aa r X i v : . [ c s . A I] J a n Merging with unknown reliability
Paolo LiberatoreJanuary 8, 2021
Abstract
Merging beliefs depends on the relative reliability of their sources. When un-known, assuming equal reliability is unwarranted. The solution proposed in thisarticle is that every reliability profile is possible, and only what holds according toall is accepted. Alternatively, one source is completely reliable, but which one isunknown. These two cases motivate two existing forms of merging: maxcons-basedmerging and arbitration.
Most of the literature on belief merging concerns sources of the information of equalreliability [41, 43, 13, 22, 36]. Such a scenario occurs, but not especially often. Twoidentical temperature sensors produce readings that are equally likely to be close to theactual value, but a difference in made, age, or position changes their reliability. Twoexperts hardly have the very same knowledge, experience and ability. The reliability oftwo databases on a certain area may depend on factors that are unknown when mergingthem.Merging under equal and unequal reliability are two scenarios, but a third exists:unknown reliability. Most previous work in belief merging is about the first [41, 43, 13,22, 36, 31, 23]; some is about the second [53, 42, 12, 35]; this one is about the third.The difference between equal and unknown reliability is clear when its implicationson some examples are shown. 1xample equal reliability unknown reliabilitytwo experts they have the very sameknowledge, experience andability they differ in knowledge,experience and ability, buthow much of these theyposses is unknowntwo sensors they are of the same kindand are in the same condi-tion (example, temperaturesensors located next to eachother, distance sensors withthe same orientation) they are of different kind, orare in different conditions,and which one is more reli-able in the current situationis not knowntwo databases they cover the very same do-main, and are equally likelyto be correct they cover different do-mains, so that a certainpiece of information mayhave been crucial to one buta detail in the secondThe assumption of equal reliability is quite strong in the example of the two experts;rather, there may be some reason to believe one more than the other; not knowing who,this scenario falls in the case of unknown reliability. For the two sensors and the twodatabases equal reliability is not unlikely, but so is the case of unknown reliability.If reliability is unknown, can it be assumed equal?When merging preferences, yes. When merging beliefs, no.Merging preferences [44, 39, 45] aims at obtaining a result that best reflects thecollective opinion of a group. A common premise is that all members of the group shouldhave the same weight on the final decision, as formalized by the condition of anonymity.In lack of information telling otherwise, equal weights are a valid assumption.A technical example shows why not when merging beliefs instead. Three scenariosare possible: A , B and C ; the two sources of information rank they unlikeliness on a scalefrom 0 to 3, with 0 being the most likely and 3 the least (unlikeliness scales are commonin belief revision [32, 18, 55], in spite of likeliness being more intuitive). The first sourcegrades A as a most unlikely scenario, the second as a most likely; numerically, these areunlikeliness 3 and 0. Both sources grade B as kind of likely (1), and C in the oppositeway of A (0 and 3).scenario unlikeliness according unlikeliness accordingto the first source to the second source A B C B is the most likely scenario, since its overall unlikeliness is minimal: 1 + 1 is lessthan 3 + 0 and 0 + 3.A further piece of information then arrives: the first source is twice as reliable as thesecond. Doubling the numbers coming from the first source changes the plausibility ofthe scenarios: 2cenario unlikeliness according unlikeliness accordingto the first source, weighted to the second source, weighed A × B × C × C now has the same unlikeliness of B (2 × × A (2 × B only. After, it was either B or C . Addinginformation makes the result less precise: before, merging uniquely identified a singlescenario; after, it is undecided between two.This happened because of the addition of some information: the first source is morereliable than the second. This new information is totally consistent with what knownbefore, since what was known before was that the relative reliability of the sources isunknown, meaning that the first could be less, equally or more reliable than the second.This is not a shift of information, only an addition. Yet, it leads to a loss of information.This example is inspired by the “penny z ” of Popper [51, pp. 425–426]: a coin isinitially assumed fair in lack of information indicating otherwise; adding the confirmationof fairness does not change its probability of falling heads or tails. In the interpretationof probability as degree of belief [29], the probabilities are the epistemic state. Addinginformation should alter the epistemic state, but the addition of fairness (which is newinformation) changes nothing.This example was used by Popper against the subjective interpretation of probabili-ties, but relies on the principle of indifference: events of unknown probability are assumedequally probable [33, 57]. The Bertrand paradox [7, 33, 57] shows it is problematic; thecoin example by Popper [51] shows another contradictory aspect of it [25].The belief merging version of the principle of indifference is the assumption of equalreliability in lack of information about the relative reliability of the sources. In thesubjective interpretation of probability, the probability of an event is the degree of beliefin that event happening [29]; in belief merging, the weight of a source is the likelinessof the formulae it provides being true, or at least close to truth [53, 42, 12, 35, 17].The event “formula F is true in the real world” provides a qualitative connection ofprobability with merging. The principle of indifference translates into the assumption ofequal reliability.The probability version of sources of unknown reliability is the lack of knowledge ofthe probability of events. Economists distinguish between risk (known probability) andKnightian uncertainty (unknown probability) [49]. An often-used example is the urncontaining twenty yellow balls and forty balls of another color, which may be either blueor green; these forty are either all blue or all green, but which of the two is not known.This scenario involves both risk (the probability of yellow or not yellow is known) andKnightian uncertainty (the presence of blue balls is unknown).3his urn suggests a way to deal with the problem in belief merging. The probabilityof drawing a yellow ball is always one third, but assuming the same for blue and green isas if the urn contained twenty balls for each color. This may be acceptable for a singledrawn, but an example shows it is not in general. The probability of drawing two ballsof the same color (putting the first ball back in the urn) under the assumption of equalprobability is instead of × + × . The first value is obtained by selecting fromthe nine possible outcomes of probability each (random first ball and random secondball) only the three where the balls are the same color: 3 × = . The second value canbe obtained by considering the second drawn not independent of the first, but also bycalculating the probability under the assumption of forty blue balls: the probability ofthe two balls being both yellow is × , that of being both blue is × . Importantly, thevery same value is obtained for forty green balls instead. Not only this probability holdsin both cases, it resists the addition of information. It holds even if it is later discoveredthat the urn is made in a factory that normally uses forty green balls, and the blue ballversion is a rare collector’s edition.In terms of belief merging, two sources of unknown reliability may have the samereliability, or one may be more reliable than the other. All cases are considered, andonly what holds in all of them is taken. This is analogous to reasoning from multipleprobability distributions [30].How does this solution work in the example of the three scenarios A , B and C respectively ranked [3 , ,
1] and [0 , A decreases. An addition of information (the relative reliability of the sources) leads to anincrease of information (from all three scenarios to only B and C ).Such a likeliness evaluation is not that easy when reliability is encoded numericallyrather than qualitatively. The first source may be 10% more likely than the second,still making B the only most likely scenario. A potentially infinite number of reliabilityindicators are involved. However, it will be proved that in all cases only a finite numberof them need to be considered.Another result in this article is that the disjunction of all maxcons [9, 3, 5, 36, 2, 27, 19]is the result of merging formulae of unknown reliability using the drastic distance. Thisresult invalidates the view that maxcons are unsuitable for merging since they do nottake into account the distribution of information [34, 36]. Rather, they do exactly whatthey should; the problem was with the assumption of equal reliability.Technically, merging is defined by selecting the models at a minimal weighted distancefrom the formulae provided by the sources. The drastic and the Hamming distancesare considered as two relevant examples. This is detailed in the next section. Thefollowing proceed by formalizing the idea of this article: weights represent the unknownreliability, so they may range arbitrarily; all models obtained from some possible weightsare selected. The results obtained in the following sections are:4 merging with arbitrary weights is related to a direct comparison of models; theresult is the same when the distance function has binary codomain, but not in thegeneral case; • the drastic distance has binary codomain; therefore, merging can be done by adirect comparison of models; a consequence is that it is the same as disjoining themaxcons; • the Hamming distance allows for a general existence result: all distances can be ob-tained from suitable formulae; in particular, some formulae require an exponentialnumber of weights; • while the definition of merging with unknown reliability involves a universal quan-tification over weights, only an exponential number of these are really required;also, selection of a model can be checked by comparing it only with m other mod-els at time, where m is the number of models to be checked; when merging twoformulae, this result allows for a graphical representation of merging as the part ofthe convex hull of a set of points that is visible from the origin; • merging with unknown reliability satisfy most of the postulate by Konieczny andPerez [36], including the original version of the arbitration postulate [37, 47]; • with a suitable restriction of the possible weights arbitration by closest pairs ofmodels [41] is recovered.The second-last section of the article briefly considers the case of sources providingmore than one formula. The last discusses the results obtained in this article. In this article, merging is done by minimizing the weighting distance of models obeyingintegrity constraints from the formulae to be merged. The integrity constraints aredenoted µ , the formulae to be merged F , . . . , F m . This is the basic settings for beliefmerge, where each source provides exactly one formula F i ; the case of multiple formulaeis considered in a following section.Models are represented by the set of literals they satisfy; for example I = { a, ¬ b, c } isthe model assigning false to b and true to a and c . The distance between two models I and J is an integer denoted by d ( I, J ). Two intuitive and commonly used distances are thedrastic and the Hamming distance. The drastic distance is defined by dd ( I, J ) = 0 if I = J and dd ( I, J ) = 1 otherwise. The Hamming distance dh ( I, J ) is the number of literalsassigned different truth values by I and J ; for example, dh ( { a, ¬ b, c } , {¬ a, ¬ b, ¬ c } ) = 2,since the two models differ on a and c . Other distances can be defined; they are assumedto satisfy d ( I, I ) = 0 and d ( I, J ) > I = J .Distance extends from models to formulae: regardless of which distance is used, d ( I, F ) is the minimal value of d ( I, J ) for every J | = F . It further extends from a5ormula to a list of them: the distance between a model and a list of formulae is thearray of integers d ( I, F , . . . , F m ) = [ d ( I, F ) , . . . , d ( I, F m )].Merging by weighed distance has been the historically first way of integrating formulaecoming from sources of different reliability. Given a vector of positive integers W =[ w , . . . , w m ], the weighted distance of a model I from the formulae is W · d ( I, F , . . . , F m ),where the dot stands as usual for the scalar product:[ w , . . . , w m ] · d ( I, F , . . . , F m ) = X ≤ i ≤ m w i × d ( I, F i )This product defines a single integer telling the aggregated distance from I to theformulae F i , weighted by the relative reliability of each as represented by the integer w i . Merging selects the models satisfying the integrity constraints µ that have minimalweighted distance from the formulae.∆ d,W, · µ ( F , . . . , F m ) = { I | = µ | W · d ( F , . . . , F m ) is minimal } This function depends on three parameters: a model-to-model distance d , a vector ofweights W = [ w , . . . , w m ] and a mechanism for distilling a single amount out of theseand d ( I, F , . . . , F m ), in this case the scalar product.Fixed weights are used when the relative reliability of the sources is known. Weights W = [1 , . . . ,
1] makes the scalar product the same as a sum, and weighted merge thesame as the usual operators based on the sum of the drastic and Hamming distances.Using the notation by Konieczny and Perez [36]:∆ dd, [1 ,..., , · = ∆ d D , Σ ∆ dh, [1 ,..., , · = ∆ d h , Σ The dh distance was first used in belief revision by Dalal [16]; for this reason, it issometimes called “Dalal distance”. Revesz [54, 53] used it with weights for belief merging,followed by Lin and Mendelzon [42, 43]. Weights reflect the reliability of the sources:the distance from a formula of large weight affects the total more than the distance froma formula of low weight.When reliability is unknown, all possible weight vectors are considered. The set ofall weight vectors is the focus of this article: W ∃ = { [ w , . . . , w m ] | w i ∈ Z > } Nevertheless, other sets of weight vectors are considered. In some scenarios a sourceis correct and the others only provide refining information. For example, a cardiologist,a pneumologist and an allergologist may have contrasting opinions about the state ofa patient; but if the illness is in fact a heart disease then the cardiologist is likely tobe right on everything, for example on the reason of the breathing problems, even if6hat contradicts the pneumologist and the allergologist; their opinion only provide someadditional insight. In the same way, if the problem is an allergy then the allergologist islikely right on everything, and the same for the pneumologist. In these cases, one sourceis totally correct, but which one is unknown. W a = { [ a, , . . . , , [1 , a, , . . . , , . . . , [1 , . . . , , a ] } The value of a for scenarios like that of the three doctors depends on the maximalpossible distance between a model and a formula. For the drastic distance, a = m + 1suffices, where m is the number of formulae to be merged. For the Hamming distance, a = n × m + 1, where n is the number of variables.If each formula is by itself consistent with the integrity constraints, this set of vectorssatisfy the disjunction property: every model of the merged formulae satisfy at least oneof the formulae.Finally, merging with fixed weights falls in this generalization as the set comprisinga single vector. For example, the case of equal reliability is captured by: W = = { [1 , . . . , } In all these cases, a set of weights W. represents all possible reliability the sourcesare considered to have. Three relevant such sets are W ∃ , W a or W = . The set W = isfor equally reliable sources; W ∃ is the other extreme: the reliability of the sources iscompletely unknown. Every W ∈ W. is an encoding of the reliability of the sources. Allof these are plausible alternatives. Every scenario (every model) that is possible whenmerging with some W ∈ W. is possible when merging with W. :∆ d,W., · µ ( F , . . . , F m ) = [ W ∈ W. ∆ d,W, · µ ( F , . . . , F m )This definition is parameterized by a distance d (which could be dd or dh ), a set ofweight vectors W. (usually W ∃ but may also be W a or W = ) and an aggregation function(always the scalar product · in this article). Given some formulae F , . . . , F m , this is howthey are merged under integrity constraints µ . If a model is further from every formula than another, the latter is always preferred tothe former regardless of the weights. The second model dominates the first. Despite theseeming triviality of the concept, a number of relevant results follow: • if a model has minimal weighted distance for some weights, it is not strictly domi-nated by another; • if the codomain is binary, the converse also holds: a model that is not strictlydominated by another has minimal weighted distance for some weights;7 for a ternary codomain, there exists two formulae such that their merge does notinclude an undominated model; • there is a corner case of a ternary codomain with two formulae where undominancestill implies minimality.Rather than defining dominance over models with respect to formulae, the conditionhas a simpler formalization over vectors of integers. It can then be carried over to thedistance vectors of two models. Definition 1
A vector of integers D dominates another D ′ , denoted D ≤ D ′ , if everyelement of D is less than or equal than the element of the same index in D ′ . Strictdominance is the strict part of this ordering: D < D ′ if D ≤ D ′ and D ′ D ′ . If the distance vector of a model is strictly dominated by that of another, the first isnever minimal regardless of the weights. This fact holds because weights are (strictly)positive.
Lemma 1
For every distance d , vector of weights W ∈ W ∃ and model I , if I ∈ ∆ d,W, · µ ( F , . . . , F m ) then d ( J, F , . . . , F m ) < d ( I, F , . . . , F m ) holds for no model J of µ . Proof.
The claim is proved in the opposite direction: d ( J, F , . . . , F m ) < d ( I, F , . . . , F m )entails I ∆ d,W, · µ ( F , . . . , F m ).Since weights are all strictly positive, d ( J, F , . . . , F m ) ≤ d ( I, F , . . . , F m ) entails W · d ( J, F , . . . , F m ) ≤ W · d ( I, F , . . . , F m ) and d ( I, F , . . . , F m ) d ( J, F , . . . , F m )entails W · d ( I, F , . . . , F m ) W · d ( J, F , . . . , F m ). These two consequences togetherare W · d ( J, F , . . . , F m ) < W · d ( I, F , . . . , F m ), which proves that I is not a model ofminimal distance weighted by W , and is not therefore in ∆ d,W, · µ ( F , . . . , F m ).This result is almost trivial, since merging select models that have a minimal valueof the sum of the distances, each multiplied by a positive weight. The converse does nothold in general, but does in a relevant case: when d ( I, F i ) can only be 0 or 1, or moregenerally when the codomain of d has size two. Lemma 2
If the codomain of d ( ., . ) is a subset of cardinality two of Z ≥ , I is a modelof µ and d ( I, F , . . . , F m ) is not strictly dominated by the vector of distances of anothermodel of µ , then there exists W such that I ∈ ∆ d,W, · µ ( F , . . . , F m ) . Proof.
Let { a, b } be the codomain of d , where a < b without loss of generality. The i -thweight of W is w i = m + 1 if d ( I, F i ) = a , and w i = 1 otherwise.For every other model J of µ , the weighted distance of J is proved to be greater thanor equal to that of I . Two cases are possible: either d ( J, F i ) = a for every F i such that d ( I, F i ) = a , or this is not the case for at least one formula F i .In the first case, d ( J, F i ) ≤ d ( I, F i ) for every F i , which implies d ( J, F , . . . , F m ) ≤ d ( I, F , . . . , F m ). Since J does not dominate I by assumption, d ( I, F , . . . , F m ) ( J, F , . . . , F m ) is false, which means that d ( I, F , . . . , F m ) ≤ d ( J, F , . . . , F m ) is true.The distance vectors of I and J are the same. Therefore, multiplying each by W producesthe same result. This proves that I is minimal.The second case is that d ( I, F i ) = a and d ( J, F i ) = b for some F i . If k is the numberof formulae F i such that d ( I, F i ) = a , the weighted distance of I is W · d ( I, F , . . . , F m ) = ( m + 1) × k × a + 1 × ( m − k ) × b = m × k × a + 1 × k × a + 1 × m × b − × k × b = m × k × a + k × a + m × b − k × b< m × k × a + m × b Only d ( J, F i ) = b is known, the distance of J from the other formulae may be either a or b . Assuming it is a for all of them leads to the minimal possible weighted distance,which is: • one formula has distance b , but since d ( I, F i ) = a the weight is m + 1; • the other formulae have all distance a , but – the k − d ( I, F i ) = a have weight m + 1; – the remaining m − k formulae have weight 1.The weighted distance of J is therefore: W · d ( J, F , . . . , F m ) ≥ ( m + 1) × × b + ( m + 1) × ( k − × a + 1 × ( m − k ) × a = m × × b + 1 × × b + m × k × a − m × × a + 1 × k × a − a +1 × m × a − × k × a = m × b + b + m × k × a − m × a + k × a − a + m × a − k × a = m × b + b + m × k × a − a> m × b + m × k × a The weighted distance of I is less than this, as shown above.The two last lemmas imply that the minimal models according to a distance of binarycodomain are exactly the models whose distance vector is not dominated by another. Theorem 1
If the codomain of d ( ., . ) is a subset of cardinality two of Z ≥ , then ∆ d,W ∃ , · µ ( F , . . . , F m ) is the set of all models of µ of minimal distance vector accordingto the dominance ordering. roof. Lemma 1 proves that models of minimal weighted distance are never strictlydominated by any other model of µ . By Lemma 2, if the codomain of d is binary thenevery model that is not strictly dominated has a weight vector W that makes its weighteddistance minimal. Since W ∃ contains all weight vectors, the claim is proved.The next question is whether this condition holds for every fixed-size codomain, orwhether a codomain of size three is sufficient for making some undominated model tobe excluded from merge. The latter is indeed the case in general. A preliminary lemmawill be useful in the sequel. Lemma 3 If µ has three models of distance [3 , , [2 , and [0 , from F and F , then ∆ dh,W ∃ , · µ ( F , F ) does not contain the model at distance [2 , . Proof.
If the model were minimal, the following set of linear inequalities would besatisfiable for some W = [ w , w ]. w × w × ≤ w × w × w × w × ≤ w × w × w × ≤ w , the second w × ≤ w : each weight is at least twicethe other. No positive values may satisfy both.This is almost the proof of the claim, but the codomain of the Hamming distancehas size unbounded, not three. However, a slight change in the definition of the distanceand a comparison of the distance vector are enough. Theorem 2
For some distance d with codomain of size three, there exists I , µ and F , . . . , F m such that I | = µ and I ∆ d,W ∃ , · µ ( F , . . . , F m ) but d ( J, F , . . . , F m ) This is shown on the codomain { , , } and a formula µ with three modelsof distance vectors [0 , , 2] e [3 , 0] from F and F . That such formulae exist islater proved by Lemma 6 for the Hamming distance. To obtain the right codomain { , , } the distance is modified by setting dh ′ ( I, F i ) = 3 for every model K such that dh ( K, F i ) 6∈ { , , } . This change does not affect the distance of the model underconsideration.None of the three distance vectors is strictly dominated by another. However, theprevious lemma shows that [2 , 2] is not minimal for any weight vector.The proof of this counterexample is based on a function of codomain { , , } , whichdoes not look very natural, as { , , } would. But the counterexample does not workin this case, at least for two formulae. The proof of this claim is not much long buttedious, and is omitted. The counterexample holds again with the same codomain butthree formulae. 10 Drastic distance Merging with all possible weights and the drastic distance is the same as disjoining allmaximal subsets of F , . . . , F m that are consistent with µ . This is proved in three steps: • dominance with the drastic distance is the same as the containment of the set offormulae F , . . . , F m satisfied by the models; • the models of the maxcons are the models that are minimal according to thatcontainment; • therefore, the models of the maxcons are exactly the undominated models; by theresults in the previous section, they are the models of minimal weighted distanceaccording to some weights.Maximal consistent subsets (maxcons) have a general definition over lists of sets offormulae, but what is necessary for this article is only the version with a list of twosets, the first comprising a single consistent formula µ and the second F , . . . , F m . Withthis limitation, the (possibly non-maximal) consistent subset and the maximal consistentsubsets are defined as:c on µ ( F , . . . , F m ) = { S ⊆ { µ, F , . . . , F m } | µ ∈ S and S = ⊥} m axcon µ ( F , . . . , F m ) = { S ∈ c on µ ( F , . . . , F m ) |6 ∃ S ′ ∈ c on µ ( F , . . . , F m ) . S ⊂ S ′ } Since µ is consistent, these sets cannot be empty. To establish the correspondencebetween models and maxcons, the subset of formulae satisfied by a model is needed. Definition 2 The set of formulae satisfied by a model I is denoted s ubsat ( I, F , . . . , F m ) = { F i | I | = F i } . The basic brick in the proof construction is that dominance of the drastic distancevectors is the same as containment of the subsets of formulae satisfied by models. Lemma 4 For every pair of models I and J and every formulae F , . . . , F m , the followingtwo conditions are equivalent, where dd is the drastic distance: dd ( I, F , . . . , F m ) ≤ dd ( J, F , . . . , F m )s ubsat ( J, F , . . . , F m ) ⊆ s ubsat ( I, F , . . . , F m ) Proof. The claim is first proved on a single formula F i , and then extended to multipleformulae. In the case of a single formula, the claim is that dd ( I, F i ) ≤ dd ( J, F i ) is thesame as s ubsat ( J, F i ) ⊆ s ubsat ( I, F i ) 11oth dd () and s ubsat () are defined in terms of I | = F i and J | = F i . Each of thesetwo conditions can be true or false, depending on the models I and J and the formula F i . The four cases are: dd ( I, F i ) dd ( J, F i ) s ubsat ( I, F i ) s ubsat ( I, F i ) I | = F i J | = F i { F i } { F i } I | = F i J = F i { F i } ∅ I = F i J | = F i ∅ { F i } I = F i J = F i ∅ ∅ Only in the third cases dd ( I, F i ) ≤ dd ( J, F i ) is false, and this is also the only casewhere s ubsat ( J, F i ) ⊆ s ubsat ( I, F i ) is false. This proves the claim for a single formula.By definition, dd ( I, F , . . . , F m ) ≤ dd ( J, F , . . . , F m ) is the same as dd ( I, F i ) ≤ dd ( J, F i ) for every F i . The claim is proved if s ubsat ( J, F , . . . , F m ) ⊆ s ubsat ( I, F , . . . , F m ) is the same as s ubsat ( J, F i ) ⊆ s ubsat ( I, F i ) for every F i .By definition, s ubsat ( I, F , . . . , F m ) = S ≤ i ≤ m s ubsat ( I, F i ), and the same for J . If s ubsat ( J, F i ) ⊆ s ubsat ( I, F i ) for every F i , then s ubsat ( J, F , . . . , F m ) ⊆ s ubsat ( I, F , . . . , F m ) holds.The converse holds because each s ubsat ( J, F i ) and s ubsat ( I, F i ) may only contain F i . As a result, s ubsat ( J, F i ) s ubsat ( I, F i ) only holds if s ubsat ( J, F i ) = { F i } ands ubsat ( I, F i ) = ∅ . Since s ubsat ( I, F i ) is the only part of s ubsat ( I, F , . . . , F m ) thatmay contain F i , this set does not contain F i . Instead, s ubsat ( J, F , . . . , F m ) contains F i because this formula is in s ubsat ( J, F i ). This proves that if s ubsat ( J, F i ) s ubsat ( I, F i )for some F i then s ubsat ( J, F , . . . , F m ) s ubsat ( I, F , . . . , F m ).Overall, s ubsat ( J, F , . . . , F m ) ⊆ s ubsat ( I, F , . . . , F m ) is equivalent to s ubsat ( J, F i ) ⊆ s ubsat ( I, F i ) for every i . These conditions have been previously proved to be equiv-alent to dd ( I, F i ) ≤ dd ( J, F i ) each. Together, these define dd ( I, F , . . . , F m ) ≤ dd ( J, F , . . . , F m ).This lemma links the dominance ordering under dd and the containment of s ubsat .What is left is a lemma that links the latter with the maxcons. Lemma 5 A model I of µ satisfies some element of m axcon µ ( F , . . . , F m ) if and only if s ubsat ( I, F , . . . , F m ) ⊂ s ubsat ( J, F , . . . , F m ) holds for no model J of µ . Proof. Let I be a model of µ such that s ubsat ( I, F , . . . , F m ) ⊂ s ubsat ( J, F , . . . , F m )holds for no model J of µ . The set s ubsat ( I, F , . . . , F m ) will be proved to be amaxcon. This set is consistent with µ because I satisfies both. It is also maxi-mally so. Otherwise, s ubsat ( I, F , . . . , F m ) ∪ { µ, F i } would be consistent for some F i s ubsat ( I, F , . . . , F m ). Consistency implies the existence of a model J | =s ubsat ( I, F , . . . , F m ) ∪ { µ, F i } . Since J satisfies all these formulae, s ubsat ( J, F , . . . , F m )contains all of them: s ubsat ( I, F , . . . , F m ) ∪ { F i } ⊆ s ubsat ( J, F , . . . , F m ). This impliess ubsat ( I, F , . . . , F m ) ⊂ s ubsat ( J, F , . . . , F m ) for a model J that also satisfies µ , contraryto assumption. 12et J be a model of µ such that s ubsat ( I, F , . . . , F m ) ⊂ s ubsat ( J, F , . . . , F m ). Theclaim is that I is not in any maxcon. By contradiction, let M be such a maxcon. Sinceall its formulae satisfy I , it holds M ⊆ s ubsat ( I, F , . . . , F m ). By assumption, this setis strictly contained in s ubsat ( J, F , . . . , F m ) for some J | = µ . Since J satisfies boths ubsat ( J, F , . . . , F m ) and µ , this other set s ubsat ( J, F , . . . , F m ) is consistent with µ ,contradicting the assumption that M is maximally consistent with µ .The lemma is the final piece for the construction of the link between maxcons anddominance under the drastic distance. Theorem 3 For every consistent formulae µ, F , . . . , F m , ∆ dd,W ∃ , · µ ( F , . . . , F m ) = _ m axcon µ ( F , . . . , F m ) Proof. By definition, all elements of m axcon µ ( F , . . . , F m ) contain µ . Therefore,the models of W m axcon µ ( F , . . . , F m ) all satisfy µ , making Lemma 5 applicable: W m axcon µ ( F , . . . , F m ) are exactly the models I of µ such that s ubsat ( J, F , . . . , F m ) ⊂ s ubsat ( I, F , . . . , F m ) holds for no other model J of µ . This is equivalent to dd ( J, F , . . . , F m ) < dd ( I, F , . . . , F m ) by Lemma 4. Therefore, these models I are themodels of µ that are not strictly dominated by other models of µ . Since the codomainof dd is binary, these are the models of ∆ dd,W ∃ , · µ ( F , . . . , F m ) by Lemma 1.Maxcons have been long used in belief revision [52, 9, 26, 24, 3, 5, 36, 2, 27, 19].Yet, they are sometimes dismissed as “unsuitable for merging” because they do not takeinto account the distribution of information among the sources [34, 36]. This theoremredeems them: maxcons are weighted merge with the drastic distance and unknownreliability. Not only they are suitable for merging, they deal with the common situationwhere the credibility of the sources cannot be assessed.An example clarifies why. If reliability is unknown, a formula ¬ x provided by twosources cannot beat a formula x provided by one source, since the one source may bemuch more reliable than the others as far as it is known. Merging by maxcons collectsas many formulae as possible while retaining consistency; each maxcon may come fromthe most reliable sources, making the number of formulae itself irrelevant. The Hamming distance dh has a codomain of more than two elements. Therefore, theprevious results about binary codomains do not apply. Some existence results are proved: • every given set of distance vectors is obtainable from some formulae µ, F , . . . , F m ; • there are µ, F , . . . , F m such that merging with all possible weight vectors is onlyequivalent to merging with some sets of weights of at least exponential cardinality.Merging with the Hamming distance does not have a simple equivalent form like forthe drastic distance, which selects the models that are not strictly dominated by others.13he same does not hold in general: a model that is undominated may still be excludedin the merging. This was proved abstractly by three distance vectors [3 , , 2] e [0 , Lemma 6 Given some vectors of distances D , . . . , D o of m elements each, all boundedby an integer n , for some formulae µ and F , . . . , F m over n × m variables the vectors ofdistance from the models of µ to F , . . . , F m are exactly D , . . . , D o . Proof. Formulae µ and F , . . . , F m are build over the set of variables { x ij | ≤ j ≤ n, ≤ i ≤ m } . Each formula F i is a conjunction of some of them. F i = x i ∧ · · · ∧ x in Given a model I , its closest model of F i has all variables x i , . . . , x in positive andthe same evaluation of I on the other variables. Therefore, dh ( I, F i ) is the number ofvariables x i , . . . , x in assigned false by I .For each distance vector [ d , . . . , d m ] among the given ones, µ has the following model: [ ≤ i ≤ m {¬ x ij | ≤ j ≤ d i } ∪ { x ij | d i < j ≤ n } For each i , this model has d i negative variables among x i , . . . , x in ; therefore, dh ( I, F i ) = d i . As a result, dh ( I, F , . . . , F m ) = [ d , . . . , d m ]. Since µ has one suchmodel for each of the given distance vectors, the claim is proved.This theorem allows for an easy way for building counterexamples: rather than pro-viding d and µ, F , . . . , F m that have a certain property, one may simply show that thesame property holds on a set of distance vectors. This method was already used toprove that some undominated models are not selected by merging, for some distance. Inparticular, it shows this being the case for the Hamming distance.Another application is the proof that an exponential number of weight vectors haveto be considered when merging. The definition itself requires all weight vectors to betaken into account: a model is selected if and only if it is selected by at least one of theinfinitely many weight vectors in W ∃ . The next lemma shows that at least exponentiallymany have to be considered. It will be later proved that an exponential number suffices. Lemma 7 There exists three formulae µ, F, F ′ on an alphabet of six variables such thatevery W r such that ∆ dh,W r , · ( F, F ′ ) = ∆ dh,W ∃ , · ( F, F ′ ) contains at least two weight vectors. Proof. By Lemma 6, given distances [3 , , 1] and [0 , µ, F , F over six variables such that the three models of µ have these distance vectors.All three distance vectors are minimal for some W ∈ W ∃ . In particular, the firsttwo are minimal for W = [2 , W = [4 , dh,W ∃ , · ( F , . . . , F m ) contains all three models of µ .14ontrary to the claim, a single weight vector is assumed to produce the same result.Since the model at distance [3 , 0] is minimal, its weighted distance is less than or equalto that of the model at distance [1 , , w , w ] · [3 , ≤ [ w , w ] · [1 , w , w ] · [0 , ≤ [ w , w ] · [1 , w × ≤ w + w w × ≤ w + w Since all weights are positive, the left-hand and right-hand sides of these inequalitiescan be added, leading to w × w × ≤ w × w × 2, which is impossible forpositive weights. This proves that no single weight vector produces the same merging of W ∃ .This lemma shows a pair of formulae that can be merged using two weight vectorsrather than infinitely many. Still better, it shows that at least two weight vectors arenecessary. The construction can be replicated over many distinct alphabets of six vari-ables each. Each alphabet doubles the number of necessary weight vectors, leading toexponentiality. Lemma 8 There exists µ, F , . . . , F m such that the size of every W r for which ∆ dh,W r , · ( F , . . . , F m ) = ∆ dh,W ∃ , · ( F , . . . , F m ) is exponential in the size of the formulae. Proof. By Lemma 7, there exists formulae µ, F, F ′ on six variables X such that W ∃ isonly equivalent to sets weight vectors of cardinality greater than or equal to two. Sincethe variables are six, these three formulae are equivalent to formulae of size at most 2 ,a constant.This construction can be replicated on m disjoint alphabets X , . . . , X m of six vari-ables each, leading to m triples µ i , F i , F ′ i of formulae with no shared variables amongdifferent triples. The size of each triple is bounded by a constant.Each µ i has three models. Therefore, V µ i has 3 m models. Each of these models hasa distance vector that is a combination of m subvectors, each being [3 , , 1] or [0 , m .This result relies on an unbounded number of formulae to be merged. With twoformulae, a number of weight vectors linear in the number of variables suffices. Such aresult is straightforward when seen on the graphical representation of merge as shown inthe next section. 15 Finiteness Merging with unknown reliability involves the infinite set of weight vectors W ∃ . However,only an exponential number of vectors matter. Furthermore, minimality of a model canbe recast as a comparison with a bounded number of other models. In summary: • given µ, F , . . . , F m , merging with a (possibly infinite) set of weight vectors is thesame as merging with at most exponentially many weight vectors; • a graphical representation of merging two formulae clarifies the difference betweendominance, minimality under fixed weights and all possible weights; • minimality of the distance from a model of µ to F , . . . , F m for some weight vectorscan be checked by comparing the model with m other models of µ at time; • merging µ, F , . . . , F m with all possible weight vectors can be expressed in termsof merging µ ′ , F , . . . , F m for all formulae µ ′ that are satisfied by at most m + 1models of µ .When merging F , . . . , F m with constraints µ , a set of weight vectors W. may beequivalent to another W r if they produce the same result. This is specific to the formulaeto be merged and integrity constraints: the same sets may not be equivalent whenchanging µ or any of the formulae F i . This is still interesting because the infinite set ofvectors W ∃ can be shown to always be the same as an exponentially sized set. Lemma 9 For every µ , F , . . . , F m and set of weight vectors W. there exists a set ofweight vectors W r of size bounded by an exponential in the size of the formulae suchthat: ∆ d,W r , · ( F , . . . , F m ) = ∆ d,W., · ( F , . . . , F m ) Proof. By definition, I ∈ ∆ d,W., · ( F , . . . , F m ) if W. contains a vector W such that I ∈ ∆ d,W, · ( F , . . . , F m ). In the same way, I ∈ ∆ d,W r , · ( F , . . . , F m ) if W r contains a vector W with the same property. If W r ⊆ W. , every model of ∆ d,W r , · ( F , . . . , F m ) is also amodel of I ∈ ∆ d,W., · ( F , . . . , F m ). The converse can be proved for a suitable choice of W r ⊆ W. .No matter what W. contains, ∆ d,W., · ( F , . . . , F m ) is always a set of models on thealphabet of the formulae µ, F , . . . , F m , and the models on this alphabet are exponentiallymany.By definition, I ∈ ∆ d,W., · ( F , . . . , F m ) if there exists at least one vector W ∈ W. suchthat I ∈ ∆ d,W, · ( F , . . . , F m ). Let (cid:22) be the lexicographic order. The set W r is: W r = [ I ∈ ∆ d,W., · ( F ,...,F m ) min (cid:22) { W | I ∈ ∆ d,W, · ( F , . . . , F m ) , } 16y construction, for every model I ∈ ∆ d,W., · ( F , . . . , F m ) this set W r containsat least a vector W such that I ∈ ∆ d,W, · ( F , . . . , F m ). As a result, I is also in I ∈ ∆ d,W r , · ( F , . . . , F m ).Lemma 8 shows a situation where the necessary number of weight vectors to consideris at least exponential. It is in a way the converse of the last theorem: no more thanexponentially many vectors are required, and in some cases that many are required.An enlightening graphical representation of the distance vectors helps to understandwhich models are selected by merging. Only two formulae F and F are considered alongwith constraints µ . Each model is drawn as a point at coordinates [ d ( I, F ) , f ( I, F )].As a result, the models of µ are points in the first quadrant of the Cartesian plane.Theorem 6 tells that every set of points represent the models of some formula µ , theircoordinates being the Hamming distances from some formulae F and F . ✻ ✲ ❢❢❢❢ ❢ ❢❢❢ The origin is the ideal position since there the distance from both formulae is zero.If a model is in the origin, it is always minimal. Otherwise, models are preferred whenclose to the origin.A model is undominated if it and the origin are the opposite corners of a rectanglefree of other models. This is a simple but crude condition that ensures that a model isclose to the origin. Still, other models may be geometrically closer than it is. ✻ ✲ ❢❢❢❢ ❢ ❢❢❢ Given an arbitrary integer a , a model has weighted distance a if w × d ( I, F ) + w × d ( I, F ) = a . Since d ( I, F ) and d ( I, F ) are coordinates in the plane, this is the equationof a line in the plane characterized by the three parameters w , w , a . In other words,a model has weighted distance a if and only if is on the line. Since the three numbers w , w , a are all positive, the line has negative slope.17 ✲ ❢❢❢❢ ❢ ❢❢❢ ❅❅❅❅❅❅❅❅❅ Every descending line that intersects [ d ( I, F ) , d ( I, F )] is a set of models having thesame weighted distance as I according to some weight vector [ w , w ]. Such a line dividesthe plane into two half-planes; the one where the origin is comprises exactly the modelsthat have a lower weighted distance than I . In the figure, three points are in this half-plane. This means that the two points the line intersects are not minimal according tothe weights [ w , w ].Merging by equal weights finds the line at 45 degrees that crosses some points butdoes not leave any other in the half-plane where the origin lies. With fixed but differingweights the slope of the line changes but the mechanism is the same. It is like shiftingan inclined ruler from the origin until it crosses one of the points. ✻ ✲ ❢❢❢❢ ❢ ❢❢❢ ❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅ The slope of the line is determined by the weights. Arbitrary weights means arbitraryslope. Merging takes the points that are first reached by shifting the ruler, inclinedarbitrarily.The visual representation offers a simpler interpretation of this process: the pointsfirst crossed by a line are the convex hull of the set of models; among those, mergingselects the models that are visible from the origin. It is like the ideal point [0 , 0] werea source of light, the models of µ formed a convex figure, and merging selected theenlightened part of it. If either formula is inconsistent with µ , the figure is closed bylines parallel to the axes, whose points are not selected.18 ✲ ❢❢❢❢ ❢ ❢❢❢ ❆❆❆❆❆ .............................................................................................. ❤❤❤❤❤ This drawing not only explains the mechanism but also motivates it: rather thanrigidly evaluating the distances as shifting an inclined ruler over the plane, mergingselects the models that are qualitatively closer to the ideal condition of zero distance.The two counterexamples previously shown are easy to grasp when seen graphically.The first proved that some non-dominated models are not minimal using the three dis-tance vectors [0 , , 2] and [3 , ❢ ❢ ❢ ✻ ✲❅❅❅❅❅ The counterexample showing that two weight vectors may be necessary for obtainingthe correct result of merging is the converse, in which the central point is closer to theorigin than the line crossing the other two. ❥ ❥ ❥ ✻ ✲▲▲▲▲▲ ❛❛❛❛❛ All three models are enlightened from the origin (visible from it) and therefore min-imal. However, no single line crosses all of them. No single weight vector selects allthree.The graphical representation also shows that two formulae have at most a linearnumber of minimal distance vectors. Indeed, two of them cannot have the same abscissa,as otherwise the one of larger ordinate would not be minimal. As a result, every abscissamay have at most one minimal distance vector. Since the points represent differingliterals, the number of different abscissas is linear. The number of weight vectors thatare necessary to obtain the minimal models is linear too.19 ✲ ❢❢❢❢ ❢ ❢❢❢ ❆❆❆❆❆ .............................................................................................. ❤❤❤❤❤ Since the number of models is finite, the enlightened points can be determined bystandard geometric algorithms: quickhull [4] isolates the segments that form the convexhull; the visible points are then singled out [56]. These algorithms are efficient when thepoints are given explicitly, but for the present case the points derive from propositionalmodels; therefore, they can be exponentially many in the number of variables and in thesize of formulae.The pictorial representation of an undominated model suggests a mechanism forestablishing whether a model is undominated: for every other model, check whether itfalls in the rectangle having the origin and the model to check as the opposite corners.This mechanism can be adapted to arbitrary weights and two formulae by checking twoother models at time instead of one. For m formulae it checks groups of m models each. Theorem 4 For every distance d , formulae µ, F , . . . , F m and model I ∈ Mod ( µ ) , itholds I ∆ d,W ∃ , · µ ( F , . . . , F m ) if and only if I ∆ d,W ∃ , · µ ′ ( F , . . . , F m ) for some formula µ ′ such that I | = µ ′ , µ ′ | = µ and | Mod ( µ ′ ) | ≤ m + 1 . Proof. If I ∆ d,W ∃ , · µ ′ ( F , . . . , F m ) then I is minimal among the models of µ ′ for noweights W ∈ W ∃ . Since µ ′ | = µ , all models of µ ′ are also models of µ . Therefore, I isnot minimal among the models of µ either. This proves one direction of the claim, therest of the proof is for the other.The assumption is that I ∆ d,W ∃ , · µ ( F , . . . , F m ). By definition, this condition holdsif no weight vector W = [ w , . . . , w m ] satisfies W > W · d ( I, F , . . . , F m ) ≤ W · d ( J, F , . . . , F m ) for every J | = µ . This is a system of inequalities in variables W = [ w , . . . , w m ]. By assumption, it is unfeasible. This condition is shown to beequivalent to the existence of a formula µ ′ satisfied by I and at most m other models of µ such that I ∆ d,W ∃ , · µ ′ ( F , . . . , F m ).Every infeasible system of linear inequalities in m variables that contains more than m + 1 is redundant: some of its inequalities are implied by the others [11]. This is anobvious consequence of a theorem by Motzkin [48, Theorem D1] on minimal unsatis-fiable system of inequalities, in modern terms called irredundant infeasible systems ofinequalities or IIS [1]. The theorem tells that every irredundant infeasible system of l inequalities has a coefficient matrix of rank l − 1. Since the rank is bounded by both thedimensions of the matrix, it holds l − ≤ m , meaning that l ≤ m + 1: an irredundantinfeasible system of inequalities in m variables contains at most m + 1 inequalities. The20riginal set of inequalities is unfeasible if and only if one of its subsets of at most m + 1inequalities is infeasible. The theorem holds for non-strict inequalities, but a simplechange turns strict inequalities into strict equalities while preserving both infeasibilityand the coefficient matrix [48, Section 67][28, Theorem 4].In the present case, the k ≤ m + 1 inequalities that form the infeasible and irre-dundant subset of the original system of inequalities comprises some w i > W · d ( I, F , . . . , F m ) ≤ W · d ( J, F , . . . , F m ). The second kind of inequalities are all satis-fied by W = [0 , . . . , w i > k ≤ m + 1 inequalities is among w i > 0, and the inequalities of theform W · d ( I, F , . . . , F m ) ≤ W · d ( J, F , . . . , F m ) are no more than m . If J , . . . , J o arethe models in the right-hand side of these o ≤ m inequalities, the system comprising W > W · d ( I, F , . . . , F m ) ≤ W · d ( J i , F , . . . , F m ) for i = 1 , . . . , o is infeasible.This defines I ∆ d,W ∃ , · µ ′ ( F , . . . , F m ) where µ ′ is the formula satisfied exactly by I and J , . . . , J o .This theorem allows reformulating merging in terms of the groups of m + 1 modelsof µ . Theorem 5 For every distance d and formulae µ, F , . . . , F m , it holds: ∆ d,W ∃ , · µ ( F , . . . , F m ) = \ | Mod ( µ ′ ) |≤ m +1 ,µ ′ | = µ { I | I = µ ′ } ∪ ∆ d,W ∃ , · µ ′ ( F , . . . , F m ) Proof. If I ∆ d,W ∃ , · µ ( F , . . . , F m ) then I ∆ d,W ∃ , · µ ′ ( F , . . . , F m ) for some µ ′ such that | Mod ( µ ′ ) | ≤ m +1, µ ′ | = µ and I ∈ µ ′ by Theorem 4. Since I is a model of µ ′ , it does notbelong to { I | I = µ ′ } either. Therefore, it is not in { I | I = µ ′ } ∪ ∆ d,W ∃ , · µ ′ ( F , . . . , F m ).Consequently, it is not in any intersection of this set with other sets. This proves thefirst direction of the claim.For the other direction, let I be a model of µ that is not in T | Mod ( µ ′ ) |≤ m +1 ,µ ′ | = µ { I | I = µ ′ } ∪ ∆ d,W ∃ , · µ ′ ( F , . . . , F m ). Since this is an intersection of sets, I cannot belong toall the sets. Let µ ′ be a formula such that I 6∈ { I | I = µ ′ } ∪ ∆ d,W ∃ , · µ ′ ( F , . . . , F m ).This is a union of sets, therefore I does not belong to any: I 6∈ { I | I = µ ′ } and I ∆ d,W ∃ , · µ ′ ( F , . . . , F m ). The first condition implies that I is a model of µ ′ . Since I | = µ ′ , | Mod ( µ ′ ) | ≤ m + 1 , µ ′ | = µ and I ∆ d,W ∃ , · µ ′ ( F , . . . , F m ) Theorem 4 applies,proving that I ∆ d,W ∃ , · µ ( F , . . . , F m ).Graphically, Theorem 4 is easier to grasp in terms of excluded models rather thanminimal models. A model is excluded (non-minimal) if and only if there exists m othermodels that exclude it: for every weight vector one of these models has lower distancethan its. With two formulae, each pair of models defines an excluded area; every modelthat they exclude is not minimal. Therefore, merging is the result of over-imposing allthese excluded areas. 21 ❢ ✻ ✲❅❅ excluded The region that is upper-right of the broken line is the area of exclusion of the twomodels (the horizontal and vertical half-lines are also excluded, but not the diagonalsegment). Each pair of models cuts out a similar area from the plane if none dominatethe other. The result of deleting all of them is the result of merging.Whether a model is in one such area is determined algebraically. The points in it areseparated from the origin by any one of the three lines of this figure: ❢ ❢ ✻ ✲❅❅❅❅❅❅❅❅ Checking whether a line separates a point from the origin is trivial: given the equationof the line ax + by + c = 0, the expression ax + by + c is zero on the points in the line andnone other. By setting x and y with same sign of a and b respectively and sufficientlylarge ( abs ( x ) > abs ( c/a ) if a = 0, the same for y ), the expression has positive value.With the opposite signs, it has negative values. Since the function is continuous and isonly zero on the line, it has the same sign in each half-plane. As a result, a point isseparated from the origin if ax + by + c and a b c have different signs. The samecondition can also be checked by computing the sign of two angles among the involvedpoints [50].That the three lines define the area of exclusion of two models is now formally proved.Let D , D ′ and D ′′ be the distance vectors of three models. The first is excluded by theother two if the following system of inequalities in two variables W = [ w , w ] has nosolution. w > w > W · D ≤ W · D ′ W · D ≤ W · D ′′ These are linear inequalities; therefore, they hold in a polygon. Each segment ofits perimeter is part of a line; this line is an extreme point for the inequalities, and is22herefore obtained by converting some of these inequalities into equations. Every linecorresponds to a single pair of weights W = [ w , w ], as each such pair defines a line inthe plan.The first conversion turns w > W · D ≤ W · D ′ into w = 0 and W · D = W · D ′ .These equations imply w d = w d ′ , which is the same as d = d ′ since w is not null.This equation defines the horizontal line that crosses D ′ . A similar restriction leads tothe horizontal line crossing D ′′ . The highest of these two lines is irrelevant, since thehalf-plane over it is all contained in that of the other. What remain is a horizontal line.In the same way, converting w > w = 0 leads to the vertical line.Converting both w > w > w = 0 and w = 0 does not lead to a line,since all points satisfy the remaining equalities in this case.The only conversion left to try is to turn W · D ≤ W · D ′ and W · D ≤ W · D ′′ into W · D = W · D ′ and W · D = W · D ′′ . These equations imply W · D ′ = W · D ′′ . If D ′ strictly dominates D ′′ this is a contradiction, and no line is generated; the same if D ′′ strictly dominates D ′ . Otherwise, W · D ′ = W · D ′′ implies that W · D = W · D ′ is thesame as W · D = W · D ′′ : these two equations with variables D are equivalent. They arethe same linear equation in two variables; therefore, they define a line. Since D = D ′ and D = D ′′ both satisfy them, this line crosses the points D ′ and D ′′ .An alternative condition for the exclusion of a model from merging is based on thelines separating other models from the origin. In particular: I | = µ is excluded if either d ( J, F , F ) < d ( I, F , F ) for some model J of µ , or two other models J and K of µ aresuch that:1. d ( J, F , F ) < d ( K, F , F );2. d ( K, F , F ) < d ( J, F , F );3. the line crossing I and J does not separate K from the origin; and4. the line crossing I and K does not separate J from the origin.Rather than proving the claim, the region of models I such that the line crossingeach and K separates J from the origin is shown on the figure for two models J and K that do not dominate each other. ❢ ❢ ✻ ✲❅❅❅❅❅✧✧✧✧✧ ✧✧✧✧✧✧✧❅❅❅ J Zexcl. excluded Adding the models in the same condition with J and removing the models that arestrictly dominated by either J or K leaves the correct area.23hese conditions allow for an algorithm that progressively excludes models. Initially,all models of µ are provisionally accepted. Then, a single model is chosen and themodels it strictly dominates removed. The procedure continues on the remaining modelsuntil nothing changes. At this point, no model dominates another. A pair of models isselected; the conditions above determine whether this pair excludes some other models,which are removed. When the procedure ends, only the models selected by merging areleft. Algorithm 1: ∆ d,W ∃ , · µ ( F , F )1. M = Mod ( µ );2. C = true 3. while C do:(a) C = false (b) for each I, J ∈ M if d ( J, F , F ) < d ( I, F , F ) theni. M = M \{ I } ii. C = true C = true 5. while C do:(a) C = false (b) for each I, J, K ∈ M if the line crossing d ( I, F , F ) and d ( K, F , F ) does not separate J fromthe origin and the line crossing d ( I, F , F ) and d ( K, F , F ) does notseparate K from the origin theni. M = M \{ I } ii. C = true 6. return M The algorithm first removes all models that are strictly dominated others. The re-moved models can be forgotten, since they are not in the part of the convex hull that isvisible from the origin; therefore, they are unnecessary for excluding other models. Whenall strictly dominated models are out from M , the algorithm proceeds by comparing eachmodel I with a pair of other models J and K . These two do not strictly dominate eachother because all models that are strictly dominated have already been removed from M . This is why no dominance check is done in Step 5b.24 Postulates Merging with unknown weights depends on the distance d and the set of weight vectors W. . Some postulates for belief merging hold for all sets of weights vectors (IC0-IC3 andIC7), others only for some including W ∃ (IC4-IC6), and one does not hold for W ∃ (IC8).Merging with completely unknown weights W ∃ cannot be expressed as a preorder, noteven a partial one.Postulates IC0-8 [36] cannot all hold, since a merging operator satisfying all of themcan be expressed as a selection of models of µ that are minimal according to some totalpreorder depending only on F , . . . , F m . Actually, not even a partial preorder expressesmerging with unknown weights. Theorem 6 No partial preorder ≤ depending on F and F only is such that ∆ dh,W ∃ , · µ ( F , F ) = min( Mod ( µ ) , ≤ ) . Proof. By Lemma 6, for every set of distance vectors there exists µ , F and F such thatthe models of µ have these Hamming distance vectors from F and F . The distancevectors that prove the claim are [3 , , 2] and [0 , µ ′ be the formula having only the models at distance [2 , 2] and [3 , W = [2 , , < [2 , µ ′′ withthe models at distance [2 , 2] and [0 , 3] shows that [0 , < [2 , ≤ is thesame since by assumption it does not depend on µ but only on F and F , which are thesame. A consequence of [3 , < [2 , 2] and [0 , < [2 , 2] is that the model at distance [2 , µ as proved by Lemma 3.The line of proof reveals which postulate is not satisfied: removing a model from µ is the same as conjoining µ with another formula; the postulate forbidding the inclusionof new models is IC8. The proof shows that IC8 does not hold.How about the others? Some hold for every set of distance vectors W. , others only forsome. Some postulates hold only if the distance d satisfies the triangle inequality, othershold even if d ( I, F ) is not defined in terms of a distance among models d ( I, J ). The latterrequires d ( I, F ) ∈ Z ≥ and d ( I, F ) = 0 if and only if I | = F . In the following summary,this case is described as “a model-formula distance”. In this section, E is sometimesused in place of F , . . . , F m following the notation by Konieczny and Perez [36]. Thissimplifies some formulae.0. ∆ d,W., · µ ( E ) | = µ holds for every model-formula distance and non-empty set of weight vectors1. if µ is consistent, then ∆ µ ( E ) is consistentholds for every model-formula distance and non-empty set of weight vectors25. if V E is consistent with µ , then ∆ d,W., · µ ( E ) = µ ∧ V E holds for every model-formula distance and non-empty set of weight vectors3. if E ≡ E and µ ≡ µ , then ∆ d,W., · µ ( E ) ≡ ∆ d,W., · µ ( E )holds for every model-formula distance and non-empty set of weight vectors4. if F | = µ and F | = µ then ∆ d,W., · µ ( F , F ) ∧ F is consistent if and only if∆ d,W., · µ ( F , F ) ∧ F is consistent.holds if W. contains every permutation of every vector it contains ( W ∃ has thisproperty, as well as W a for every a ∈ Z > ) and d satisfies the triangle inequality: d ( I, K ) + d ( K, J ) ≥ d ( I, J ) (both dd and dh have this property); if any of thesetwo conditions do not hold, a counterexample shows that the postulate does nothold5. ∆ d,W. ′ , · µ ( F , . . . , F k ) ∧ ∆ d,W. ′′ , · µ ( F k +1 , . . . , F m ) | = ∆ d,W., · µ ( F , . . . , F m )requires W. = W. ′ × W. ′′ 6. if ∆ d,W. ′ , · µ ( F , . . . , F k ) ∧ ∆ d,W. ′′ , · µ ( F k +1 , . . . , F m ) is consistent, it is entailed by∆ d,W., · µ ( F , . . . , F m )requires W. = W. ′ × W. ′′ µ ′ ∧ ∆ d,W., · µ ( E ) | = ∆ d,W., · µ ∧ µ ′ ( E )holds for every model-formula distance and non-empty set of weight vectors8. if µ ′ ∧ ∆ d,W., · µ ( E ) is consistent, then ∆ d,W., · µ ∧ µ ′ ( E ) | = ∆ d,W., · µ ( E )does not hold for the Hamming distance dh and the set of all weight vectors W ∃ The formal proofs of these claims follow. First, postulates IC0-IC3 and IC7 hold forevery non-empty set of weight vectors W. and model-formula distance. Lemma 10 For every model-formula distance d ( I, F ) and non-empty set of weight vec-tors W. , the merging operator ∆ d,W., · satisfies postulates IC0-IC3 and IC7. Proof. The claim is proved one postulate at time.0. ∆ d,W., · µ ( E ) | = µ by definition, ∆ d,W., · µ ( E ) is a subset of the models of µ ;1. if µ is consistent, then ∆ µ ( E ) is consistentby assumption, W. contains at least a vector of weights W ; for this vector, ∆ d,W, · µ ( E )is the set of models of µ at minimal weighted distance from F , . . . , F m ; if µ isconsistent, it has at least a minimal model;26. if V E is consistent with µ , then ∆ d,W., · µ ( E ) = µ ∧ V E since d ( I, F i ) = 0 when I | = F i , the distance vectors of the models of V E are[0 , . . . , E ≡ E and µ ≡ µ , then ∆ d,W., · µ ( E ) ≡ ∆ d,W., · µ ( E )equivalence between profiles E ≡ E holds when there exists a bijection such thatthe associated formulae are equivalent; this postulate holds because merging isdefined from the models of the formulae;7. µ ′ ∧ ∆ d,W., · µ ( E ) | = ∆ d,W., · µ ∧ µ ′ ( E )the models of µ ′ ∧ ∆ d,W., · µ ( E ), if any, are the models that satisfy µ ′ , and also satisfy µ and no other model of µ has a lower distance from E weighted by some W ∈ W. ;each such model satisfies µ ∧ µ ′ , and no other model of µ ∧ µ ′ has lower distanceweighted by W , since the models of µ ∧ µ ′ are a subset of those of µ ;This lemma does not even require d ( I, F ) to be defined in terms of a distance betweenmodels. The next one does, and additionally needs the triangle inequality. Lemma 11 If W. contains every permutation of every vector it contains and d satisfiesthe triangle inequality ∀ I, J, Kd ( I, K ) + d ( K, J ) ≥ d ( I, J ) , then IC4 holds. For some setof weight vectors that does not include a permutation of one of its elements IC4 does nothold. The same for some distance not satisfying the triangle inequality. Proof. Postulate 4 is: if F | = µ and F | = µ then ∆ d,W., · µ ( F , F ) ∧ F is consistent ifand only if ∆ d,W., · µ ( F , F ) ∧ F is consistent.This postulate does not hold in general. For example, the single weight vector [2 , µ = true , F = x and F = ¬ x would selectthe model of F only. Both distances satisfy the triangle inequality.This counterexample suggests that the postulate holds if the set W. has some sort ofsymmetry: if it contains a weight vector, it also contains all its permutations. This isnot however not sufficient, as shown by the following counterexample: ds ( I, J ) = dh ( I, J ) = 01 if dh ( I, J ) = 12 if 2 ≤ dh ( I, J ) ≤ 45 if 5 ≤ dh ( I, J ) W. = { [5 , , [2 , } F = x ∧ x ∧ x ∧ x ∧ x F = ¬ x ∧ ¬ x ∧ ¬ x ∧ ¬ x ∧ ¬ x µ = F ∨ F ∨ ( x ∧ ¬ x ∧ ¬ x ∧ ¬ x ∧ ¬ x )27he distance ds may look unnatural, but has a rationale: instead of measuring thedistance between models by the exact number of differing literals, it roughly approximatesit by aggregating certain groups of consecutive values into one, so that only a finitenumber of possible distances would result.The models of µ have distance vectors [0 , , 1] and [5 , F and F , respectively. The weight vector [5 , 2] turns these distancevectors into the weighted distances [5 , · [0 , 5] = 10, [5 , · [2 , 1] = 12, [5 , · [5 , 0] = 25;only the model of F is minimal. For the weight vector [2 , , · [0 , 5] = 25,[2 , · [2 , 1] = 9, [2 , · [5 , 0] = 10; the only minimal model is the second, which is nota model of F . This is a case in which both F and F imply µ and ∆ ds,W., · µ ( F , F ) isconsistent with F but not with F .Note that Lemma 6 does not apply to this case. It tells how to obtain certain distancevectors with formulae µ, F , F , but these do not necessarily obey F | = µ and F | = µ .To the contrary, the proof itself shows that both F and F have models that falsify µ .Postulate 4 requires not only W. to be symmetric, but also d to satisfy the triangleinequality: for every three models I , J and K it holds d ( I, K ) + d ( K, J ) ≤ d ( I, J ).Since ∆ d,W., · µ ( F , F ) ∧ F is consistent, there exists a weight vector [ a, b ] and a model I ∈| = F with distance vector [0 , c ] such that [ a, b ] · [0 , c ] is minimal (the zero is because I | = F ).By definition, d ( I, F ) = c means that d ( I, J ) = c for some J ∈ Mod ( F ). This impliesthat d ( J, F ) ≤ d ( J, I ) = c ; if d ( J, F ) < c then d ( J, K ) < c for some K ∈ Mod ( F ), whichimplies that d ( K, F ) < c = d ( I, F ), contradicting the assumption that I is minimal;therefore, d ( J, F ) = c .Since J satisfies F , it also satisfies µ . It is therefore a candidate for being in theresult of merge. If a < b , then the weighted distance of J would be [ a, b ] · [ c, 0] = a × c
0] = b × c . Contrary to the claim, let K be a model with distancevector [ e, f ] such that [ b, a ] · [ e, f ] < b · c .The triangular property implies e + f ≥ c . In details: e + f < c implies the existenceof I ′ and J ′ of respectively F and F such that d ( K, I ′ ) = e , d ( K, J ′ ) = f and d ( I ′ , J ′ ) ≤ e + f < c . This contradicts the assumption of minimality of I . This property e + f ≥ c ,together with a ≥ b , makes the following inequalities valid:[ b, a ] · [ e, f ] = b × e + a × f = b × e + b × f + ( a − b ) × f = b × ( e + f ) + ( a − b ) × f ≥ b × c + ( a − b ) × f ≥ b × c b, a ] · [ e, f ] ≥ b × c . This proves that no such model K mayexists, and that J has minimal distance weighted by [ b, a ]. Since J | = F , the requiredconsistency of ∆ d,W., · µ ( F , F ) ∧ F is proved.Since W ∃ is symmetric and both dd and dh satisfy the triangle inequality, Postu-late 4 holds in these two cases. Actually, for W ∃ the distance does not matter, and∆ d,W ∃ , · ( F , . . . , F m ) is always consistent with every F i that is consistent with µ . If themaximal value of the distance from F and from F is k , the weight vectors [ k + 1 , 1] and[1 , k + 1] suffice. The first guarantees that every model of F is always better than oneof ¬ F , no matter how close the second is to F . The same for the second weight vector.This lemma shows an effect of the triangle inequality on belief merging. It is a quitenatural requirement and is obeyed by both the drastic and the Hamming distance, but ismostly useless in belief merging [36]; so far it only seemed to affect the infinite-alphabetcase [10] and the application of belief revision to case-based reasoning [15].Postulates 5 and 6 require special care to be even formulated. Informally, they tellthat merging F , . . . , F k , F k +1 , . . . , F m is the same as merging F , . . . , F k , then merging F k +1 , . . . , F m and finally conjoining the two results if they do not conflict. This is simpleto express if no weights are involved, otherwise each of these three integrations is definedover its set of weights. If these are unrelated, like [1 , . . . , , . . . , 1] for the overall mergeand [10 , , . . . , 1] and [1 , . . . , 10] for merging the two parts, the three results cannot beexpected to be coherent.This is why Postulates 5 and 6 cannot be said to be obeyed plain and simple. Rather,they are satisfied only when the sets of weights are related in the appropriate way. Lemma 12 If ∆ d,W. ′ , · µ ( F , . . . , F k ) ∧ ∆ d,W. ′′ , · µ ( F k +1 , . . . , F m ) is consistent, it is equivalentto ∆ d,W., · µ ( F , . . . , F m ) , where W. = W. ′ × W. ′′ (postulates IC5 and IC6). Proof. Let I be a model of ∆ d,W. ′ , · µ ( F , . . . , F k ) ∧ ∆ d,W. ′′ , · µ ( F k +1 , . . . , F m ). By as-sumption, there exists W ′ ∈ W. ′ and W ′′ ∈ W. ′′ such that the distance vector d ( I, F , . . . , F k ) weighted by W ′ is minimal among the models of µ , and the distancevector d ( I, F k +1 , . . . , F m ) weighted by W ′′ is minimal among the models of µ . This isequivalent to d ( I, F , . . . , F k , F k +1 , . . . , F m ) being minimal when weighted by W ′ W ′′ ; thisis the vector obtained by concatenating W ′ and W ′′ , and is therefore in W. = W. ′ × W. ′′ .In the other way around, a model that is not minimal on its weighted dis-tance to F , . . . , F k or to F k +1 , . . . , F m is not minimal on its weighted distance to F , . . . , F k , F k +1 , . . . , F m .IC8 does not hold. The following counterexample shows that for the Hamming dis-tance dh and the set of all weight vectors W ∃ . Theorem 7 There exists µ , µ ′ , F and F such that µ ′ ∧ ∆ dh,W ∃ , · µ ( E ) is consistent but ∆ dh,W ∃ , · µ ∧ µ ′ ( E ) = ∆ dh,W ∃ , · µ ( E ) . Proof. Let µ , F and F be such that µ has three models with distance vectors d ( I, F , F ) = [1 , d ( J, F , F ) = [0 , 1] and d ( K, F , F ) = [0 , dh,W., · µ ( F , F ) are I and J , since these two models have minimaldistance weighted by [1 , J dominates K , by Lemma 1 K is not in the result ofmerge for any weights.Let µ ′ be the formula with models I and K . Since I also satisfies ∆ dh,W ∃ , · µ ( F , F ), theconjunction of this and µ ′ is consistent. When merging under constraints µ ∧ µ ′ , model K is now minimal with weights [2 , I of µ ∧ µ ′ have bothweighted distance 3.This counterexample completes the analysis of the basic postulates IC0-IC8. Twoadditional ones exist: majority and arbitration. The first tells that a formula repeatedenough times is entailed by the result of merging; the second was initially defined as theirrelevance of the number of repetitions, and has a newer definition that is difficult tosummarize in words.Majority does not hold with W ∃ . Not that it should. No matter how many times aformula is repeated, regardless of how many sources supports it, its negation may comefrom a single source that is more reliable than all the others together. When reliability isuncertain, this case has to be taken into account. It is not even uncommon in practice:many commonly held belief are in fact false.Many commonly held belief are in fact false: Napoleon was short [20]; diamonds hadbeen typical gemstones for engagement rings since a long time [21], the red telephone isa telephone line, and one of its end is in the White House [14]; meteorites are always hotwhen they reach the Earth’s surface; flowering sunflowers turn to follow the sun (only thegems do); the Nazis issued an ultimatum before the Ardeatine massacre (something evenwitnesses of the time believe) [46, p. 155]; fans in closed rooms kill people (many peoplein Korea believed this). A page on Wikipedia list more than a hundred of commonlybelieved facts that are in fact false [59]. The material was enough for a 26-episodes TVshow [58].All of this shows that no matter how many times a fact is repeated, if the reliabilityof its sources is unsure it may still be falsified by a single reliable source. This is whatthe following theorem formally proves. Theorem 8 There exists F , F such that ∆ d,W ∃ , · true ( F , F , . . . , F ) = F , where F is re-peated an arbitrary number of times. Proof. The formulae are F = a and F = ¬ a . For every number of repetitions n , thereexists W such that ∆ d,W, · µ ( F , F , . . . , F ) contains the model { a } , which does not satisfy F . In particular, the weights are W = [ n, , . . . , { a } fromthe formulae is n , the same as the weighted distance of the only other model {¬ a } . Asa result, { a } is minimal.Arbitration was initially defined as the opposite condition of irrelevance of the numberof repetitions [37, 47]. This property holds for W ∃ . The following theorem proves anequivalent formulation of it. Lemma 13 For every µ, F , . . . , F m it holds: ∆ d,W ∃ , · µ ( F , . . . , F m ) = ∆ d,W ∃ , · µ ( F , . . . , F m , F m )30 roof. By definition, ∆ d,W ∃ , · µ ( F , . . . , F m ) is the union of ∆ d,W, · µ ( F , . . . , F m ) for every W ∈ W ∃ , and the same for merging with a duplicated F m . The claim is proved by showingthat for each W = [ w , . . . , w m − , w m ] there exists W ′ = [ w ′ , . . . , w ′ m − , w ′ m , w ′′ m ] suchthat ∆ d,W, · µ ( F , . . . , F m ) is equal to ∆ d,W ′ , · µ ( F , . . . , F m , F m ), and vice versa.The distance of a model from F , . . . , F m weighted by W = [ w , . . . , w m − , w m ] isexactly half of the distance of the same model from F , . . . , F m , F m weighted by W ′ =[2 × w , . . . , × w m − , w m , w m ], since each distance is multiplied by two. Therefore, theminimal models are the same.Vice versa, the distance of a model from F , . . . , F m , F m weighted by W ′ =[ w , . . . , w m − , w m , w ′ m ] is exactly the same as the distance of the same model from F , . . . , F m weighted by W = [ w , . . . , w m − , w m + w ′ m ]. In this case, the weighteddistances are exactly the same, and the minimal models coincide.A newer version of the arbitration postulate is expressed in terms of the preorderbetween models as: if I < F J , I < F J ′ and J ≡ F ,F J ′ then I < F ,F J . As provedby Theorem 6, merging under unknown reliability cannot be expressed as a preorder,total or otherwise. The expression of the postulate in terms of formulae is even moreconvoluted, and is not clear whether it makes sense when merging is not expressible interms of a preorder. A further condition merging may or may not meet is the disjunctive property, defined ontwo formulae as Postulate 7 by Liberatore and Schaerf [41] and later generalized to anarbitrary number of formulae with integrity constraints by Everaere et al. [22]. In termsof models it has a simple intuitive expression. Every model is a possible state of theworld; merging only selects worlds that at least one of the sources consider possible. Informulae, a model I is in the result of merging only if I | = F i for at least one of the mergedformulae F i . Since I must also satisfy the integrity constraints µ , this requirement islifted when none of the formulae F i is consistent with µ .This condition is not satisfied by ∆ dh,W = , · ; as a result, is not satisfied by ∆ dh,W ∃ , · either since W = ⊂ W ∃ . However, a suitable set of weight vectors allows for a specificform of disjunctive merging, that based on closest pairs of models. Definition 3 ([41]) Merging by closest pairs of models is defined from the orderingbetween pairs of models h I, J i ≤ d h h I ′ , J ′ i if and only if dh ( I, J ) ≤ dh ( I ′ , J ′ ) by selectingthe models in all minimal pairs: F ∆ D F = { I, J | h I, J i ∈ min( Mod ( F ) × Mod ( F ) , ≤ d h ) } The set of weight vectors used for obtaining this definition is W n +1 = { [1 , n + 1] , [ n +1 , } , the specific form of W a when a is the number of variables increased by one andmerging is between two formulae. 31 heorem 9 For every pair of satisfiable formulae F and F over an alphabet of n variables, it holds F ∆ D F = ∆ dh,W n +1 , · true ( F , F ) . Proof. By definition, I ∈ F ∆ D F if and only I | = F and there exists J | = F such that h I, J i is minimal according to ≤ d h , or the same with F and F swapped. What is nowproved is that the first condition is equivalent to I ∈ ∆ dh, [ n +1 , , · true ( F , F ). By symmetry,the condition with the two formulae swapped is equivalent to I ∈ ∆ dh, [1 ,n +1] , · true ( F , F )without the need of a proof.The relevant cases are: I | = F and h I, J i is minimal for some J | = F , I | = F and h I, J i is minimal for no J | = F , and I = F . The claim holds if I ∈ ∆ dh,W n +1 , · true ( F , F )holds exactly in the first case.1. I | = F and h I, J i is minimal for some J | = F ; since I ∈ F , the distance from I to F is zero: dh ( I, F ) = 0; therefore, the weighted distance from I to the formulae is[ n + 1 , · [0 , dh ( I, F )] = dh ( I, F ), which is at most n ; the negation of the claim isthat the weighted distance ( n + 1) × dh ( K, F ) + 1 × dh ( K, F ) of some other model K is less than that; for it being less than n implies dh ( K, F ) = 0; as a result, theweighted distance of K is dh ( K, F ); if it were less than the weighted distance of I then dh ( K, F ) < dh ( I, F ); by definition, this means that there exists K ′ suchthat dh ( K, K ′ ) is less than dh ( I, I ′ ) for every I ′ | = F , including I ′ = J ; this meansthat dh ( K, K ′ ) < dh ( I, J ), contrary to the assumption that h I, J i is minimal;2. I | = F and h I, J i is minimal for no J | = F ; by assumption, there exists K, K ′ such that K | = F , K ′ | = F and dh ( K, K ′ ) < dh ( I, J ) for every J | = F ; thismeans that dh ( K, F ) < dh ( I, F ); since both I and K satisfy F , it also holds dh ( I, F ) = dh ( K, F ) = 0; as a result, the weighted distances of these models are[ n + 1 , · [0 , dh ( I, F )] = dh ( I, F ) and [ n + 1 , · [0 , dh ( K, F )] = dh ( K, F ); since dh ( K, F ) < dh ( I, F ), the model I is not at a minimal weighted distance;3. I = F ; since F is by assumption satisfiable, it has a model K ; since dh ( K, F ) =0, the weighted distance for this model is [ n + 1 , · [ dh ( K, F ) , dh ( K, F )] = dh ( K, F ), which is at most n ; the weighted distance of I is instead [ n + 1 , · [ dh ( I, F ) , dh ( K, F )] = ( n + 1) × dh ( I, F ) + dh ( I, F ), which is greater than n since dh ( I, F ) > I has minimal weighted distance from F and F in the first case but not in thesecond and the third, the claim is proved.A disjunctive operator on m formulae is obtained similarly when all formulae areconsistent and the integrity constraints are void: µ = true . Theorem 10 For every distance d bounded by k and such that d ( I, F ) = 0 if and only if I | = F , if F , . . . , F m are satisfiable then ∆ d,W k × m , · true ( F , . . . , F m ) is a disjunctive mergingoperator. roof. Let I be a model satisfying no formula F i . The disjunctive property holds if I is not in ∆ d,W k × m , · true ( F , . . . , F m ). This holds if I is not in ∆ d,W, · true ( F , . . . , F m ) for any W ∈ W k × m .By assumption, I does not satisfy any of the formulae. Therefore, its distance vector isless than or equal to [1 , . . . , W , the result is k × m +( m − i be the index such that the i -th element of W k × m is k × m . Since F i is satisfiable,it has a model J . The distance vector of J is at most [ k, . . . , k, , k, . . . , k ] where 0 is atindex i . The result of multiplying it by W is ( m − × k .The upper bound for the weighted distance of J is ( m − × k , which is less than k × m + ( m − I . This provesthat I is not minimal.The set of weight vectors used in this and the previous theorem provides an alter-native view of the disjunctive property. One line of reasoning when merging conflictinginformation is to assume that a single source i is completely right; consequently, theinformation that one provides is accepted in full, and is integrated with that comingfrom the other sources only as long as this is consistent with it. Since the actual relia-bility is unknown, no unique choice of i is warranted. Technically, every formula that isconsistent with the integrity constraints could come from the completely reliable source;the weight vectors W a for a sufficiently large a formalize this mechanism.This is a correct way of implementing the principle of indifference in belief merg-ing: rather than assuming that all sources are equally reliable, one of them is taken ascompletely right, but this is done for each of them at time. Indifference is realized bysymmetry, not equality. The previous sections are about combining a number of independent formulae. This isthe basic problem of belief merging: each formula comes from a different source, so thattheir reliabilities are independent. This is formalized by the weights being unconstrainedin the set W ∃ .When a source provides more than one formula, each of them is as reliable as itssource. The same mechanism employing the weighted sum of the drastic or Hammingdistance can be used, but the weights are associated to the sources rather than to theformulae. All formulae from the same source have the same reliability and thereforethe same weight (this condition is close in spirit to the unit partitions by Booth andHunter [8]).Technically, each source is represented by a set of formulae S i . Its reliability isencoded by a positive integer w i . Given a set { S , . . . , S m } of such sources, merging isdone by selecting the minimal models of the integrity constraints µ according to thisevaluation: v ( I ) = X S i w i × X F i ∈ S i d ( I, F i )33his is the DA operator [35] with the sum as intra-source aggregation and theweighted sum as the inter-source aggregation. The sum is subject to the problem ofmanipulation: a source may provide the same formula multiple times in order to influencethe final result [13]; this is a problem especially when merging preferences, but not whenmerging beliefs with unknown reliability. Even if a source provides the same formula athousand times, one of the considered alternatives is that the weight of this source is athousand times smaller than the others, making such a manipulation ineffective.The only technical result in this section is that merging with the drastic distance isnot the same as disjoining the maxcons. This is proved by the following sources with µ = true . S = { x, y, z } S = {¬ x, ¬ y } S = {¬ x, ¬ z } One of the maxcons of { x, y, z, ¬ x, ¬ y, ¬ z } is { x, ¬ y, ¬ z } , which is not obtained whenmerging with unknown reliabilities. Intuitively, to include the formula x from S in theresult that formula needs to count at least twice as much as each formula ¬ x from S and S , but this implies the same for y and z , which excludes ¬ y and ¬ z .Formally, let the weights of the sources be w , w , w . The weighted distance of somerelevant models are: I = { x, ¬ y, ¬ z } v ( I ) = w × w × w × w × w + w J = {¬ x, ¬ y, ¬ z } v ( J ) = w × w × w × w × K = { x, y, z } v ( K ) = w × w × w × w × w × I to be minimal v ( I ) must be less than or equal to v ( J ) and v ( K ): w × w + w ≤ w × w × w + w ≤ w × w × w + w ≤ w , which makesthe right-hand side of the second become less than or equal to w × 2, while it shouldinstead be greater than or equal than w × w + w , and therefore greater than w × { x, ¬ y, ¬ z } is not a minimal model for any weight vector.A similar example shows that merging does not result in the maxcons of the con-junctions of the sources. Let S = { x, y } and S = {¬ x, ¬ y } . The only maxcons of {∧ S , ∧ S } are x ∧ y and ¬ x ∧ ¬ y , but merging with unknown weights selects all fourmodels over x and y , since this is the result when W = [1 , Sometimes the information to be merged comes from sources of equal reliability. In suchcases, merging with equal weights is correct. But if reliability is unknown, assumingweights equal is unwarranted. The difference is not only conceptual but also technical.Theorem 6 shows that merging with unknown reliability cannot in general be reducedto a preorder among models, not even a partial one.A result emerged by the study of this setting is a motivation for merging by max-cons [3]. This mechanism has sometimes been considered unsuitable for merging becauseit disregards the distribution of information among sources [34, 36]. Theorem 3 shows itthe same as merging with the drastic distance when reliability is unknown. The numberof repetitions of a formula is irrelevant to this kind of merging—as it should. A formulaonly occurring once may come from a very reliable source, while its negation is supportedonly by untrustable sources. Without any knowledge of the reliability of the sources, thisis a situation to take into account.This article not only backs merging by maxcons, but more generally merging by the(MI) postulate: the number of repetitions of a formula is irrelevant to merging. Of course,there are many cases where this postulate should not hold; whenever reliability is known,two sources providing a formula give twice the support for it. But when reliability isunknown every numeric evaluation becomes irrelevant, including doubling the supportfor a formula like in this case. As already discussed by Meyer [47], Postulate (MI) maysometimes be right; it is inconsistent with the other postulates IC0-IC8, but the fault ison them. While Meyer blames Postulate 4 of merging without integrity constraints [37],merging with unknown reliability conflicts with IC8.A minor technical contribution of this article is a case for the distance function to obeythe triangle inequality. This property had only a couple of applications in belief revisionand merging so far [10, 15], but is generally not required [36]. The new consequence ofit shown in this article is that it allows satisfying Postulate (IC4) when merging withunknown reliability.Some problems of merging with unknown reliability are left open.The graphical representation of merging by a convex hull has been shown to work inthe bidimensional case, which corresponds to merging two formulae only. It looks like itworks for an arbitrary number of formulae, but this is still left to be proved.Another open problem is the characterization of merging with unknown reliabilitywhen sources provide more than one formula each. This is the case treated by theDA merging operators [35]. It has been only touched in the previous section, but thatpreliminary analysis already shows that this case cannot in general be reduced to thatof one-formula sources.While the weighed sum is one of the basic mechanism for merging [53, 42, 43, 35],it is not the only one. The Max [53], Gmax [38] and quota merging [22] semantics arealternatives to be used under different conditions; for example, Gmax is the system touse when sources are assumed to be unlikely to be very far from truth.35 comparison with related work follows.The weighted distance from a set of formulae was first used for merging by Revesz [53],and investigated by Lin [42]. Lin and Mendelzon [43] and Konieczny and Perez [37] usedthe unweighted sum for merging. These articles assume either equal or fixed weights,not varying weights like is done in the present one.Benferhat, Lagrue and Rossit [6] considered the related problem of commensurability:when the sources themselves assess the reliability of the formulae they provide, they maynot use the same scale; this is related to a similar issue in social choice theory. Their studyand the present one differ in formalism (ranked bases instead of formulae with distancefunctions), but they share the principle of considering a set of alternative reliabilityassessments. There is however an early point of departure: Benferhat, Lagrue andRossit [6] distill a single preorder and then select the models that are minimal accordingto it; Theorem 6 shows that the same cannot be done in general in the settings of thepresent article. A point of contact is the case of drastic distance: Theorem 1 could bealternatively proved from certain results by Benferhat, Lagrue and Rossit [6, Propositions1,2,8].Accepting what is true according to all possible relative reliabilities is analogous todrawing the consequences that hold in all probability measures in a set [30], and can beseen as the formal logic version of the “worst scenario” in economics: “ the firm may notbe certain about the “relative plausibility” of these boom probabilities. [...] if the firmacts in accordance with certain sensible axioms, then its behavior can be characterizedas being uncertainty-averse: when the firm evaluates its position, it will use a probabilitycorresponding to the “worst” scenario ” [49]. Belief revision and merging aim at themost knowledge that can be justifiably and consistently obtained; therefore, minimalknowledge takes the place of the least profit, and the worst scenario for a formula is onewhere it is false. References [1] E. 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