Consensus Determining with Dependencies of Attributes with Interval Values

Transcription

1 Jounal of Univesal Compute Science, vol. 13, no. (007), submitted: 31/7/06, accepted: 15/1/07, appeaed: 8//07 J.UCS Consensus Detemining with Dependencies of Attibutes with Inteval Values Michal Zgzywa (Technical Univesity of Woclaw, Poland Abstact: In this pape the autho consides some poblems elated to attibute dependencies in consensus detemining. These poblems concen the dependencies of attibutes epesenting the content of conflicts, which cause that one may not teat the attibutes independently in consensus detemining. It is assumed that attibute values ae epesented by intevals. In the pape the autho consides the choice of pope distance function. Next, the limitations guaantying detemining a coect consensus despite teating the attibutes independently ae pesented. Additionally, the algoithm of calculating the pope consensus in cases when these limitation ae not met is intoduced. Keywods: Consensus theoy, Conflict, Attibute dependency, Agent, Distibuted system Categoies: E.1, H..1, I..4, I Intoduction Conflict esolution is one of the most impotant aspects in distibuted systems and multi-agent systems. The esouces of conflicts in these kinds of systems come fom the autonomy featue of thei sites (nodes). This featue means that each site of a distibuted o multi-agent system pocesses its tasks independently and may collect diffeent knowledge [Helpen, 01]. Thee ae seveal easons to oganize a system in such an achitectue [Coulouis, 96]. Fist of all, infomation collected in the system is easie to obtain some sites may be neae o not as busy as othes. Also the eliability of such systems is bette the failue of one node may be compensated by using othes. Finally, the tustwothiness of the system may be inceased when seveal agents ae investigating the same issue. Unfotunately, thee may aise such a situation that fo the same task, diffeent sites may geneate diffeent solutions. Thus, one deals with a conflict. In distibuted and multi-agent systems thee oigins of conflicts can be found: insufficient esouces, diffeences of data models and diffeences of data semantic [Pawlak, 98]. Fo a semantic conflict one can distinguish the following thee its components: conflict body, conflict subject and conflict content. Consensus models, among othes, seem to be useful in semantic conflicts solving [Tessie, 01]. The oldest consensus model was woked out by such authos as Condocet, Aow and Kemeny [Aow, 63]. This model seves to solving such conflicts in which the content may be epesented by odes o ankings. Models of Bathelemy and Janowitz [Bathelemy, 91], Bathelemy and Leclec [Bathelemy, 95] and Day [Day, 88] enable to solve such conflicts fo which the stuctues of the conflict contents ae n-tees, semillatices, patitions etc. These models ae still being developed, e.g. in woks

2 330 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes... [Wang, 01] o [McMois, 03]. Also detemining consensus fo fuzzy opinions is widely consideed (among othes in [Yage, 01] and [Lee, 0]). The common chaacteistic of these models is that they ae one-attibute, it means that conflicts ae consideed only efeing to one featue. Multi-featue conflicts have not been investigated. In woks [Nguyen, 0a] and [Nguyen, 0b] the autho pesents a consensus model, in which multi-attibute conflicts may be epesented. Futhemoe, in this model attibutes ae multi-valued, what means that fo epesenting an opinion on some issue an agent may use not only one elementay value (such as,, o 0) [Pawlak, 98] but a set of elementay values. The autho consides also conflicts with opinions epesented by intevals of values. Intevals ae simulated by sets of neighboing elements. This model enables to pocess multi-featue conflicts, but attibutes ae mainly teated as independent. Howeve, in many pactical conflict situations some attibutes ae dependent on othes. Fo example, in a meteoological system attibute Wind_powe (with values: weak, medium, stong) is dependent on attibute Wind_speed, the values of which ae measued in unit m/s (of couse, a set of values may be poposed fo each attibute when a meteoological station is not sue about a foecast). This dependency follows that if the value of the fist attibute is known, then the value of the second attibute is also known. It is natual that if a conflict includes these attibutes then in the consensus the dependency should also take place. The uestion is: Is it enough to detemine the consensus fo the conflict efeing to attibute Wind_speed? In othe wods, is it tue that if some value is a consensus fo the conflict efeing to attibute Wind_speed, then its coesponding value of attibute Wind_powe is also a consensus fo the conflict efeing to this attibute? And if it is not tue, how one should calculate the pope consensus that fulfils the dependency? In this pape we conside the answes fo mentioned above uestions. Fo this aim we assume some dependencies between attibutes and show thei influence on consensus detemining. We will focus on conflicts whee agents opinions ae epesented by intevals of values. The oiginal contibution of the aticle is: adjustment of distance function intoduced in [Nguyen, 0a] to the inteval stuctue; intoduction of consensus calculating algoithm togethe with theoem consideing its popeties; intoduction of theoem consideing conditions allowing to calculate consensus fo each attibute independently. The Outline of Consensus Model The consensus model which enables pocessing multi-attibute and multi-valued conflicts has been discussed in detail in woks [Nguyen, 0a] and [Nguyen, 0b]. In this section we pesent only some of its elements with extensions needed fo the consideation of attibute dependencies. We assume that some eal wold is commonly consideed by a set of agents that ae placed in diffeent sites of a distibuted system. The inteest of the agents consists of events which occu (o have to occu) in this wold. The task of the agents is based on detemining the values of attibutes descibing these events. If seveal agents conside the same event then they may geneate diffeent desciptions (which consist of, fo example, scenaios, timestamps etc.) fo this event. Thus we say that a conflict takes place. Let us

3 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes conside the simple example. Let the meteoological system intoduced in Section 1 give infomation about tempeatue foecast fo the next day. Evey agent meteoological station foesees the weathe fo a few county egions. Conseuently, many stations made a foecast fo the cental egion. The following pedictions wee gatheed: thee stations claimed that the tempeatue will be 4 Celsius degees and one foecast 5 Celsius degees. Which pediction should be pesented in an official communicate? The intuition suggests that 4 is the coect value, but how could it be fomally chosen? In othe wods, how to calculate the consensus? Fo epesenting ontologies of potential conflicts we use a finite set A of attibutes and a set V of attibute elementay values, whee V U a A V a (V a is the domain of attibute a). Let Π(V a ) denote the set of all possible intevals of elements fom set V a (o single elements when elements in V a can not be odeed) and Π(V B ) U b B Π(V b ). Let B A, a tuple B of type B is a function B : B Π(V B ) whee B (b) V b fo each b B. Empty tuple is denoted by symbol φ. The set of all tuples of type B is denoted by TYPE(B). The conflict ontology is defined as a uaduple: Conflict_ontology <A, X, P, F>, whee: A is a finite set of attibutes, which includes a special attibute Agent; a value of attibute a whee a Agent is an inteval of elements fom V a (o single elements when elements in V a can not be odeed); values of attibute Agent ae singletons which identify the agents. X{Π(V a ): a A} is a finite set of conflict caies. P is a finite set of elations on caies fom X, each elation P P is of some type T P (fo T P A and Agent T P ). Relations belonging to set P ae classified into goups of two, identified by symbols " " and " " as the uppe index to the elation names. Fo example, if R is the name of a goup, then elation R is called the positive elation (contains positive knowledg and R is the negative elation (contains negative knowledg. Positive elations contain tuples epesenting such desciptions which ae possible fo events. Negative elations, on the othe hand, contain tuples epesenting such desciptions which ae not expected fo events. When thee is only a positive elation, the uppe index may be omitted. Finally, F is a set of function dependencies between sets of attibutes (futhe descibed in Section 3). The stuctues of the conflict caies ae defined by means of a distance function between tuples of the same type. In this chapte we will use distance functions that measue the distance between two values (intevals o singleton elements) as the minimal costs of the opeation which tansfoms the fist value into the second value. The symbol δ will be used fo distance functions. A consensus is consideed within a conflict situation, which is defined as a pai s <{P,P }, (A,B)> whee A,B A, A B, and A φ holds fo any tuple P P (P,P ae elations of TYPE({Agent} A B)). The fist element of a conflict situation (i.e. set of elations {P,P }) includes the domain fom which consensus should be chosen, and the second element (i.e. pai (A,B)) pesents the schemas of consensus. Fo a subject e (as a tuple of type A, included in P o P ) thee should be assigned

4 33 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes... only one tuple of type B. A conflict situation yields a set Subject(s) of conflict subjects which ae epesented by tuples of type A. Fo each subject e two conflict pofiles, i.e. pofile( and pofile(, as elations of TYPE({Agent} B) may be detemined. Pofile pofile( contains the positive opinions of the agents on the subject e, while pofile pofile( contains agents negative opinions on this subject. Definition 1. Consensus on a subject e Subject(s) is a pai (C(s,,C(s, ) of tuples of type A B which fulfill the following conditions: a) C(s, AC(s, Ae and C(s, B C(s, Bφ, b) The sums δ ( B, C( s, B) and δ ( B, C( s, ) ae minimal. B pofile( pofile( Any tuples C(s, and C(s, satisfying the conditions of Definition 1 ae called consensuses of pofiles pofile( and pofile(, espectively. Example 1. Let us conside the meteoological system fom the beginning of the fist section. The ontology of that conflict is the uaduple: <{Agent, Region, Tempeatue}, {Π(Silesia, Geat Poland, Little Poland, Pomeania), Π(-30,- 9,,34,35)}, {Weathe }, φ>. We can distinguish one conflict situation: <{Weathe,φ}, {Region} {Tempeatue}>. Infomation about the conflict fo the subject Silesia is gatheed in Table 1. Agent Region Tempeatue station 1 Silesia 5 station Silesia 5 station 3 Silesia 4 Table 1: Relation Weathe. Column Tempeatue is in fact the pofile(silesia). To compae the values of diffeent opinions fo Tempeatue attibute function δ Temp will be used: δ Temp (t 1,t ) t 1 t. Now, afte calculating all the necessay distances, we can use the Definition 1b to detemine the consensus fo this subject (5 Celsius degees). 3 Detemining Consensus fo Attibutes with Inteval Values In this aticle we will focus on detemining consensus with inteval values. Such a poblem may occu when agents disagee about the scope of some object popety o some phenomenon. A good example is a conflict whee agents ague about the timing of some action. Agents descibe thei knowledge showing the moment of the action beginning and the moment of action ending. The esult of such a conflict should be a time inteval, which is the most simila to all agents popositions. To allow constuction of intevals, an attibute domain must fulfill an impotant condition. Thee must exist elation which odes elements of the domain. Additionally, in this aticle we impose anothe condition: all domain elements must be located on one axis. This condition guaantee that the following euation is tue: v i v j v j v k v i v k, when v i v j v k. It will help us to focus on inteesting domains, fo example numbes, which ae the most commonly used fo intevals.

5 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes Intevals will be epesented by odeed pais [i,i ], whee i is the beginning of the inteval and i is the ending of the inteval, i i. Fo example, [3,5] epesents an inteval with scope fom 3 to 5, and [5,5] epesents a single value 5. Attibutes which domains can not be odeed will only have single values: [i,i], o shotly: i. Let s see a case of conflict with inteval-value attibutes. Example. Suppose that out meteoological system gives also a foecast fo ain. Stations pedict when it is going to ain and popose intevals of time. Again, infomation about the conflict fo the subject Silesia is gatheed in Table. Agent Region RainingTime station 1 Silesia [10:00, 1:00] station Silesia [11:00, 14:00] station 3 Silesia [11:30, 13:00] Table : Relation Rain with intevals of values. Having a conflict situation, we would like to calculate an optimal consensus. Unfotunately, befoe detemining consensus, thee is always an impotant issue to conside: the choice of a distance function. A distance function calculates how diffeent (o simila) ae values poposed by agents. Fo the same conflict situation, algoithms facilitating diffeent distance functions may etun completely diffeent esults. It is cucial to choose such a function than is not only easy to calculate but also always etuns intuitive esults. Calculating consensus fo inteval-value attibutes is not diffeent in this aspect. Let s conside function δ Sym-Dif (inteval 1,inteval ) defined as length of span coveed by only one of the intevals. It woks just fine when intevals ae intesecting, e.g.: δ Sym-Dif ([1,4],[,5]) < δ Sym-Dif ([1,4],[3,5])3 o δ Sym-Dif ([1,4],[,5]) < δ Sym-Dif ([1,4], [3,6])4. But it stats to wok unintuitive when intevals has no common span. It does not notice the distance between intevals: δ Sym-Dif ([1,3],[4,5]) δ Sym-Dif ([1,3],[5,6]) δ Sym-Dif ([1,3],[6,7]) 3. Because it is wong to assume that all popositions in conflict will always have common span, we have to use a little moe complex distance function. Let s conside function δ R, defined as follows: Definition. Distance δ R between two intevals and euals the sum of: half of the length of the pat of which is outside of, half of the length of the pat of which is outside of, the length of span between and. Function δ R suits vey well fo calculating diffeence between intevals. Let s conside some of its popeties (consideed also in [Nguyen, 0a] and [Nguyen, 0b], but fo intevals simulated by sets of neighboing elements): the distance inceases while one of the intevals expands on the diection opposite to the second inteval: δ R ([1,3], [5,7]) < δ R ([1,3], [5,8]), the distance deceases while one of the intevals expands on the diection to the second inteval: δ R ([1,3], [5,7]) > δ R ([1,3], [4,7]),

6 the distance inceases while span between the intevals inceases: δ R ([1,3], [5,7]) < δ R ([1,3], [6,8]), the distance between intevals with length 0 points euals the egula distance between numbes: δ R ([3,3], [5,5]) 3-5, the function keeps its popeties fo both intesecting and sepaated intevals. As one can see, the function etuns vey intuitive esults. Fotunately, it also is vey easy to calculate. Let s intoduce the following theoem: Theoem 1. Let set V contain elements which can be odeed. Let the elements fom set V fulfill the euation: v i v j v j v k v i v k, when v i v j v k. Let [, ] and [, ] be intevals of elements fom set V. The distance δ R between intevals and euals: ), ( R δ. Poof. In such conditions, thee ae thee possible vaiants of locating two intevals: a) intevals ae sepaated:, b) intevals have a common span, c) one inteval contains the othe. In vaiant a) we have: ] [ ] [ )] ( ) [( )] ( ) [( ) ( ) ( ) ( ) (. The obtained esult coesponds to the definition of function δ R (Definition 4). In vaiant b) we have: ] [ ] [ 0. Again, the esult matches the definition. In vaiant c) we have: ] [ ] [ 0 0 ] [ ] [. Finally, also in this vaiant the definition is fulfilled. Anothe advantage of using function δ R is vey uick pocedue of detemining consensus fo single attibute. Let s conside the following theoem: 334 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes...

7 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes Theoem. Lets have a conflict situation descibed by at least one attibute x. Let attibute domain V x contain elements which can be odeed. Let the elements fom both set V x fulfill the euation: v i v j v j v k v i v k, when v i v j v k. Let the popositions in the conflict be intevals of elements fom set V x. Let function δ R be used fo measuing distance between intevals. An inteval which stats fom a cental element among popositions fo inteval opening and ends in a cental element among popositions fo inteval closing is a consensus in the conflict situation. Poof. Inteval [i,i ] is a consensus if its distance to all popositions (pofile X) is the smallest. When we use δ R, the distance may be tansfomed to: δ R([ i, i ],X) δ R([ i, i ],[ x, x ]) [ x, x ] X 1 [ x, x ] X [ x, x ] X [ x, x ] X ( x i 1 x i x i. x i ) In wok [Zgzywa, 04] the autho poved, that in the same conditions (odeed elements of V which can be located on one axis) consensus calculated fo poposition that ae singleton elements ( x i ) is always the cental element. (If thee is an x X even numbe of popositions, any element between both cental popositions is a consensus). Now, ou esult sum is the smallest when both subsumes ae the smallest. The fist subsume is in fact the distance of the inteval stat to all the popositions fo inteval stat. So it will be the smallest if we choose the cental element. The second subsume is the distance of the inteval end to all the popositions fo inteval end. Again - it will be the smallest if we choose the cental element. Thus, it is tue that if an inteval stats fom a cental element among popositions fo inteval opening, and ends in a cental element among popositions fo inteval closing then it is a consensus. Now, since we have chosen a satisfying distance function, we can etun to Example. Using Theoem, we can easily detemine consensus fo the conflict situation in elation Rain. The foecast fo egion Silesia should contain infomation about ain fom 11:00 to 13:00. 4 Some Aspects of Attibute Dependencies In Definition 1, condition b) is the most impotant. It euies the tuples C(s, B and C(s, B to be detemined in such a way thus the sums ( B, C( s, B ) and ( B pofile( B pofile(, C( s, ) ae minimal. These tuples could be calculated in the following way: fo each attibute b B one can detemine sets C(s, b and C(s, b, which minimize sums ( b, C( s, b ) and ( b, C( s, b ) espectively. This way is pofile( pofile(

8 336 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes... an effective one, but it is coect only if the attibutes fom set B ae independent (Fφ). In this section we conside consensus choice assuming that some attibutes fom set B ae dependent on some othes. The definition of attibute dependency given below is consistent with those given in the infomation system model ([Pawlak, 93], [Skowon, 9]): Definition 3. Attibute b is dependent on attibute a if and only if thee exists a sujective function f b a : V a V b fo which in conflict ontology <A,X,P,F> (f b a F) fo each elation P P of type T P and a,b T P the fomula ( P)( ( a [a,a ]) > ( b [f a b (a ), f a b (a )]) ) is tue. The dependency of attibute b on attibute a means that in the eal wold if fo some object the value of a is known then the value of b is also known. In pactice, owing to this popety fo detemining the values of attibute b it is enough to know the value of attibute a. Instead of [f a b (a ), f a b (a )] we will wite shotly f b a ([a,a ]). The given definition of attibute dependency is vey simila to the concept of function dependencies in the elational model. Howeve, thee is an impotant diffeence that should be emphasized. The dependency function fom Definition 3 tansfoms single element fom the domain of attibute a into one element fom the domain of attibute b. The dependency function in elational model tansfoms an inteval of elements fom the domain of attibute a (whole poposition a ) into an inteval of elements fom the domain of attibute a (whole poposition b ). Additionally, it is not possible to decompose a elation P into two elations accoding to a dependency function (like in elation model) because it is not even in the Fist Nomal Fom. Conside now a conflict situation s<{p,p },A B>, in which attibute b is dependent on attibute a whee a,b B. Let pofile( be the positive pofile fo given conflict subject e Subject(s). The poblem elies on detemining consensus fo this pofile. We can solve this poblem using two appoaches: 1. Notice that pofile( is a elation of type B {Agent}. Thee exists a function fom set TYPE(B {Agent}) to set TYPE(B {Agent}\{b}) such that fo each pofile pofile( one can assign exactly one set pofile'( { B {Agent}\{b} : pofile( }. Set pofile'( can be teated as a pofile fo subject e in the following conflict situation s' <{P,P }, A B\{b}>. Notice that the diffeence between pofiles pofile( and pofile'( elies only on the lack of attibute b and its values in pofile pofile(. Thus one can expect that the consensus C(s, fo pofile pofile( can be detemined fom the consensus C(s,' fo pofile pofile(' afte adding to tuple C(s,' attibute b and its value which is eual to f b a (C(s,' a). In the simila way one can detemine the consensus fo pofile pofile(.. In the second appoach attibutes a and b ae teated independently. That means they play the same ole in consensus detemining fo pofiles pofile( and pofile(. The consensus fo pofiles pofile( and pofile( ae defined as follows: Definition 4. The consensus fo subject e Subject(s) consideed in situation s<{p,p },A B> is a pai of tuples (C(s,,C(s, ) of type A B, which satisfy the following conditions:

9 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes a) C(s, AC(s, Ae and C(s, B C(s, Bφ, b) C(s, bf b a (C(s, a) and C(s, bf b a (C(s, a), c) The sums ( B, C( s, B) and ( B, C( s, B) ae minimal. pofile( pofile( We ae inteested in the cases when conditions b) and c) of Definition 4 can be satisfied simultaneously. The uestion is: Is it tue that if set C(s, a is a consensus fo pofile pofile( a (as the pojection of pofile pofile( on attibute a) then set f b a (C(s, a) will be a consensus fo pofile pofile( b (as the pojection of pofile pofile( on attibut? 5 Poblems of Detemining Consensus with Attibute Dependencies It has been shown that consensus fo pofiles consisting of independent attibutes may be detemined sepaately fo each attibute. But is this appoach also suitable when thee ae some dependencies between attibutes? Let us conside the following example. Example 3. Let the meteoological system descibed in Section 1 give also some infomation about ai humidity and elated foecast of comfot level fo meteopaths (people who ae vey sensitive to weathe conditions). Thee is a possibility of conflict in knowledge about ai (Ai ) which is descibed using attibutes: humidity (fom 0 to 100%) and comfot level (bad, good). Indeed, a conflict situation takes place efeing to the county egion Silesia at 10 AM. Table 3 contains data collected by meteoological stations. Agent Region Time Humidity Comfot level station 1 Silesia 10:00 AM 5 Bad station Silesia 10:00 AM 55 Good station 3 Silesia 10:00 AM 85 Bad Table 3: Relation Ai. Additionally, attibute Comfot level depends on attibute Humidity: Humidity Comfot level 0 h 30 Bad 30 < h 80 Good 80 < h 100 Bad Table 4: Dependency function between Comfot level and Humidity. Now, let us detemine the consensus sepaately fo each attibute. The distance function fo attibute Humidity etuns the diffeence between two values, e.g. δ Humidity (75%,35%) 40. The function fo attibute Comfot level etuns the

10 338 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes... numbe of anks between comfot levels, e.g. δ Comfot_level (bad, good) 1 and δ Comfot_level (bad, bad) 0. Afte assembling the esults, the following tuple may be constucted: Region Time Humidity Comfot level Silesia 10:00 AM 55 Bad Table 5: The esult calculated fo independent attibutes. Table 4 shows that when humidity is about 55% the comfot level should be good, not bad. Appaently, the obtained esult does not fulfill the dependencies between attibutes. Theefoe it cannot be consideed as a consensus fo this situation. It would also be incoect to detemine the consensus fo humidity attibute and to calculate the comfot level attibute using the dependency function. As it was shown, dependencies between attibutes ae not always tue in calculated consensus. The limitations guaantying detemining a coect consensus fo single-element values of attibutes wee consideed in pevious woks: [Zgzywa, 04], [Zgzywa, 05]. In [Zgzywa, 05_] and [Zgzywa, 06] the autho found these limitations fo values epesented by sets. A few diffeence functions wee consideed thee. In this aticle we will focus only on detemining consensus with dependencies of inteval values of attibutes. We will conside the following poblems: 1. What is the geneal algoithm of calculating the optimal consensus.. What ae the limitations guaantying that dependencies takes place in consensus calculated fo sepaate attibutes. 6 Consensus Detemining with Dependencies of Inteval-Value Attibutes The fist issue to conside is if consensus calculated fo evey attibute sepaately will always fulfill attibutes dependencies. The following example poves that it is not tue. Example 4. We have a conflict situation in elation P, descibed in Table 6. Agent a b agent 1 [1,] [11,15] agent [1,3] [10,11] agent 3 [4,5] [11,16] Table 6: The conflict situation in elation P. Additionally, attibute b depends on attibute a. The dependency is descibed in Table 7. V a V b

11 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes Table 7: Dependency function between b and a. We will stat fom attibute a. Accoding to Theoem, the consensus fo sepaate attibute a is [1,3]. Unfotunately, when we epeat the pocedue fo attibute b it will esult with inteval [11,15]. As we can see, the dependency is not tue (f a b ([1,3]) [11,15]) so ou consensus is inconsistent. The example showed, that it is wong to detemine consensus in the uickest way: to calculate consensus only fo attibute a (C a) and to use f a b (C a) as consensus fo b. This pai is not necessay optimal maybe pai (f a b ) -1 (C b) and C b would be bette? O some othe pai? Unfotunately, without imposing futhe conditions, a naive algoithm must be used to find an optimal consensus. Algoithm 1, pesented below, ties evey possible inteval fo attibute a and its image fo attibute b and etuns the best pai. But the algoithm is bette than simple naive algoithm. It does not iteate though whole V a it naows the set to elements which ae possible to give the best esults. It ejects elements fo witch eithe the element and its image ae befoe the ealiest poposition o afte the latest poposition. Algoithm 1. Algoithm of calculating consensus. Input: X conflict pofile, f a b dependency function, δ R similaity function. Output: C a consensus fo attibute a, C b consensus fo attibute b. Step 1. Sum maximum, i 1, j 1. Step. Pepae set PopSpan b containing all the elements fom V b gate o inside the smallest poposition fo attibute b (min(x b )) and lesse o inside the biggest poposition fo attibute b (max(x b )): PopSpanb { x x Vb min(xb ) [ min, min ] max(xb ) [ max, max ] min x max }. Step 3. Pepae set Pops a containing all the elements fom V a that geneates image fom PopSpan b : a Popsa { x x Va fb ( x) PopSpanb}. Step 4. Pepae ode PopSpan a containing ascending odeed elements fom V a gate o eual than min(pops a ) and lesse o eual than max(pops a ): PopSpana { x x Va min( Popsa ) x max( Popsa )}. Step 5. If j > count(popspan a ) then i i 1,

12 340 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes... j i. Step 6. If i > count(popspan a ) then go to step 11. Step 7. newsum δ R ([PopSpan a (i), PopSpan a (j)],x a ) δ R (f b a [PopSpan a (i), PopSpan a (j)],x b ) Step 8. If newsum < Sum then: Sum newsum, C a [PopSpan a (i), PopSpan a (j)], C b f b a [PopSpan a (i), PopSpan a (j)]. Step 9. j j 1. Step 10. Go to step 5. Step 11. Retun C a, C b. The pesented algoithm has the following popeties: Theoem 3. Lets have a conflict situation descibed by at least two attibutes a and b. Let attibute b depend on attibute a accoding to function f b a. Let attibutes domains V a and V b both contain elements which can be odeed. Let the elements fom both sets V a and V b fulfill the euation: v i v j v j v k v i v k, when v i v j v k. Let the popositions in the conflict be intevals of elements fom sets V a and V b. Let function δ R be used fo measuing distance between intevals. It is tue that: a) The complexity of Algoithm 1 is O(n ), whee n is the size of V a. b) The algoithm etuns an optimal consensus. Poof. a) The algoithm consists of two pats. In the fist pat, the algoithm pepaes a span of elements which should contain the esult. To accomplish this task, it iteates once though V b and twice though V a. The complexity of this pat is then O(n). In the second pat, the algoithm iteates though pepaed span, checking all the possible inteval ends fo all the possible inteval stats. As the opeation of calculating distance between two intevals is vey simple, the complexity of steps 5-11 is O(n ). Thus, the complexity of whole algoithm is also O(n ). b) The algoithm ties evey possible inteval in the pepaed span. Thus, if the optimal consensus lays in the span, the algoithm will find it. Then, it is only necessay to pove that the optimal consensus does not lay outside the span. In steps, 3, 4 the algoithm ejects elements fo witch both the element and its image ae befoe the ealiest poposition o afte the latest poposition. If an inteval and its image (tuple i ) stat and end befoe the ealiest poposition, then the distance fom tuple i to all popositions is bigge than the distance fom the ealiest poposition to all popositions. The situation is analogical fo tuples afte the latest poposition they geneate bigge distances to all popositions than the latest poposition. As the ealiest and the latest poposition lay in the span and will not be ejected, then it is cetain that the algoithm will find bette tuples than these outside the span. This poves that the algoithm gives the optimal esults. As one can see, the algoithm complexity is not vey high only O(n ). Unfotunately, n is not the count of agents popositions but is the size of V a (o its pat). It will wok vey fast if all popositions in a conflict ae close to each othe. But what if the popositions ae scatteed though whole V a? O elements fom V a ae not natual numbes but eal numbes with high pecision? In the next section we will ty to find conditions, which allow to use less calculations duing detemining an optimal consensus.

13 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes Conditions Sufficient fo Teating Attibutes Independently On the pevious section we consideed the example, whee consensus calculated fo each attibute sepaately did not fulfilled the dependency function. It bought us to the conclusion that geneally detemining optimal consensus needs time-consuming calculations fo both attibutes. But is it possible to educe the complexity of the poblem? Let s conside anothe example. Example 5. We have a conflict situation in elation P, descibed in Table 8. Agent a b agent 1 [,4] [1,16] agent [3,3] [14,14] agent 3 [,5] [1,19] agent 4 [3,4] [14,16] agent 5 [7,8] [,3] Table 8: The conflict situation in elation P. Additionally, attibute b depends on attibute a. The dependency is descibed in Table 9. V a V b Table 9: Dependency function between b and a. When we use Algoithm 1 fo the conflict, we will get the esult [3,4] fo attibute a and [14,16] fo attibute b. But when we calculate consensus fo each attibute sepaately (using Theoem ), the esult will be the same. What makes the diffeence? Let s conside the following theoem. Theoem 4. Lets have a conflict situation descibed by at least two attibutes a and b. Let attibute b depend on attibute a accoding to function f a b. Let attibutes domains V a and V b both contain elements which can be odeed. Let the elements fom both sets V a and V b fulfill the euation: v i v j v j v k v i v k, when v i v j v k. Let the popositions in the conflict be intevals of elements fom sets V a and V b. Let function δ R be used fo measuing distance between intevals. If function f a b does not change the ode of elements ([v i v j ]>[f a b (v i ) f a b (v j )]) then the dependency takes place in consensus calculated fo sepaate attibutes. The theoem claims, that if a dependency function does not change the ode of elements then it is not necessay to calculate consensus using Algoithm 1. In such a

14 34 Zgzywa M.: Consensus Detemining with Dependencies of Attibutes... case it is enough to calculate consensus only fo attibute a (C a) and use f a b (C a) fo attibute b. Poof. Theoem claims, that an inteval which stats fom the cental poposition fo inteval openings and ends in the cental poposition fo inteval closings is a consensus. This means that if [a,a ] is a consensus fo attibute a than a is the cental poposition fo inteval openings fo attibute a and a is the cental poposition fo inteval closings fo attibute a. As the dependency function does not change the ode of elements, it is tue that f a b (a ) is the cental poposition fo inteval openings fo attibute b and f a b (a ) is the cental poposition fo inteval closings fo attibute b. Thus it is cetain that f a b ([a,a ]) is a consensus fo attibute b. As it was shown, it is vey easy to detemine an optimal consensus when conditions fom Theoem 4 ae met. Not only it is necessay to calculate consensus fo just one attibute, but thanks to Theoem also the detemining pocess itself is vey uick. When the conditions fom Theoem 4 ae not met, Algoithm 1 must be used. 8 Conclusions In this pape we descibed how dependencies of inteval attibutes influence the possibilities of consensus detemining. Assuming that δ R distance function is used, the limitations of dependency functions wee shown, guaantying detemining a coect consensus despite of teating attibutes independently. Using such functions povides the following pofits. Fist of all, they enable detemining a consensus fo only a pat of the attibutes (the est may be calculated using dependency functions). Secondly, they pevent fom detemining an incoect consensus, which does not fulfill some of the dependencies of attibutes. Additionally, the algoithm of consensus detemining was shown, which may be used when the limitations ae not met. Algoithms of consensus detemining fo conflicts with attibute dependencies may be used to solve many pactical poblems. One good example is the poblem of best book ecommendation in intenet bookstoes. Afte a custome chooses one book, an algoithm should ecommend anothe. The algoithm uses infomation about choices of othe buyes of the fist book. But it could also facilitate the fact, that thee is an attibute dependency {book}->{domain}. Sometimes it is bette to ecommend a book fom a vey popula domain than a book with a slightly highe numbe of copies sold but fom an unpopula domain. A bookstoe with an algoithm facilitating attibute dependencies may geneate highe income fom coss-selling. The theoems pesented in this pape do not solve all of the poblems of this aea. The following issues, among othe, need to be consideed: which othe limitations ae necessay when many attibutes may depend on many attibutes? does using the suae of distance in consensus detemining influences on pesented theoems? how to calculate coect consensus fo diffeent element stuctues (odes, divisions, tees)?

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