations forL. An in- terpretationIfromHis defined as a pairhDI,MIi, where:

• DI is a mapping which associates to every typetj ofT a tuple (nj, Hj, Dj, dj), where:

- njis a positive integer,

- Hjis a finite non-empty set of hyperplanes ofR^nj,

- Dj ⊆ R^nj results from some finite unions, intersections and complements of closed half-spaces associated to hyperplanes ofHj,

- djis a metric onDjinduced by a norm onR^nj.

• MI is mapping which associates to every predicate symbol X ∈ P of type tj a closed half-space MI(X) associated to a hyperplane of Hj, i.e., MI(X) = h^≤ or MI(X) = h^≥ with h ∈ Hj. We require that for every Dj, there exists X1, . . . , Xn ∈ P such thatDj is equal to a finite combina- tion ofMI(X1), . . . ,MI(Xn)where allowed combinations are unions, intersections and complements.

We also define the classHSof interpretations as being the subclass ofHsatisfying the two following conditions, for every typetj:

• everyHjis a finite set of rectilinear hyperplanes ofR^nj,

• for every predicate symbolX of typetj such thatMI(X) = h^≤withh ∈ Hj, there exists a predicate symbolY such that MI(Y) =h^≥, and vice-versa,

• every metricdj is induced by the Manhattan normNM onR^nj defined for every e = (e1, . . . , en_j) ∈ R^nj as NM(e) = P{|ek| |k∈ {1, . . . , nj}}.

For each variablexⁱ∈ Vof typetj∈ T,Djis called thedomain ofxⁱ, also noteddom(xⁱ). LetIbe an interpretation fromH. AnI- assignmentω(onV) is a mapping which associates to each variable xⁱ ∈ Van element of its domain. The semantics of an atom of the form(xⁱ ∈ X) for a givenI-assignmentωis defined as[[(xⁱ ∈ X)]](I)(ω) =true ifω(xⁱ)∈ MI(X), false otherwise.

In the rest of the paper, anI-assignmentωis also considered as the vector(ω(x¹), . . . , ω(x^|V|))∈Q

{dom(xⁱ)|xⁱ∈ V}.

WIdenotes the set of allI-assignments. AnI-assignmentωis an I-modelof a formulaφ(denotedω|=Iφ) iff it makes the formulaφ true. A formula is said to beI-consistentif it admits anI-model. The set ofI-models of a formulaφis denotedM odI(φ). Two formulasφ andψareI-equivalent(denotedφ≡ψ) iffM odI(φ) =M odI(ψ).

Notice that using a formulaφofLin the context of an interpreta- tionI ∈ H, for every variablexⁱof typetjand for every hyperplane h ∈ Hj, if there exists two predicate symbolsX andY such that MI(X) =h^≤andMI(Y) =h^≥, then we can easily express the fact thatxⁱbelongs to one of the closed half-spacesh^≤andh^≥as well as the open half-spaces associated toh, the hyperplanehitself or the whole spaceR^njby use of the logical connectives¬,∧and∨.

Acube is a finite conjunction of literals. A formula is said to be indisjunctive normal form(DNF) if it is a disjunction of cubes (also viewed as a set of cubes). Any formulaφ ∈ Lcan be trans- formed into anI-equivalent DNF formula denotedDN F(φ), in a finite number of steps (though exponential in the size of the formula).

Example.We take the case drafted in the introduction to illustrate an interpretationI^∗ofH. We consider the variablesV ={x¹, x², x³} wherex¹represents the kind of accomodation (house or flat),x²rep- resents its price in thousands of euros (k C) ranging over the interval [0,+∞[andx³ represent the location ranging over the planeR². Since the variables range over different domains, we consider three typesT ={t1, t2, t3}such that for everyi∈ {1,2,3},xⁱhas the typeti. We consider a set of predicatesPcomposed of18symbols denotedX1¹, . . . , X4¹,X1², . . . , X5²,X1³, . . . , X9³, such that for every Xjⁱ∈ P,Xjⁱhas the typeti.

Assume that the family is composed of three members Helen, David and Marc. Figure 1 depicts some regionsA,BandCofR² considered by the family for the location of the accomodation, the half-spaces (i.e., half-planes) defining these regions, denotedh^≤_i and h^≥_i,i ∈ {1, . . . ,9}, themselves associated to9hyperplanes (i.e., lines); the presence of the dashed region will be explained in the next section. The equations of every hyperplane considered are defined as follows:

h1: 2x³1−x³2=−4, h2:x³1= 2, h3:x³2= 4, h4:x³2= 6, h5:x³1+x³2= 11, h6: 2x³1+x³2= 21, h7:x³1= 8, h8:x³2= 6, h9:x³1= 4.

We ask each member of the family to express their wishes in terms of a formula. Helen prefers a flat between90and 100kC in the regionA. David wants a flat below80kC in the regionBor a house

C

1 9

10 9 5 7

4 1 3

2 5 7 8

A B

h^≥₆ h^≤₁

h^≥₁ h^≥₂ h^≤₈

h^≥₈

h^≥₄ h^≥₃

h^≤₃

h^≤₉ h^≥₉

x³₁

h^≤₂ h^≤₄ h^≤₇ h^≥₇ h^≥₅

x³₂

h^≤₅ h^≤₆

Figure 1. Nine linesh1, . . . , h9forming the setH3.

above130kC in the regionA. Marc expects an accomodation in the regionCregardless of its price and of its type.

The interpretationI^∗ =hDI^∗,MI^∗ion which we focus is de- fined as follows:

• x¹ is a one-dimensional binary variable. We setn1 = 1,H1 = {0,1},D1 = {0,1}andd1 the restriction onD1 of the usual distance over reals.0represents the flat and1the house. Notice that elements ofH1are points as hyperplanes ofR¹.

• x² is a one-dimensional variable ranging over the half-space [0,+∞[. We set n2 = 1,H2 = {0,80,90,100,130},D2 = [0,+∞[andd2 the restriction onD2of the usual distance over reals.

• x³is a two-dimensional variable ranging overR². We setn3= 2, H3 ={h1, . . . , h9},D3 =R²andd3the euclidean distance of the plane.

• MI^∗is defined as follows:

- predicate symbols of typet1:

MI^∗(X1¹) =]− ∞,0], MI^∗(X2¹) = [0,+∞[, MI^∗(X3¹) =]− ∞,1], MI^∗(X4¹) = [1,+∞[, - predicate symbols of typet2:

MI^∗(X1²) = [90,+∞[, MI^∗(X2²) =]− ∞,100], MI^∗(X3²) =]− ∞,80], MI^∗(X4²) = [130,+∞[, MI^∗(X5²) = [0,+∞[,

- predicate symbols of typet3:

MI^∗(X1³) =h^≥₁, MI^∗(X2³) =h^≤₂, MI^∗(X3³) =h^≥₃, MI^∗(X4³) =h^≥₄, MI^∗(X5³) =h^≥₅, MI^∗(X6³) =h^≤₆, MI^∗(X7³) =h^≤₇, MI^∗(X8³) =h^≤₈, MI^∗(X9³) =h^≤₉. Notice that for every Dj there exists a formula φ such that M odI^∗(φ) = Dj. Indeed, for instance,M odI^∗(((x¹ ∈ X1¹)∧ (x¹∈X2¹))∨((x¹∈X3¹)∧(x¹∈X4¹))) ={0,1}=D1. The formulas encoding the information provided by Helen, David and Marc are respectively:

• φ1= (x¹∈X₁¹)∧(x¹∈X₂¹)∧(x²∈X₁²)∧(x²∈X₂²)∧(x³∈ X1³)∧(x³∈X2³)∧(x³∈X3³),

• φ2= ((x¹∈X1¹)∧(x¹∈X2¹)∧(x²∈X3²)∧(x³∈X4³)∧(x³∈ X5³)∧((x³ ∈ X6³)∨(x³ ∈ X7³)))∨((x¹ ∈ X3¹)∧(x¹ ∈ X₄¹)∧(x²∈X₄²)∧(x³∈X₁³)∧(x³∈X₂³)∧(x³∈X₃³)),

• φ3= (x³∈X8³).

3 THE MERGING PROCESS

In this sectionIis any interpretation fromH. Recall thatWIdenotes the set of allI-assignments. Abelief/goal baseis a finite set of for- mulasφ1, φ2, . . .fromL, also viewed as the formula that is conjunc- tion of its elements. Aprofileis a finite multisetK={K1, . . . , Km} ofI-consistent belief/goal bases. A merging operator∆is a mapping which associates to a profileKand aI-consistent formulaICrepre- senting integrity constraints a subset∆IC(K)ofWI.

As in the PL case [8] we define a distance-based merging method of a profile via a three-step process: we first define the distanced between twoI-assignmentsω, ω⁰∈ WIas follows:

d(ω, ω⁰) =X

{dj(ω(xⁱ), ω⁰(xⁱ))|xⁱ∈V, τ(xⁱ) =tj}.

Given anI-assignmentω, its “local distance” to a belief/goal base K is defined as follows:dK(ω) = inf{d(ω, ω⁰) | ω⁰ |=I K}.

Then we use an aggregation function to compute the “global dis- tance” between ω and the profile K. As argued in the introduc- tion, we focus in this paper on the aggregation functionP

which supports the majority point of view of the belief/goal bases [10].

The global distance betweenω and Kis then denoteddK(ω) = P{dK_k(ω)|Kk ∈ K}. Lastly, the result of the merging satisfies

∆IC(K) ={ω∈ ◦M odI(IC)|dK(ω)is minimal}.

In PL the set∆IC(K)is obviously a non-empty set satisfyingIC.

Indeed every formula of PL admits a finite set of models over its set of variables, thus there exists at least one modelωofICfor which the distancedK(ω)reaches a minimum. However, in our classHof interpretations ofL,M odI(IC)is usually an infinite set and it can be the case that∆IC(K)is empty or that someI-assignmentωin it is not anI-model ofIC. To circumvent these problems, we consider I-assignments of ◦M odI(IC) as candidates. This choice ensures

∆IC(K)to be a non-empty set whenM odI(IC)is a bounded set.

Furthermore, it does not question the natural distance requirement w.r.t.IC: eachI-assignment of◦M odI(IC)is at distance0ofIC w.r.t.dK. The following proposition holds:

Proposition 1. (1) AssumeM odI(IC)is a bounded non-empty set.

Then∆IC(K)is a closed and bounded non-empty set.

(2) Assume in addition that M odI(IC) is a closed set. Then

∆IC(K)is a subset ofM odI(IC).

Proof. (1) By definition the set◦M odI(IC)is a closed subset of WIand it is bounded sinceM odI(IC)is bounded. Let us prove that

∆IC(K)is a non-empty set. A norm onRⁿis a continuous function.

Since for everyxⁱ∈ Vof typetj,djis a distance induced by a norm and sincedKresults from operations ondjpreserving continuity,dK

is a continuous function onWI. Since◦M odI(IC)is a closed and bounded set, the minimum ofdK on◦M odI(IC)is reached in at least one point (from the Weierstrass extreme value theorem). This means that∆IC(K)is a non-empty subset of◦M odI(IC), it is also closed since it is the inverse image of a closed set (a singleton) by a continuous function.

(2) IfM odI(IC)is a closed set, it is equal to◦M odI(IC). There- fore,∆IC(K)is a subset ofM odI(IC)by definition.

Example (continued). LetK = {K1, K2, K3}with for every Kk ∈ K,Kk = {φk}. LetIC = (x³ ∈ X9³), i.e., the integrity

constraints only bear on the location of the accomodation: available accomodations must be contained in the half-plane h^≤₉. Let D be the closed region of R² which is dashed in Figure 1. Then

∆IC(K) ={ω∈ WI|ω(x¹) = 0, ω(x²)∈[80,90], ω(x³)∈D}.

In PL every variablexⁱ ∈ V is binary, i.e., of typet1 according to our running example. Hence, for two given assignmentsω, ω⁰of PL we simply haved1(ω(xⁱ), ω⁰(xⁱ)) = 0ifω(xⁱ) = ω⁰(xⁱ),1 otherwise; this implies that the distance between two assignments ω, ω⁰ defined asd(ω, ω⁰) =P

{d1(ω(xⁱ), ω⁰(xⁱ))|xⁱ ∈ V}cor- responds to the Hamming distancedH between assignments of PL.

This shows that in the restricted PL setting,∆IC(K)corresponds to the propositional majority merging operator∆^dH,

P

IC (K)[10, 8].

Notice also that in PL the metric is the same for every variable.

Yet in our framework the variables can range over different domains and be associated to different metrics. Due to the incommensurability of these metrics, our majority merging operator does not garantee to give the same importance to every variable. If one would like to overcome this problem, one would need to add an upstream step of normalization of the different metrics considered.

Finally, in PL the set of all possible assignments (propositional worlds in this case) is finite, hence the result of the merging process can be computed in a finite number of steps and be represented as a formula of PL. However, in our caseWI is not finite so that

∆IC(K) cannot always be expressed as a formula of L(see the example above). We intend in the sequel to overcome this problem.

In the rest of the paper,IC∗will denote a cube andK∗a profile in which every belief/goal base is a cube. We first point out two prelim- inary results. The first one states that computing∆IC_∗(K∗)comes down to computing componentwise some∆ⁱ_IC∗(K∗) ⊆ dom(xⁱ) for everyxⁱ ∈ V. The second result exploits the first one and pro- vides a generic method to compute∆IC(K)in a piecewise fashion, whenIC and formulas ofKare given in DNF (which can be as- sumed without loss of expressiveness).

Let xⁱ a variable of type tj. Given a cube φ, the pro- jection of φ on xⁱ, denoted φⁱ, is the conjunction of lit- erals appearing in φ bearing on xⁱ; Sⁱ(φ) denotes the set T{MI(X) | (xⁱ ∈ X)is a literal ofφⁱ} ∩ T

{dom(xⁱ) \ MI(X)| ¬(xⁱ∈X)is a literal ofφⁱ}.∆ⁱIC_∗(K∗)is defined in a three-step process. The i-local distancedⁱ_S between an elemente of dom(xⁱ) and a subset S of dom(xⁱ) is defined as dⁱ_S(e) = inf{dj(e, e⁰) | e⁰ ∈ S}. The i-global distance dⁱK_∗ between an element e of dom(xⁱ) and the multiset {Sⁱ(Kk) | Kk ∈ K∗} is defined as dⁱK_∗(e) = P

{dⁱ_Si(K_k)(e) |Kk ∈ K∗}. The sub- set∆ⁱ_IC∗(K∗)ofdom(xⁱ)is then defined as∆ⁱ_IC∗(K∗) = {e ∈

◦Sⁱ(IC∗)|dⁱK_∗(e)is minimal}. The following proposition holds:

Proposition 2. ∆IC∗(K∗) =Q

{∆ⁱ_IC∗(K∗)|xⁱ∈ V}.

Proof. Letω∈ WIandKk∈ K∗. By definition, the local distance betweenωandKkisdK_k(ω) = inf{P

{dj(ω(xⁱ), ω⁰(xⁱ))|xⁱ ∈ V, τ(xⁱ) =tj} |ω⁰|=I Kk}. Yet sinceKkis a cube,M odI(Kk) = Q{Sⁱ(Kk)|xⁱ∈ V}. Hence,

dK_k(ω) =P

{inf{dj(ω(xⁱ), e⁰)|e⁰∈Sⁱ(Kk)} |xⁱ∈ V, τ(xⁱ) =tj}

=P

{dⁱ_Si(K_k)(ω(xⁱ))|xⁱ∈ V, τ(xⁱ) =tj}.

Then we have:

dK_∗(ω) =P {P

{dⁱ_Si(K_k)(ω(xⁱ))|xⁱ∈ V} |Kk∈ K∗}

=P {P

{dⁱ_Si(K_k)(ω(xⁱ))|Kk∈ K∗} |xⁱ∈ V}

=P

{dⁱ_K∗(ω(xⁱ))|xⁱ∈ V}.

For every ω∆ ∈ WI, ω∆ ∈ ∆IC_∗(K∗) iff dK_∗(ω∆) = min{P

{dⁱK_∗(ω(xⁱ))|xⁱ ∈ V} |ω ∈ ◦M odI(IC∗)}. Yet since IC∗is a cube,◦M odI(IC∗) =Q

{◦Sⁱ(IC∗)|xⁱ∈ V}.

Hence,ω∆∈∆IC_∗(K∗)iffdK_∗(ω∆) =P

{min{dⁱ_K∗(e)|e∈

◦Sⁱ(IC∗)} | xⁱ ∈ V}. This means that ∆IC∗(K∗) = Q{∆ⁱIC_∗(K∗)|xⁱ∈ V}.

Proposition 2 shows that computing∆IC∗(K∗)comes down to computing componentwise a set∆ⁱIC_∗(K∗)for everyxⁱ ∈ Vsince the former results from the Cartesian product of the latters. This property holds for our class of majority merging operators since the aggregation functionP

commutes with itself (see the proof of Proposition 2). Contrastingly, arbitration operators [11, 7] using e.g.

M AXas the aggregation function would not satisfy this property, sinceM AXdoes not commute withP

.

In [10] the authors proposed a syntactic characterization of the re- sult of the merging process for propositional majority merging oper- ators. They assumed that every propositional belief/goal base of the profile to be merged is given in DNF. In the following we generalize their approach. For this purpose, we assume now thatICand every belief/goal base ofKare formulas ofLgiven in DNF. We exploit Proposition 2 and provide a generic algorithm to compute∆IC(K) in a piecewise fashion.

Proposition 3.

∆IC(K) =S

{∆IC_∗(K∗)|

IC∗∈IC, K∗∈K1×. . .×Km,

dK(ω)is minimal, for someω∈∆IC∗(K∗)}.

Proof. LetRbe the set on the right side of the equality.

⊆: letω ∈ ∆IC(K). Letω1 |=I K1, . . . , ωm |=I Kmsuch that dK(ω) =P

{d(ω, ωk)|Kk ∈ K}. There exists a cubeIC∗ ∈IC such thatω |=I IC∗and for everyKk ∈ Kthere exists a cube T ∈Kksuch thatωk|=IT. Thusω∈R.

⊇: letω ∈ ∆IC_∗(K∗), withIC∗ ∈ IC andK∗ ∈ K1 ×. . .× Km such thatdK(ω)is minimal. Assuming that there existsω⁰ ∈

∆IC(K)withdK(ω⁰)< dK(ω)would contradict the minimality of dK(ω). Thereforeω∈∆IC(K).

Taking advantage of Propositions 2 and 3, we get Algorithm 1 for computing∆IC(K):

Proposition 4. Assume we are given an algorithm to compute

∆ⁱIC_∗(K∗)for everyxⁱ ∈ V, and letf(IC∗,K∗)be its time com- plexity. Letα= max{|IC|,max{|Kk| |Kk∈ K}}. Then the space and time complexities of Algorithm 1 are respectively inO(|V|α^m) andO(|V|α^mf(IC∗,K∗)).

Proof. Obvious.

Example (continued). φ1, φ3 and IC are cubes and thus are already in DNF. A DNF formula I-equivalent to φ2 is φ⁰2 = {T₁^φ02, T₂^φ02, T₃^φ02}where:

• T₁^φ02 = (x¹ ∈X1¹)∧(x¹∈X2¹)∧(x²∈X3²)∧(x³∈X4³)∧(x³∈ X₅³)∧(x³∈X₆³),

• T₂^φ02 = (x¹ ∈X1¹)∧(x¹∈X2¹)∧(x²∈X3²)∧(x³∈X4³)∧(x³∈ X5³)∧(x³∈X7³),

• T₃^φ02 = (x¹ ∈X₃¹)∧(x¹∈X₄¹)∧(x²∈X₄²)∧(x³∈X₁³)∧(x³∈ X2³)∧(x³∈X3³).

Algorithm 1: Computing∆IC(K) Input : a profileKand a formulaIC Output:∆IC(K)

begin

1

∆IC(K) =∅;

2

dmin= +∞;

3

forallIC∗∈IC,K∗∈K1×. . .×Kmdo

4

forallxⁱ∈ Vdo

5

Compute∆ⁱIC_∗(K∗);

6

Pick up anyωⁱ∈∆ⁱIC_∗(K∗);

7

end

8

ifP

{dK∗(ωⁱ)|xⁱ∈ V} ≤dminthen

9

ifP

{dK_∗(ωⁱ)|xⁱ∈ V}< dminthen

10

∆IC(K) =∅;

11

∆IC(K) = ∆IC(K)∪Q

{∆ⁱIC_∗(K∗)|xⁱ∈V};

12

end

13

end

14

return∆IC(K);

15

end

16

Since|IC×K1×K2×K3|= 3, the main loop of Algorithm 1 (lines 4 to 14) is performed three times. The definition of∆IC(K) previously reported in this paper is given by∆IC(K∗)∪∆IC(K⁰∗) withK∗={K1, T₁^φ02, K3}andK⁰∗={K1, T₂^φ02, K3}.

The algorithm proposed in [10] for computing∆^dH,Pis a special case of Algorithm 1. When all variables ofVare Boolean ones, com- puting∆ⁱIC_∗(K∗)simply consists in electing using strict majority the truth value(s) of the literals bearing onxⁱappearing inK∗.

4 COMPUTING ∆

(K

) UNDER H

In this section we consider interpretations ofHSforLand show that in this context∆ⁱIC_∗(K∗)can be expressed as a formula ofL. In our running example, the considered interpretationI^∗does not belong toHS. So we need to switch to a particular interpretationIS∈ HS. Example (continued). Figure 2 depicts three regionsA⁰,B⁰andC⁰ considered by the family for the location of the accomodation. Each one of these regions is a cuboid ofR²(i.e., a rectangle), thus all the lines defining them are rectilinear hyperplanes as required, i.e., every line ofH3is parallel to the axisx³1orx³2.

Their equations are defined as follows:

h1:x³1= 0, h2:x³1= 2, h3:x³2 = 4, h4:x³1= 6, h5:x³2= 1, h6:x³1 = 10, h7:x³2= 8, h8:x³2= 6, h9:x³1 = 4.

ISrequires that for every predicate symbolXof typetjsuch that MI_S(X) = h^≤withh ∈ Hj, there exists a predicate symbolY such thatMI_S(Y) =h^≥, and vice-versa. Following this condition, we consider18predicate symbolsX1³, . . . , X18³ of typet3.MI_S is defined as follows, for every predicate symbol of typet3, for every j∈ {1, . . . ,9}:

MI_S(X_2j−1³ ) =h^≤_j and MI_S(X_2j³) =h^≥_j.

Let K = {K1, K2, K3}. Let IC∗ be a cube of IC and for every Kk ∈ K, let Tk be a cube of Kk such that by project- ing formulasT1, T2, T3, IC∗onx³, we get respectively four cubes T1³, T2³, T3³, IC∗³bearing onx³, defined as

A’

9

10 9 5 7

4 1 3

2 5 7 8

3

C’

B’

h^≤₂ h^≥₂ h^≤₈

h^≥₈

h^≤₄ h^≥₄ h^≥₃

h^≤₃

h^≤₉ h^≥₉

x³₂

x³₁

h^≤₁ h^≥₁ h^≤₅

h^≥₅ h^≥₇ h^≤₇

h^≤₆ h^≥₆

Figure 2. Nine rectilinear linesh1, . . . , h9forming the setH3.

• IC∗³= (x³∈X17³ ),

• T1³= (x³ ∈X2³)∧(x³∈X6³)∧(x³∈X3³)∧(x³∈X13³ ),

• T2³= (x³ ∈X8³)∧(x³∈X10³)∧(x³ ∈X11³ ),

• T3³= (x³ ∈X15³ ).

Notice that for everyk ∈ {1,2,3},T_k³ is a cube, hence every set S³(Tk³)represents a cuboid ofR²(i.e., a rectangle). In fact, the sets S³(T1³),S³(T2³)andS³(T3³)respectively correspond to the regions A⁰, B⁰ and C⁰. LetK∗ = {T1, T2, T3}. Then∆³_IC∗(K∗) is the closed set of points of◦S³(IC∗)the “closest” ones to regionsA⁰, B⁰andC⁰(i.e., the dashed region in Figure 2).

LetI ∈ HSandxⁱa variable ofVof typetj. For everyKk∈ K∗, recall thatK_kⁱ denotes the conjunction of literals appearing inKk

bearing onxⁱ. For everyl∈ {1, . . . , nj}, letK_kⁱ(xⁱ_l)be the conjunc- tion of literals ofK_kⁱdefined on predicate symbolsXwhereMI(X) is a half-space which the associated hyperplane is orthogonal to the axisxⁱl(namely, the only axis which is not parallel to the hyperplane associated toMI(X)). In our running example, the linesh1andh2

are orthogonals to the axisx³1 but noth3andh7, henceK1³(x³1) = (x³∈X₂³)∧(x³∈X₃³)andK₁³(x³₂) = (x³∈X₆³)∧(x³∈X₁₃³ ).

Algorithm 2 allows us to compute∆ⁱIC_∗(K∗)for a givenxⁱ∈ V of typetj. The key of this algorithm is that inHS,◦Sⁱ(IC∗)is a cuboid, as well asSⁱ(Kk)for everyKk∈ K∗. Hence, for comput- ing∆ⁱ_IC∗(K∗), it is enough, for each axisxⁱ_l, to project the cuboids onxⁱ_l, then to “merge” locally the set of intervals resulting from the projections onxⁱ_l, and lastly to return the Cartesian product of all the results. Notice that this decomposition on each axis is only fea- sible when the metricdjis induced by the Manhattan norm, as it is required byHS. For the sake of clarity, we use the following nota- tion in Algorithm 2. LetR∞=R∪ {−∞,+∞}andα, β∈ R∞, (α, β)denotes the interval in which each bound is either open or closed and[α, β]denotes a (possibly unbounded) closed interval; for a given finite setE⊆R∞, if there existsγ∈Esuch thatγ=−∞, thenmin{γ|γ ∈E}=−∞, and if there existsγ ∈Esuch that γ= +∞, thenmax{γ|γ∈E}= +∞.

Proposition 5. Letxⁱ∈ Vof typetj.

(1) Algorithm 2 computes∆ⁱIC_∗(K∗)inO(nj|V|)time.

(2) There exists a formulaφofLsuch thatSⁱ(φ) = ∆ⁱIC_∗(K∗).

We omit the proof of this proposition for space reasons.

Algorithms 1 and 2 allow us to compute∆IC(K)when for every