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https://arxiv.org/pdf/1605.00495v1.pdf

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{
  "title": "Coalition Formability Semantics with Conflict-Eliminable Sets of  Arguments",
  "authors": [
    "Ryuta Arisaka",
    "Ken Satoh"
  ],
  "abstruct": " We consider abstract-argumentation-theoretic coalition formability in this\nwork. Taking a model from political alliance among political parties, we will\ncontemplate profitability, and then formability, of a coalition. As is commonly\nunderstood, a group forms a coalition with another group for a greater good,\nthe goodness measured against some criteria. As is also commonly understood,\nhowever, a coalition may deliver benefits to a group X at the sacrifice of\nsomething that X was able to do before coalition formation, which X may be no\nlonger able to do under the coalition. Use of the typical conflict-free sets of\narguments is not very fitting for accommodating this aspect of coalition, which\nprompts us to turn to a weaker notion, conflict-eliminability, as a property\nthat a set of arguments should primarily satisfy. We require numerical\nquantification of attack strengths as well as of argument strengths for its\ncharacterisation. We will first analyse semantics of profitability of a given\nconflict-eliminable set forming a coalition with another conflict-eliminable\nset, and will then provide four coalition formability semantics, each of which\nformalises certain utility postulate(s) taking the coalition profitability into\naccount.\n",
  "pdfurl": "https://arxiv.org/pdf/1605.00495v1",
  "references": [
    "] I taking the coalition prof itability into account. A . 1 Introduction s c Coalition formation among agents is an important topic in many ",
    "[  domains including economics, political science, and computer sci- 1 ence. Two groups of agents, by teaming up together, could achieve v a task which they cannot on their own. Exploring abstract argumen- 5 tation theory for finding an apt characterisation of coalition forma- 9  bility looks specially rewarding, and there are already a few papers 4  in the literature looking at this subject matter: with preference-based 0 argumentation frameworks and task allocations [1]; with a coopera- 0 tive goal generation and fulf illing ",
    "[7] respecting the property of reci- . 5 procity, i.e.agents give to the coalition they are in and benef it from 0 it; and with dialogue games and pay-offs ",
    "[21]. Shared by them is the 6 theme of identifying a group of individual agents who optimise ben- 1 efits/social welfare to themselves by being in the group. The optimal : v group thus formed is free of internal conflicts, and the participating i  agents do not have to give up anything by being a part. We consider X these types of coalitions supportive. r  The sort of coalition formationwehaveinmind,ontheotherhand, a  is one that may be found in political alliance among political parties. Such alliance is motivated if, for example, political parties want  to reach the voting threshold of passing certain bills. A politicalalliance-like coalition exhibits the following unique characteristics:  More organisational than individual It is not possible to freely  move agents across multiple political parties such as to connect  1 National Institute of Informatics, Japan, email<EMAIL_ADDRESS>2  National Institute of Informatics, Japan, email<EMAIL_ADDRESS>those having close interests together for formation of optimal political parties. A participant to a political party is often expected  to stay in the party during the parliament term. The assumption of  self-interested agents as studied in ",
    "[21] does not fit very well here.  There are repercussions to prof itability of a coalition, too, in that it  is primarily for a party’s, or parties’, benef its than the participants’  benef its that a political alliance is formed. Partial internal conflicts Agents in a political party should support  the party’s agendas and policies. Hence they do not defeat each  other about them, broadly spoken. It is common, however, that  there are smaller factions within a political party arguing against  one another over details. This often entails that some participating  agents cannot proclaim their opinions on certain policies as the  party’s opinions. Put another way, some individual opinions may  be toned down for the party. Asymmetry in attacks to and from a coalition For a political alliance to retain any credibility for the arguments it expresses, it  must argue only by the conflict-free, i.e. self-contradiction-free,  portion of the arguments of the party’s participants. But the other  political parties not in the alliance are unhindered by the personal  circumstance of the alliance. If a political party A is in alliance  with another political party B, then an external party C can argue  against any argument of the individual participants in A or in B as  an argument against the coalition. Better larger than smaller In ",
    "[1], the rate of defections is associated to the number of agents in a coalition, thus a smaller set preferred. That does not carry over here: if one single political party  or one single political alliance dominates the parliament, it has total freedom in policy making, which is clearly desirable. While  not primarily on abstract argumentation, there is a work ",
    "[8] on  alternating-temporal logic incorporating the framework of ",
    "[1]. In  the logic, a larger set is better. In this work, we contemplate abstract-argumentation-theoretic char- acterisations of prof itability, and then formability, of a coalition of this kind. More specif ically we consider the following questions: (1) supposea set of arguments that may contain partial internal conflicts (as an abstract representation of a political party) and suppose also rational criteria of prof itability, with which other (disjoint) sets of ar- guments that may also contain partial internal conflicts (i.e. represen- tations of other political parties) can it prof itby forming a coalition?; and (2) suppose the prof itability relation, supposesome rational prin- ciples to judge the goodness of a coalition, and suppose such a set of arguments, which other (disjoint) such sets of arguments can it actu- ally form a coalition?  The following are particularly interesting technicalities in the derivation of these semantics. First and foremost, the above- described coalition formability is not simply about whether the re- sulting coalition is acceptable, which could be handled by adapting the standard acceptability semantics in abstract argumentation the- ory, but about whether a set of arguments potentially with partial internal conflicts forms a coalition with another similar set. The for- mer concerns the state of the resulting set, while the latter must be parametrised by coalition prof itabilities of both sets. Secondly, be- cause of the presence of partial internal conflicts and of the asymme- try in attacks to and from a coalition, we have to: (1) obviously need a weaker notion than conflict-freeness asa property that a set of ar- guments should primarily satisfy - conflict-eliminability as we term it, which permits members of a set to attack other members of the same set so long as none of them is completely defeated; (2) obtain intrinsic arguments of a conflict-eliminable set, which are the argu- ments that would remain if any partial internal conflicts within the  3 set were resolved away ; and (3) use the intrinsic arguments to de- termine which external arguments are being attacked by the conflict- eliminable set, while still keeping the original arguments of the set in order to determine if it is being attacked by external arguments (see Asymmetry in attacks to and from a coalition above). Here again, our task is not just whether some set of arguments satisf ies conflict- eliminability, and we must consider if any attacks are strong enough to defeat an argument, and, in case an argument is attacked but not defeated, how much it would be weakened/compromised by the at- tacks. To cope with these, we attach argument capacity, a numerical value, to each argument, and an attack strength, again a numerical value, to each attack. The idea is: a set of arguments is conflict- eliminable just when none of the members of the set attack an ar- gument of the same set with a greater numerical value than the argu- ment’s capacity. As the argument capacities and attack strengths are both numerical, it iseasy to derive its intrinsic arguments and their attacks on external arguments. 1.1 Related work We are not aware of other works in the literature of abstract ar- gumentation theory dealing with this kind of political-alliance-like coalition formation. Also, to the best of our knowledge, the previous approaches proposed in the abstract argumentation literature are not self-suff icient for dealing with the two above-mentioned technicali- ties. We have already mentioned the key works on coalition forma- tion ",
    "[1, 7, 21]. They apply Dung’s acceptability semantics ",
    "[13] for characterising acceptability of a coalition that is individual-benef it- oriented, that is conflict-free, and that generally prefers a smaller set. In Section 4 of ",
    "[7], a coalition is associated to a set of tasks/goals, and conflicts between coalitions are measured such as by competition which occurs when two coalitions share the same tasks/goals. We, however, focus on political-alliance-like coali- tion prof itability/formability semantics with conflict-eliminable sets of arguments. We do not consider the meta-knowledge of tasks or goals. Instead, we measure prof itability of a coalition for a conflict- eliminable set of arguments by: (1) the size of the coalition; (2) whether the number of attackers to the conflict-eliminable set of arguments increases or decreases in the coalition; and (3) how de- fended the coalition is from external arguments. Coalition forma- bility is relativised to prof itabilities of two conflict-eliminable sets. A rather different perspective of coalition formation: calculation of 3  This should not be confused. We are not meaning that some arguments  would disappear as the result. The internal conflicts are assumed nondefeating. We rather mean that some arguments may be weakened of their  potency. See Partial internal conflicts. What would remain are then those  arguments unaffected by the partial internal conflicts and those weakened  arguments. probabilistic likelihood of a coalition formation capable of achieving sometask,givenpriorprobabilityofagents’joininginacoalitionand of preventing other members from joining in the coalition, was high- lighted in probabilistic argumentation frameworks ",
    "[19]. Such quan- titative judgement is out of the scope of this work.  We mention other works relevant to ours. Attack-tolerant abstract argumentation Characterisation of acceptability semantics for a non-conflict-free set of arguments is  gaining attention. Conflict-tolerant semantics ",
    "[3] relaxes conflictfreeness by using four values (accept, reject, no opinion, and  mixed feeling) for labelling arguments for para-consistent abstract argumentation. However, this approach does not incorporate numerical values, which makes it diff icult to reason about  the strength of attacks. Weighted argument systems ",
    "[15] attach  numerical values to attack relations. There is also a system-wide  numerical value called the inconsistency budget. In their systems,  conflicts in a set of arguments are quantif ied as the sum of numerical values given to the attack relations appearing in the set. If  the sum does not exceed the given inconsistency budget, then the  set is considered para-conflict-free. Their acceptability semantics  is relative to the global inconsistency budget. In our setting, having numerical attack strengths alone is not suff icient, aswe must  know intrinsic arguments of a conflict-eliminable set, which relies  both on attack strengths and on argument strengths. We do not use  any global and uniform budget. Further, substantively we do not  tolerate any inconsistency: intrinsic arguments must be conflictfree. Social abstract argumentation frameworks ",
    "[18], to which an  equational approach by Gabbay and Rodrigues ",
    "[16] also relates,  attach numerical values to arguments in the form of for votes and  against votes. They allow for fine-grained para-consistency. We  could potentially adapt their approach for characterisation of our  conflict-eliminability. However, their numerical attack characterisations by votes are quite specif ic. We choose more abstract, axiomatic characterisations. In the literature, axiomatic approaches  have been considered to for instance ensure logical consistency of  an argumentation framework ",
    "[2,9]. Classif ications of attack relations by axioms they satisfy have been also done ",
    "[17]. The axiomatic approaches help regulate an abstract argumentation system from a general standpoint. Dynamic abstract argumentation From one perspective, our  framework possessesa certain kind of dynamic nature. So far  dynamic changes have been considered within the literature  of abstract argumentation to: assume structural argumentation  (e.g. Assumption-based argumentation ",
    "[14]) and modify nonfalsif iable facts ",
    "[22]; add a new argument ",
    "[10]; revise attack  relations ",
    "[12]; revise an argumentation framework by encoding  it into propositional logic ",
    "[11]; and revise an argumentation  framework with an argumentation framework ",
    "[4].4 Given an  argumentation framework, these works calculate a revised  argumentation framework. That is, they derive a post-state  from a pre-state given some input. In our setting, however, we  require interactions between the initial set of arguments (the  pre-state) and intrinsic arguments of a conflict-eliminable set  (post-states) due to the asymmetry in attacks to and from a  coalition. The pre-state/post-state coordinations are, as far as we  are able to fathom, not dealt with in the above-mentioned studies.  Meanwhile, one of the works on coalition formation mentioned  earlier, namely ",
    "[7], allows for coalitional and non-coalitional 4  There are also analysis on mutability of acceptable arguments by adding or  removing an argument and/or an attack ",
    "[5,6].  views ofagentsFrom anon-coalitionview ofagents,anumber  of coalitionviews may bederived.Stilwhat theyconsiderare  conflictamong coalitionviews (fogoal fulf illmewhich is  a problempossessingadifferennature. 1.2 The remaining sections In therestwe willrecallNielsen-Parsonsargumentationframe- works ",
    "[20]thatgeneralisDung’s oneswithgroup attack(Section 2);introduceourargumentationframeworksfor conflict-eliminable setsofarguments(Sectio3);and developsemanticforprof itability, and thenformabilityofcoalitioformation,atthe same timepre- sentingtheoreticresults(Sectio4),beforedrawing conclusions. We omit triviproofs. 2 Preliminaries WhileDung’sargumentationframeworks",
    "[13]arethemostimportant in thabstracargumentationliteratuNielsen-Parsonsgeneralised versionwith groupattackareprobablyclosertoour own.Letus re- callthekeydefinitionoftheiframeworks.  An argument isanabstracentityandtheclassofallargumentsis A..AnnaarmeennttaatffmwoorrkkiiattuupA,G)e whheeAre⊆⊆ffAA  A\\∅) in and G :(2 × A.    We refer toany element of S by swith or withouta subscriptEach of them representsan argument. Not justany setof arguments will we be interestein, however. Definition 1 (Coherent sets of arguments) Let S1 be a subset of S. We say thatS1 is coherent ifS1 satisf ithefollowing condi- tions. 1. S1 isafinitesubsetof S. 2. For any (a,n) ∈ S1,it holdsthatn > 0. 3. For any (a,n) ∈ S1, thereisno m 6= n such that(a,m) ∈ S1 . An argumentation framework usuallysatisf ithefirstconditionTo explain the second condition,we mention thateach argument has argument capacity.For(a,n) ∈ S, a is its identifand,n is itca- pacity.An argument having argument capacityof0 basicallymeans thatthe argument has no utilitThe thirdpropertyisto ensurethat each argument identityisused at mostby one argument in achosen subset of S.From hereon, by S with orwithout a subscriptwe de- note acoherent setof arguments. S  Also let R beapartialfunction2 × S * N such thatit satisf ies  the following conditions (or axioms). Informally R(S,s) represents  the attack strength of S’s attack on s. In the below, ‘R is defined for  (S,s)’ is synonymous to ‘R(S,s) is defined’. 1. R is undef ined for (∅,s) for any s ∈ S ",
    "[Coherence]. 2. For any S1 ⊆ S ⊆f in S and for any s ∈ S, if R is defined for  (S1 ,s), then R is defined for any (S2 ,s) for ∅ ⊂ S2 ⊆ S1 .  ",
    "[Quasi-closure by subset relation]. 3. For any S1 ,S2 ⊆ S ⊆f S and for any s ∈ S, if R is defined both  in  for (S1 ,s) and for (S2 ,s), then R is defined also for (S1 ∪S2 ,s)  ",
    "[Closure by set union] 4. For any S1 ⊆ S ⊆f in S and for any s ∈ S such that R is defined  for (S1 ,s), it holds that R(S1 ,s) > 0 ",
    "[Attack with a positive  strength]. 5. For any (a,n),(a,m) ∈ S such that n ≤ m, the following holds  true: if R(S1 ,s) for some s ∈ S and for some S1 ⊆f in S such that  (a,n) ∈ S1 is defined, then R(S2 ,s) for S2 = (S1 \\(a,n)) ∪  (a,m) is defined and is such that R(S1 ,s) ≤ R(S2 ,s) ",
    "[Attack  relation monotonicity (source)]. 6. For any S1 ,S2 ⊆ S ⊆f S and for any s ∈ S, if R is defined for  in  (S1 ,s), (S2 ,s) and (S1 ∩ S2 ,s), then R(S1 ∩ S2 ,s) ≤ R(Si ,s)  for both i = 1 and i = 2 ",
    "[Attack strength monotonicity (source)]. 7. For any (a,n),(a,m) ∈ S such that n ≤ m, it holds that  if R is defined for (S1 ,(a,n)) for some S1 ⊆fin S such that  SSNl∈∈N  S1 ∩ {(a,l)} = ∅, then it is defined for (S1 ,(a,m)), and,  moreover, R(S1 ,(a,n)) ≤ R(S1 ,(a,m)) ",
    "[Attack relation and  strength monotonicity (target)] 8. For any S1 ⊆fin S aand for any s ∈ SS, R is undef ined for (S1 ,s)  if s ∈ S1 .",
    "[No self attacks].  ",
    "[Coherence] ensures that an attack must come from some argu- ment(s). ",
    "[Quasi-closure by subset relation] ensures that there is a group attack from a set of arguments on an argument just because each member of the set is attacking the argument. This can be con- trasted with the group attacks in Nielsen-Parsons’ argumentation frameworks. But it must be noted, as we are to mention shortly, that that there is an attack of an argument (a1 ,n1 ) on another argument (a2 ,n2 ) does not mean that (a1 ,n1 ) defeats (a2 ,n2 ) in our frame- work. ",
    "[Closure by set union] is the reverse of ",
    "[Quasi-closure by sub- set relation]. The purpose of ",
    "[Attack with a positive strength] is as follows: we mentioned earlier that R(S1 ,s) is the strength of attack by S1 on s, measured in N. The value being positive means that S1 is indeed attacking s. The value being 0 would mean that S1 is not at- tacking s. The purpose of an attack relation in abstract argumentation frameworks is to know which arguments attack which arguments. Consequently, we only consider positive values for R. ",
    "[Attack rela- tion monotonicity] expresses the following reasonable property: an attack may be occurring from some S1 ⊆ S ⊆fin S on some s ∈ S; now,increase theattackcapacityof justoneargument s1 ∈ S1 ,keep- ing all else equal; then the attack which occurred before the capacity increase should still occur. ",
    "[Attack strength monotonicity] expresses the property that ifan attack occurs from a set of arguments on an ar- gument with some strength, then any superset does not decrease the attack strength. To explain ",
    "[Attack relation and strength monotonic- ity], let us say that a set of arguments is attacking an argument with certain argument capacity. That intuitively means that the set intends to tone down the argument. Now, if the argument capacity of the argument increases, the set still intends to tone down the argument just as strongly or even more strongly, but not less strongly, for there are more materials in the argument that the set could attack. This is the direct reading of the condition. In the technical development to follow, the converse reading will be more useful: ifthe argument ca- pacity of the argument on the other hand decreases, the set may no longer intend to tone it down further. Finally, ",
    "[No self attacks] pre- vents self-contradictory arguments from being present.  Additivity of attack strengths is not postulated for R: if an ar- gument s1 is attacked by s2 and s3 such that R({s2 },s1 ) = n1 and such that R({s3 },s1 ) = n2 , it is not necessary that R({s2 ,s3 },s1 ) = n1 + n2 . The additivity holds good when each attack can be assumed independent, but may not in other cases. A generalisedversionof",
    "[Attackrelationmonotonicity]holdsgood.Let π be a projection function that takes a natural number and an ordered set List and that outputs a set member, such that π(n,List) = {the nn--ttmeembbeerr ooff LLiissIItt iiss uunnddeeffiifnniisgrreeaattetthhaantthhee ssiooffe the ordered set. Proposition 1 (Generalised attack relation monotonicity) Let S1 ⊆fin S and s ∈ S be such that R(S1 ,s) is defined. Then if  SSssxx SSssxx S2 ⊆fin S is such that: (1) ∈S1 π(1,sx ) = ∈S2 π(1,sx ); and that (2) (a,n) ∈ S1 materially implies (a,m) ∈ S2 for n ≤ m, then R(S2 ,s) is defined. Proof. By induction on the number of arguments s1 ∈ S1 and s2 ∈ S2 for which π(1,s1 ) = π(2,s2 ) and π(2,s1 ) < π(2,s2 ). Here and everywhere, we may make use of and (having the semantics of classical logic conjunction), distinguishing ‘and’ in formal contexts from ‘and’ in natural contexts for greater clarity. The base case is vacuous. Use ",
    "[Attack relation monotonicity] for inductive cases. 2 Our argumentation framework is (S,R) for some coherent set of ar- guments S and for some R. 3.1 Attacks  We distinguish complete attacks (defeats) from partial attacks (attacks).  Definition 2 (Attacks and defeats) WesaythatS1 ⊆ S attackss ∈  S iff there exists S2 ⊆⊆ SS11such that R is defined for (S2 ,s). We say  5  that S1 ⊆ S defeats s ∈ S iff S1 attacks s and R(S1 ,s) ≥ π(2,s).  Informally, an attack defeats its target when the attack strength surpasses the target’s argument capacity.  Definition 3 (Maximum attack strengths) We define V max (S1 ,s)  to be: 0 if S1 does not attack s; otherwise, R(S2 ,s) for some S2 ⊆  S1 such that: (1) R is defined for (S2 ,s); and such that (2) if R is  defined for (Sx ,s) for Sx ⊆ S1 ,then R(Sx ,s) ≤ R(S2 ,s).  3.2 Conflict-eliminable sets  Definition 4 (Conflict-eliminable sets of arguments) We say that  S1 ⊆⊆ S iiss ccoonnffmiinnaabblleeliimiifftthheerreexxiissnoo ss ∈∈ S11 such that  S1 defeats s.  Conflict-eliminability is a weaker notion of the usual conflictfreeness, i.e. the property that there exists no s ∈ S1 such that S1  attacks s. For the rationale behind obtaining this definition and using it as a primitive entity in our argumentation framework, let us  consider one mock example of a peer review which, just as political  alliance we saw does, exhibits partial internal conflicts: 1. (Authors) We have addressed a gap in the literature about LMN in  this paper. Our approach is entirely new. 5  See in the proof of Proposition 1 for what and is. 2. (A reviewer) My overall impression is that the paper contains suf-  ficiently new and interesting materials to be warranted acceptance.  Nonetheless, their claim that the approach considered in the paper  is entirely new is misleading. There are for instance papers by  XYZ et al that use a similar method to solve problems ABC. I  suggest that they include the references. 3. (The handling editor) I accept the paper on the condition that the  authors modify the part mentioned by the reviewer.  Now, if not wholly, the reviewer is stillattacking the authors. Despite  that, the decisions by the reviewer as well as by the handling editor  are overall favourable ones to them. They could take the expert opinion of the reviewer into account, weaken their claim; and then their  paper will be accepted. In this example, (1) that the authors have addressed some problem about LMN, (2) that a similar method was  used by XYZ et al. to solve problems ABC, and (3) that the paper  is accepted, are basically intrinsic arguments of the three arguments  given. The authors’ contribution is not completely defeated by the  reviewer, and the modif ication is a necessary compromise for the acceptance of their argument (that is, their paper). Intrinsic arguments  are the arguments that result from resolving, in a conflict-eliminable  set, internal non-defeating attacks.  S S  Definition 5 (Intrinsic arguments) Let α : 2 * 2 be such that  it is defined for S1 ⊆ S iff S1 is conflict-eliminable. If α is de- fined for S1 ⊆ S, then we define that α(S1 ) = {(π(1,s),n) | s ∈  max  S1 and n = π(2,s)−V (S1 ,s)}. We say that α(S1 ) are intrinsic  arguments of S1 .  Proposition 2 (Well-definedness) For any S1 ⊆ S, if α is defined  for S1 ,then every member of α(S1 ) is a member of S: in particular,  there exists no a ∈ A and no n ∈ N such that (a,−n) ∈ α(S1 ).  Intrinsic arguments of a conflict-eliminable set must be substantively  conflict-free.  Definition 6 (The view of intrinsic arguments) Let DelR (S,Sx )  bbee{SyyS ,,ss)||ss ∈∈ SSxx and Sy ⊆⊆ SSxx and R(Sy ,,ss)iiss ddeeffiinneedd..}},,  which is the set of attack relations within Sx . Now, let S1 be a  subset of S. If α is defined for S1 , then we say that ((S\\S1 ) ∪  α(S1 ),R\\Del(S,S1 )) is the view that S1 has about S, or simply  S1 ’s view of S. We denote S1 ’s view of S by ViewR (S,S1 ).  Proposition 3 Let S1 be a subset of S. If α(S1 ) is defined, then  α(S1 ) is conflict-free in ViewR (S,S1 ).  3.3 Coalition attacks, c-admissible sets and  c-preferred sets  A coalition may attack external arguments only by its intrinsic arguments.  Definition 7 (C-attacks and c-defeats) We say that S1 ⊆⊆ S c-attacks s ∈ S iff α is defined for S1 and there exists some  S2 ⊆ α(S1 ) such that π(2,ViewR (S,S1 )) is defined for (S2 ,s).  We say that S1 c-defeats s ∈ S iff S1 c-attacks s and  π(2,ViewR (S,S1 )))(α(S1 ),s) ≥ π(2,s).  There are no self c-attacks; Cf. Proposition 3.  Proposition 4 The following are equivalent. 1. S1 ⊆ S c-attacks s ∈ S. 2. S1 ⊆ S c-attacks s ∈ π(1,ViewR (S,S1 )).  Also, the following are equivalent. 1. S1 ⊆ S c-defeats s ∈ S. 2. S1 ⊆ S c-defeats s ∈ π(1,ViewR (S,S1 )).  Proof. By definition, π(2,ViewR (S,S1 ))(Sx ,s) is undef ined for  any Sx ⊆ S1 ∪ α(S1 ) and for any s ∈ S1 ∪ α(S1 ). Hence if  S1 ⊆ S c-attacks s ∈ S, then s 6∈ S1 and s 6∈ α(S1 ). Meanwhile,  (S\\S1 ) = (π(1,ViewR (S,S1 ))\\α(S1 )). 2  We now def ine the notions of c-admissible and c-preferred sets,  which are analogous to admissible and preferred sets we touched  upon in Section 2, but which are for conflict-eliminable sets. One  part in the def inition could appear diff icult at first. We italicise the  part, and elaborate it later.  Definition 8 (C-admissible/c-preferred sets) We say that S1 ⊆⊆  S is c-admissible iff α is defined for S1 and if S2 ⊆  π(1,ViewR (S,S1 )) attacks s ∈ S1 and if Sx ⊆ S2 is such that  R(Sx ,s) is defined, then there exists some S3 ⊆ α(S1 ) such that S3  c-defeat some sx ∈ Sx . We say that S1 ⊆ S is c-preferred iff S1  is c-admissible and there exists no S1 ⊂⊂ SSyy ⊆⊆ S such haat Syy is  c-admissible.  We explain why in the italicised part in the above definition it is  S2 ⊆ π(1,ViewR (S,S1 )) and not S2 ⊆ S; and why it is s ∈ S1  and not s ∈ α(S1 ). The notion of c-admissibility is intuitively the  sameas that of admissibility in Section 2: subsets of S2 are attacking  a conflict-eliminable set S1 ; and so S1 is not admissible unless some  member of S1 defeat all the attacking subsets of S2 . Now, because  we are presuming that S1 is conflict-eliminable and not necessarily  conflict-free, S2 ⊆ S would include any partial conflict in S1 as  an attack. However, c-admissibility, which is the admissibility of a  conflict-eliminable set in the view of the set, should not be defined to  defeat it. This explains why S2 ⊆ π(1,ViewR (S,S1 )) which compiles away all those purely internal partial conflicts. On the other  hand, the reason that it should be s ∈ S1 and not s ∈ α(S1 ) is because of the asymmetry in attacks to and from a conflict-eliminable  set. Recall Asymmetry in attacks to and from a coalition in Section 1.  3.4 Reduction to Nielsen-Parsons’ argumentation  frameworks  ↓  Theorem 1 (Restricted (S,R) as Nielsen-Parsons’) Let R be  such that it satisfies all but ",
    "[Quasi-closure by subset relation]. Let  S be such that for any S1 ⊆⊆ S aannddffooranyy s ∈ SS, iiRf R is defined  for (S1 ,s), then S1 defeats s. Then (S,R ↓) is Nielsen-Parsons’  argumentation framework, and the following all hold good. 1. Any conflict-eliminable set in S is a conflict-free set. 2. If α is defined for S1 ⊆ S, then α(S1 ) = S1 and  π(1,ViewR (S,S1 )) = S. 3. A c-attack by S1 ⊆ S on s ∈ S is an attack by S1 on s. 4. A c-attack is a c-defeat. 5. A c-admissible set is an admissible set, and a c-preferred set is a  preferred set.  Proof.  ↓ 1. Bydef inition,ifR isdefinedfor(S1 ,s),thenS1 defeatss.Hence  it isnecessary that a conflict-eliminable set be a conflict-free set. 2. By 1., it is vacuous that α(S1 ) = S1 . Further, Del(S,S1 ) = ∅,  and so π(1,ViewR (S,S1 )) = S. 3. By 2., both are trivial. 4. By definitionof (S,R ↓). 5. By 2.,both are trivial.  With these, itis straightforward to see that (S,R ↓) is NielsenParsons’ argumentation framework. 2  4 Semantics for coalition profitability and  formability  Itisof interestto learnmeaningful conflict-eliminablesets ofarguments.Inthissectionweshowsemanticcharacterisationsofcoalition  formabilityout of conflict-eliminablesets.Since coalition formation  presupposes at leasttwo groups of arguments, what we are characterisingisnot whether a set ofarguments is admissible and how good  anadmissible setover otheradmissible sets is, but whetheraconflicteliminable setcan form a coalitionwith another conflict-eliminable  set,and how good a coalitionover othercoalitions is.We presume  some conflict-eliminable set at fronand will talkof coalitionformabilitysemantics relativeto the conflict-eliminable set.We will first  discussprof itabilitrelationof a coalition,will show itstheoretical  properties,and willthen present four coalitionformability semanticseach of which formalises certain utilitypostulate(s)taking the  prof itabiliintoaccount. We assume the following notations.  Definition 9 (One-directional attacks) Let S1 ⊆ S be such that α(S1 ) is defined.We say that S1 is one-directionally attacked iff  thereexistsSx ⊆ π(1,ViewR (S,S1 ))such thatSx attackss ∈ S1  and S1 does not c-attackany sx ∈ Sx .  Definition 10 (States ofa conflict-eliminable set) Let ?: 2 S × 2S  be a binary relationsuch that (S1 ,S2 ) ∈?, written also S1 ? S2 ,  iffα is defined both for S1 and S2 and any of the three conditions  below is satisf ied: 1. S2 is c-admissible. 2. S1 is one-directionally attacked. 3. neither S1 nor S2 is c-admissible or one-directionally attacked.  Informally if Sx ⊆ S is c-admissible, then it is fully defended from  external attacks and is good. If Sx is one-directionally attacked, then  that means that Sx does not have any answer to external attacks,  which is bad. Any conflict-eliminable set that does not belong to either of them is better than being one-directionally attacked but is  worse than being c-admissible. Consequently, if S1 ? S2 , then S2 is  in a better state or in at least as good a state as S1 .  Definition 11 (Coalition permission) We say that coalition is permitted between S1 ⊆ S and S2 ⊆ S iiff S1 ∩ S2 = ∅ and α is  defined for S1 ∪ S2 .  The following results tell that this definition of coalition permission  isnot underspecif ied.  Lemma 1 (Defeats are unresolvable) Let S1 and s be such that  S1 ⊆ S and s ∈ S1 . If S1 defeats s, then for all S2 such that  S1 ⊆ S2 ⊆ S it holds that S2 defeats s.  Proof. By ",
    "[Attack strength monotonicity] of R. 2  Proposition 5 (Coalition and conflict-eliminability) If coalition is  permitted between S1 and S2 , then α is necessarily defined for S1  and S2 .  Proof. Suppose otherwise, then by conflict-eliminability of a set,  there must be an argument in Si ,i ∈ {1,2}, such that Si defeats  s ∈ Si .The required result obtains via Lemma 1. 2  We make one notion formally explicit for convenience, and then de- fine coalition prof itability.  S S  Definition 12 (Attackers) Let Attacker : 2 → 2 be such that  Attacker(S1 ) = {s ∈ S | there exists some s1 ∈ S1 such that s  attacks s1 .}. We say that Attacker(S1 ) is the set of attackers to S1 .  S S  Definition 13 (Coalition profitability) Let ? : 2 ×2 be such that  S1 ? S2 (or (S1 ,S2 ) ∈ ?) satisf iesthree axioms below. 1. S1 ⊆ S2 (larger set). 2. S1 ? S2 (better state). 3. |{s ∈ Attacker(S1 ) | S1 does not c-defeat s and s 6∈ S1 }| ≥  |{s ∈ Attacker(S1 ) | S2 does not c-defeat s and s 6∈ S2 }|  (fewer attackers).  Due to (better state), S1 ?S2 implies that α is definedboth for S1 and  S2 .By (larger set), a set that contains more arguments is a better set.  By (better state), a set that is in a better state is a better set. Finally,  by (fewer attackers), a set that is attacked by a smaller number of  arguments is a better set. The S1 in Attacker(S1 ) on the second line  is not a typo: this criterion is for measuring the prof its of coalition  formation for S1 .  Proposition 6 A (S,R) can be chosen in such a way that a pair of  S1 ⊆ S and S2 ⊆ S do not satisfy more than one axioms.  Proof. (larger set)  s3 s3  (a1,2) (a2,2) (a1,1) (a2,1)  Let S be {(a1 ,2),(a2 ,2),s3 }, and let R be such that it is defined only for any combination that matches the attack arrows  in the two drawings above. Def ine that R({(a1 ,2)},(a2 ,2)) =  R({(a2 ,2)},(a1 ,2)) = R({s3 },(a1 ,2)) = R({s3 },(a1 ,1)) =  R({(a2 ,2),s3 },(a1 ,2)) = 1, and that R({(a1 ,2)},s3 ) ≥  π(2,s3 ) (, among others implicit by the conditions of R). Then  the left and the right drawings represent (S,R) and respectively  ViewR (S,{(a1 ,2),(a2 ,2)}). Now, let S1 be {(a1 ,2)} and let  S2 be {(a1 ,2),(a2 ,2)}. Then clearly S1 ⊆ S2 . However, it  does not satisfy (better state): S1 is neither c-admissible nor onedirectionally attacked but S2 is one-directionally attacked. It does  not satisfy (fewer attackers), either: Attacker(S1 )) = {{ss3}},S11 cdefeats s3 ,and S2 c-defeats none. (better state)  s1 s2 s3  Let S be {s1 ,s2 ,s3 }, and let R be such that it is defined only mutually for s1 and s2 as shown above, and such that R({s1 },s2 ) ≥  π(2,s2 ) and R({s2 },s1 ) ≥ π(2,s1 ). Now, let S1 be {s1 } and  let S2 be {s3 }. Then S1 ? S2 because S2 is c-admissible. However, clearly it is not the case that S1 ⊆ S2 .Also, Attacker(S1 ) =  {s2 }, S1 c-defeats s2 , but S2 does not c-defeat it. (fewer attackers) Let S be {s1 ,s2 ,s3 }, and let R be such that it  is defined as is previously, and such that R({s1 },s2 ) < π(2,s2 )  aanndR(({{ss2}},,s)11< π(2,s1 ). Now, let S1 bbee {{ss3aanndlleeS2 be  {s2 }.ThenAttacker(S1 ) = ∅,andso(fewerattackers)issatisf ied  trivially.However, clearly it is not the case that S1 ⊆ S2 . And it  is not the case that S1 ? S2 ,because S1 is c-admissible, while S2  is neither c-admissible nor one-directionally attacked. 2  4.1 Theoretical results around profitability  It is triviato show that there exists some (S,R) and some S1 ,S2 ⊆  S such that S1 ? S1 ∪ S2 .We show other results.  Theorem 2 (Profitable coalition for subsets of a c-admissible set)  Let S1 ⊆ S be such that α(S1 ) is defined, and let Sx ⊆ S be a  c-admissible set.If S1 ⊂ Sx , then the following all hold good. 1. α is defined for S2 = Sx \\S1 . 2. Coalition is permitted between S1 and S2 . 3. S1 ? Sx .  Proof. 1. Suppose otherwise, then S2 would defeat at least one s ∈  S2 suchthat S2 woulddefeat s.Bylemma 1, Sx wouldthendefeat  s, for S2 ⊆ Sx . But Sx ,being c-admissible, does not defeat s. 2. α is defined for Sx by definition; and S1 ∩∩ SS22= ∅∅,aso by def-nition. 3. S1 ⊆ Sx ; Sx is a c-admissible set; and  |{s ∈ AAttacker(S1 ) | Sx does not c-defeat s and s 6∈ Sx }| = 0.  2  Theorem 3 (Existence theorem) If,for any S1 ⊆ S, there exists some S2 ⊆⊆ S su hh that coalion is perm ted between S11 and S2 and S1 ? S1 ∪ S2 and S1 ∪ S2 is c-admissible, then there exists some S3 ⊆ S such that coalition is permitted between S1 and S3 and S1 ? S1 ∪ S3 and S1 ∪ S3 is c-preferred and S2 ⊆ S3 . Proof. As S is of a finite size, it is straightforward to see that if S1 ∪ S2 is c-admissible, then there must exist some Sx such that it is c-preferred and that S1 ∪ S2 ⊆ Sx . We show that the choice of Sx \\S1 for S3 ensures the requirement to be satisf ied. By Theorem 2 coalition is permitted between S1 and S3 and S1 ? S1 ∪ S3 . By assumption, S1 ∪ S3 = Sx is c-preferred and S2 ⊆ S3 . 2 Theorem 4 (A c-preferred set as a mutually maximal coalition) Let S1 ⊆ S be such that α(S1 ) is defined, and let Pref(S1 ) be the set of all c-preferred sets that contain S1 as their sub- set. If Pref(S1 ) 6= ∅, then the following holds good: for any Sx ∈ Pref(S1 ), S1 ? Sx and (Sx \\S1 ) ? Sx and there exists no Sx ⊂ Sy ⊆ S such that S1 ? Sy or such that (Sx \\S1 ) ? Sy . Proof. By Theorem 2 and by definition of a set being c-preferrable. 2In  general, though, mutual prof itability is not guaranteed. Theorem 5 (Asymmetry of profitabilities) There exists an argu- mentation framework (S,R) such that there are disjoint subsets: S1 and S2 ,of S such that S1 ? S1 ∪ S2 , but such that it is not the case that S2 ? S1 ∪ S2 . Proof.  s3 s1 s3 s1  (a2,2) s4 (a2,1) s4  s5 s5  Let S be {s1 ,(a2 ,2),s3 ,s4 ,s5 }, and let R be such that it is defined only for any combination that matches the attack arrows in the drawings above, and such that: R({s1 },(a2 ,2)) = 1; R({s1 ,(a2 ,1)},s3 ) ≥ π(2,s3 ); R({s1 },s3 ) < π(2,s3 ); R({(a2 ,2)},s3 ) < π(2,s3 ); R({(a2 ,1)},s4 ) < π(2,s4 ); R({(a2 ,1)},s5 ) < π(2,s5 ); R({(a2 ,2)},s4 ) ≥ π(2,s4 ); R({(a2 ,2)},s5 ) ≥ π(2,s5 ); R({s3 },s1 ) ≥ 2; R({s4 },(a2 ,1)) ≥ 2; and R({s5 },(a2 ,1)) ≥ 2, among others. Now, let S1 be {s1 }, and let S2 be {(a2 ,2)}. Then S1 ? S1 ∪ S2 , for S1 and S1 ∪ S2 are both neither c-admissible nor one-directionally attacked. The axiom (fewer attackers) is also satisf ied. However, it is not the case that S2 ? S1 ∪ S2 , for it does not satisfy (fewer attackers). 2 Moreover, ? does not satisfy what we may at first expect hold good. Definition 14 (Continuation property of ?) Let S1 ⊆ S be such that α(S1 ) is defined, and let Max(S1 ) be the set of all Sx ⊆ S such that S1 ? Sx and such that if Sx ⊂ Sy ⊆ S, then it is not the case that S1 ? Sy . Then we say that ? is weakly continuous for S1 iff there exists some Sz ∈ Max(S1 ) such that, for any Sw ⊆ Sz , if coalition is permitted between S1 and Sw \\S1 , then S1 ?Sw .We say that ? is continuous for S1 iff it is weakly continuous for S1 for any Sz ∈ Max(S1 ). Theorem 6 (Coalition profitability discontinuation theorem) There exist S1 ,S2 ,Sx ⊆ S such that: (1) α is defined for S1 ,S2 and Sx ; (2) Sx ∈ Max(S1 ); and (3) S2 ⊆ Sx ,but such that S1 ? S2 does not hold good. Moreover, Max(S1 ) = Pref(S1 ) would not change this result. Proof. It suffices to consider the case where Max(S1 ) = Pref(S1 ).  s3  s3  s5 s5  s4 s4  (a1, 2) (a2,2) (a1,1) (a2,1) Let S be {(a1 ,2),(a2 ,2),s3 ,s4 ,s5 }, and let R be such that it is defined only for any combination that matches the attack arrows in the drawings above, and such that: R({(a1 ,2)},s4 ) ≥ π(2,s4 ); R({(a2 ,2)},s5 ) ≥ π(2,s5 ); R({(a1 ,2)},(a2 ,2)) = 1; R({(a2 ,2)},(a1 ,2)) = 1; R({s3 },s4 ) ≥ π(2,s4 ); R({(a2 ,1),s3 },s5 ) ≥ π(2,s5 ); R({s4 },(a1 ,1)) ≥ 2; and R({s5 },(a2 ,1)) ≥ 2, among oth- ers. Clearly α is defined for S1 = {(a1 ,2)}, Sa = {(a2 ,2)} S2 = S1 ∪ Sa , and Sx = {(a1 ,2),(a2 ,2),s3 }, among others.  Further, Sx is c-preferrable. However, it is not the case that  S1 ? S2 . In fact, it is also not the case that Sa ? S2 , since S2 is  one-directionally attacked (see the right drawing for the attacks  in ViewR (S,{(a1 ,2),(a2 ,2)})), whereas S1 and Sa are neither  c-admissible nor one-directionally attacked (see the left drawing). 2  Coalition prof itability continuation property satisf ies in certain special cases, however.  Theorem 7 (Coalition profitability continuation theorem) Let  Sx ⊆ S be a c-preferred set. Then ? is weakly continuous for  any S1 ⊂ Sx iff any disjoint pair Sy ,Sz of subsets of Sx satisfy  Sy ? Sy ∪ Sz .  Proof. If: Suppose, by way of showing contradiction, there are  three disjoint subsets of Sx : Sa ,Sb and Sc , such that Sa ?  Sa ∪ Sb 6?(Sa ∪ Sb ) ∪ Sc .By assumption we have (Sa ∪ Sb ) ?  (Sa ∪ Sb ) ∪ Sc ,contradiction.  Only if: By definition of weak continuation property of ?. 2  4.2 Coalition formability semantics  We use the prof itability relation to express our coalition formability  semantics. We set forth three rational utility postulates:  I Coalition is good when it is prof itable at least to one party.  II Coalition is good when it is prof itable to both parties. III Coalition is good when maximal potential future prof its are expected from it.  Of these, the first two can be understood immediately with the  prof itability relation. Our interpretation for the last postulate is as  follows. Suppose a party, some conflict-eliminable set S1 ⊆⊆ SS in our context, considers coalitionformation with another conflict- eliminable setS2 .We know thatS2 issome subset of S1 ⊆ S\\S1 . Before S1 forms a coalition with S2 , we have Max(S1 ) as the set of maximal coalitionspossible for S1. Once the coalition is formed, we have Max(S1 ∪ S2 )as theset ofmaximal coalitionspossible for the coalition. Here clearly Max(S1 ∪ S2 ) ⊆ Max(S1 ). What this means is thata particular choice of S2 blocks any possibilitiesin Max(S1 )MaxSS1 ∪∪ SS22 they become unrealisable from S1 ∪∪ SS.2 Hence S1 has an incentive not to form a coalition with such a S2 if all the members of Max(S1 ∪S2 ) are strictland comparatively less prof itablethan some member of Max(S1 ). We reflectthisintuition. Definition 15 (Maximal profitabilityrelation) Let ≤ll≤bb ≤ff : 2 S × 2 S be such thatthey satisfyall thefollowing: 1. S1 ≤l S2 iff|S1 | ≤|S2 |. 2. S1 ≤b S2 iffS2 is at leasas good by (better state)asS1 . 3. S1 ≤f S2 iffS2 is atleastas good by (fewer attackers)as S1 . We writeS1 <β S2 for each β ∈ {l,b,f} justwhen it is thecase that S1 ≤ββ S2 and it is not thcase that S2 ≤ββ S1 .Then we deine ?m :2S ×2 S to be such thatifS1 ?m S2 ,then both of thefollowing conditions satisfy: 1. S1 ? S2 . 2. Some Sx ∈ Max(S2 ) is such that, for all Sy ∈ Max(S1 ),  if Sx <β Sy for some β ∈ {l,b,f}, then there exists γ ∈  ({l,b,f}\\β) such that Sy <γ Sx . Intuitively,ifS1 ?m S2 ,then at leasone set iMax(S1 ) maximal in the three criteriathe setsize,the statequality and the number of ex- ternal attackers,is reachablefrom S2.The maximality here is judged by the principle that if S2 ,S3 ∈ Max(S1 ) are equal by two criteria, then S3 is better ifit is better than S2 by the remaining criterion.  We define four formability semantics: W which respects I, M which respects II (and implicitly also I), WS which respects I and III, and S which respects II and III. Definition 16 (Coalition formability semantics)  W(S1 ) = {S2 ⊆ S | S1 ? S1 ∪ S2 or S2 ? S1 ∪ S2 }.  M(S1 ) = {S2 ⊆ S | S1 ? S1 ∪ S2 and S2 ? S1 ∪ S2 }.  WS(S1 ) = {S2 ⊆ S | S1 ?m S1 ∪ S2 or S2 ?m S1 ∪ S2 }.  S(S1 ) = {S2 ⊆ S | S1 ?m S1 ∪ S2 and S2 ?m S1 ∪ S2 }. Here, or has the semantics of classical logic disjunction. Intuitively, ρ(S1 ) for ρ ∈ {W,M,WS,S} means that S1 is comfortable with forming a coalition with S2 ∈ ρ(S1 ) under the given criteria. Theorem 8 (The relation among coalition formability semantics) Let S1 ⊆ S be such that α(S1 ) is defined. The following all hold good. (1) M(S1 ) ⊆ W(S1 ). (2) WS(S1 ) ⊆ W(S1 ). (3) S(S1 ) ⊆ M(S1 ). (4) S(S1 ) ⊆ WS(S1 ). Meanwhile, neither WS(S1 ) ⊆ M(S1 ) nor M(S1 ) ⊆ WS(S1 ) is necessary. We provide one example to illustrate the semantic differences.  (a3, 2) s1 (a3, 2) s1 (a3, 1) s1  s4 s4 s4  (a2, 2) (a2, 1) (a2, 2)  s5 s5 s5  s6 s6 s6  s7 s7 s7 Let S = {{ss11,(a2 ,2),(a3 ,2),s4 ,s5 ,s6 }},, aanndd lleett R bbee ssuucchh that it is defined only for any combination that matches the attack arrows in the above drawings, and such that: RR(({{((aa22 ,,22))}},,((aa33,2)) = 1; RR(({{((aa22 ,,22))}},,ss44 )) ≥≥ ππ((22,,ss44 ); RR(({{((aa22 ,,22))}},,ss55)) ≥≥ ππ((22,,ss55 ); RR(({{((aa22 ,1),s1 }},,((aa33,,22)))) ≥≥ 22;; RR(({{((aa22 ,,11))}},,ss44) < π(2,s4 ); RR(({{((aa22 ,,11))}},,ss55 ) < π(2,s5 ); RR(({{ss11 }},,((aa33,2)) = 1; RR(({{((aa33 ,,11))}},,ss11 )) ≥≥ ππ((22,,ss11 ); RR(({{((aa33 ,,11))}},,ss66)) ≥≥ ππ((22,,ss66 ); RR(({{ss66 }},,((aa33,1)) = 2; R({s6 },s7 ) ≥ π(2,s7 ); R({s7 },s6 ) ≥ π(2,s6 ); RR(({{ss77 }},,ss55)) ≥≥ ππ((22,,ss55 ); RR(({{ss77 }},,ss44 )) ≥≥ ππ((22,,ss44 ); RR(({{ss44 }},,ss77) < π(2,s7 ); RR(({{ss44 }},,((aa22,1)) = 2; RR(({{ss55 }},,((aa22,1)) = 2; RR(({{ss55 }},,ss77 ) < π(2,s7 ); RR(({{ss44 ,s5 }},,ss77)) ≥≥ ππ((22,,ss77 ), among others that are implicit by the conditions of R. The left drawing, the middle drawing, and the right drawing represent (S,R), ViewR ((SS,,{{ss11 ,(a2 ,,22))}})),, and respectively ViewR (S,{(a2 ,2),(a3 ,2)}). We have:  W({(a2 ,2)}) = {{(a3 ,2)},{s1 },{s6 },{s7 },{s1 ,s6 },{s1 ,s7 },  {(a3 ,2),s7 }}.  M({(a2 ,2)}) = {{(a3 ,2)},{s7 },{s1 ,s6 },{s1 ,s7 },{(a3 ,2),s7 }}. WS({(a2 ,2)}) = {{(a3 ,2)},{s1 },{s7 },{s1 ,s7 },{(a3 ,2),s7 }}.  S({(a2 ,2)}) = WS({(a2 ,2)}).  The first two equalities can be checked one by one. For WS({(a2 ,2)}), we note that {s1 ,s6 } is excluded because: (1) for {(a2 ,2)}, {(a2 ,2),(a3 ,2),s7 } and {s1 ,(a2 ,2),s7 } are both strictly better in ?m than {s1 ,(a2 ,2),s6 }; and (2) for {s1 ,s6 }, {s1 ,s6 ,s4 ,s5 } is strictly better in ?m than {s1 ,(a2 ,2),s6 }. Similarlyl {s6 } is excluded. Finally, in this particular example, both {s1 ,(a2 ,2),s7 } and {(a2 ,2),(a3 ,2),s7 } are c-preferred, and, moreover, we have: {(a3 ,2)} ?m {(a2 ,2),(a3 ,2),s7 }; {s1 } ?m {s1 ,(a2 ,2),s7 }; {s7 } ?m {s1 ,(a2 ,2)s7 }; and {s7 } ?m {(a2 ,2),(a3 ,2),s7 }. By these, together with the subsumption of S({(a2 ,2)}) in WS({(a2 ,2)}) (Theorem 8), the last equality for S({(a2 ,2)}) follows. 5 Conclusion We proposed abstract-argumentation-theoretic coalition prof itability and formability semantics for conflict-eliminable sets. Further, we showed theoretical results around the semantics. We also showed how our framework can be reduced to Nielsen-Parsons’ argumen- tation framework. Our work has a connection to several important subf ields of abstract argumentation theory such as postulate-based abstract argumentation, attack-tolerant abstract argumentation and dynamic abstract argumentation. 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    "[18] Jo ao Leite and Jo ao Martins, ‘Social Abstract Argumentation’, in IJCAI, pp. 2287–2292, (2011). ",
    "[19] Hengfei Li, Nir Oren, and Timothy J. Norman, ‘Probabilistic Argumentation Frameworks’, in TAFA, pp. 1–16, (2012). ",
    "[20] Søren Holbech Nielsen and Simon Parsons, ‘A generalization ofDung’s  Abstract Framework for Argumentation’ , in Argumentation in MultiAgent Systems, pp. 54–73, (2006). ",
    "[21] Luke Riley, Katie Atkinson, and Terry Payne, ‘A Dialogue Game  for Coalition Structure Generation with Self-Interested Agents’, in  COMMA, pp. 229–236, (2012). ",
    "[22] Nicol as Rotstein, Mart ın O. Moguillansky, Alejandro J. Garc ´ıa, and  Guillermo R. Simari, ‘An abstract Argumentation Framework for Han-  dling Dynamics’, in Proceedings of the Argument, Dialogue and Decision Workshop in NMR 2008, pp. 131–139, (2008)."
  ]
}

following lines are unnecessary:
 "references": [
    "] I taking the coalition prof itability into account. A . 1 Introduction s c Coalition formation among agents is an important topic in many ",
    "[  domains including economics, political science, and computer sci- 1 ence. Two groups of agents, by teaming up together, could achieve v a task which they cannot on their own. Exploring abstract argumen- 5 tation theory for finding an apt characterisation of coalition forma- 9  bility looks specially rewarding, and there are already a few papers 4  in the literature looking at this subject matter: with preference-based 0 argumentation frameworks and task allocations [1]; with a coopera- 0 tive goal generation and fulf illing ",

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    "[Coherence]. 2. For any S1 ⊆ S ⊆f in S and for any s ∈ S, if R is defined for  (S1 ,s), then R is defined for any (S2 ,s) for ∅ ⊂ S2 ⊆ S1 .  ",
    "[Quasi-closure by subset relation]. 3. For any S1 ,S2 ⊆ S ⊆f S and for any s ∈ S, if R is defined both  in  for (S1 ,s) and for (S2 ,s), then R is defined also for (S1 ∪S2 ,s)  ",
    "[Closure by set union] 4. For any S1 ⊆ S ⊆f in S and for any s ∈ S such that R is defined  for (S1 ,s), it holds that R(S1 ,s) > 0 ",
    "[Attack with a positive  strength]. 5. For any (a,n),(a,m) ∈ S such that n ≤ m, the following holds  true: if R(S1 ,s) for some s ∈ S and for some S1 ⊆f in S such that  (a,n) ∈ S1 is defined, then R(S2 ,s) for S2 = (S1 \\(a,n)) ∪  (a,m) is defined and is such that R(S1 ,s) ≤ R(S2 ,s) ",
    "[Attack  relation monotonicity (source)]. 6. For any S1 ,S2 ⊆ S ⊆f S and for any s ∈ S, if R is defined for  in  (S1 ,s), (S2 ,s) and (S1 ∩ S2 ,s), then R(S1 ∩ S2 ,s) ≤ R(Si ,s)  for both i = 1 and i = 2 ",
    "[Attack strength monotonicity (source)]. 7. For any (a,n),(a,m) ∈ S such that n ≤ m, it holds that  if R is defined for (S1 ,(a,n)) for some S1 ⊆fin S such that  SSNl∈∈N  S1 ∩ {(a,l)} = ∅, then it is defined for (S1 ,(a,m)), and,  moreover, R(S1 ,(a,n)) ≤ R(S1 ,(a,m)) ",
    "[Attack relation and  strength monotonicity (target)] 8. For any S1 ⊆fin S aand for any s ∈ SS, R is undef ined for (S1 ,s)  if s ∈ S1 .",
    "[No self attacks].  ",
    "[Coherence] ensures that an attack must come from some argu- ment(s). ",
    "[Quasi-closure by subset relation] ensures that there is a group attack from a set of arguments on an argument just because each member of the set is attacking the argument. This can be con- trasted with the group attacks in Nielsen-Parsons’ argumentation frameworks. But it must be noted, as we are to mention shortly, that that there is an attack of an argument (a1 ,n1 ) on another argument (a2 ,n2 ) does not mean that (a1 ,n1 ) defeats (a2 ,n2 ) in our frame- work. ",
    "[Closure by set union] is the reverse of ",
    "[Quasi-closure by sub- set relation]. The purpose of ",
    "[Attack with a positive strength] is as follows: we mentioned earlier that R(S1 ,s) is the strength of attack by S1 on s, measured in N. The value being positive means that S1 is indeed attacking s. The value being 0 would mean that S1 is not at- tacking s. The purpose of an attack relation in abstract argumentation frameworks is to know which arguments attack which arguments. Consequently, we only consider positive values for R. ",
    "[Attack rela- tion monotonicity] expresses the following reasonable property: an attack may be occurring from some S1 ⊆ S ⊆fin S on some s ∈ S; now,increase theattackcapacityof justoneargument s1 ∈ S1 ,keep- ing all else equal; then the attack which occurred before the capacity increase should still occur. ",
    "[Attack strength monotonicity] expresses the property that ifan attack occurs from a set of arguments on an ar- gument with some strength, then any superset does not decrease the attack strength. To explain ",
    "[Attack relation and strength monotonic- ity], let us say that a set of arguments is attacking an argument with certain argument capacity. That intuitively means that the set intends to tone down the argument. Now, if the argument capacity of the argument increases, the set still intends to tone down the argument just as strongly or even more strongly, but not less strongly, for there are more materials in the argument that the set could attack. This is the direct reading of the condition. In the technical development to follow, the converse reading will be more useful: ifthe argument ca- pacity of the argument on the other hand decreases, the set may no longer intend to tone it down further. Finally, ",
    "[No self attacks] pre- vents self-contradictory arguments from being present.  Additivity of attack strengths is not postulated for R: if an ar- gument s1 is attacked by s2 and s3 such that R({s2 },s1 ) = n1 and such that R({s3 },s1 ) = n2 , it is not necessary that R({s2 ,s3 },s1 ) = n1 + n2 . The additivity holds good when each attack can be assumed independent, but may not in other cases. A generalisedversionof",
    "[Attackrelationmonotonicity]holdsgood.Let π be a projection function that takes a natural number and an ordered set List and that outputs a set member, such that π(n,List) = {the nn--ttmeembbeerr ooff LLiissIItt iiss uunnddeeffiifnniisgrreeaattetthhaantthhee ssiooffe the ordered set. Proposition 1 (Generalised attack relation monotonicity) Let S1 ⊆fin S and s ∈ S be such that R(S1 ,s) is defined. Then if  SSssxx SSssxx S2 ⊆fin S is such that: (1) ∈S1 π(1,sx ) = ∈S2 π(1,sx ); and that (2) (a,n) ∈ S1 materially implies (a,m) ∈ S2 for n ≤ m, then R(S2 ,s) is defined. Proof. By induction on the number of arguments s1 ∈ S1 and s2 ∈ S2 for which π(1,s1 ) = π(2,s2 ) and π(2,s1 ) < π(2,s2 ). Here and everywhere, we may make use of and (having the semantics of classical logic conjunction), distinguishing ‘and’ in formal contexts from ‘and’ in natural contexts for greater clarity. The base case is vacuous. Use ",
    "[Attack relation monotonicity] for inductive cases. 2 Our argumentation framework is (S,R) for some coherent set of ar- guments S and for some R. 3.1 Attacks  We distinguish complete attacks (defeats) from partial attacks (attacks).  Definition 2 (Attacks and defeats) WesaythatS1 ⊆ S attackss ∈  S iff there exists S2 ⊆⊆ SS11such that R is defined for (S2 ,s). We say  5  that S1 ⊆ S defeats s ∈ S iff S1 attacks s and R(S1 ,s) ≥ π(2,s).  Informally, an attack defeats its target when the attack strength surpasses the target’s argument capacity.  Definition 3 (Maximum attack strengths) We define V max (S1 ,s)  to be: 0 if S1 does not attack s; otherwise, R(S2 ,s) for some S2 ⊆  S1 such that: (1) R is defined for (S2 ,s); and such that (2) if R is  defined for (Sx ,s) for Sx ⊆ S1 ,then R(Sx ,s) ≤ R(S2 ,s).  3.2 Conflict-eliminable sets  Definition 4 (Conflict-eliminable sets of arguments) We say that  S1 ⊆⊆ S iiss ccoonnffmiinnaabblleeliimiifftthheerreexxiissnoo ss ∈∈ S11 such that  S1 defeats s.  Conflict-eliminability is a weaker notion of the usual conflictfreeness, i.e. the property that there exists no s ∈ S1 such that S1  attacks s. For the rationale behind obtaining this definition and using it as a primitive entity in our argumentation framework, let us  consider one mock example of a peer review which, just as political  alliance we saw does, exhibits partial internal conflicts: 1. (Authors) We have addressed a gap in the literature about LMN in  this paper. Our approach is entirely new. 5  See in the proof of Proposition 1 for what and is. 2. (A reviewer) My overall impression is that the paper contains suf-  ficiently new and interesting materials to be warranted acceptance.  Nonetheless, their claim that the approach considered in the paper  is entirely new is misleading. There are for instance papers by  XYZ et al that use a similar method to solve problems ABC. I  suggest that they include the references. 3. (The handling editor) I accept the paper on the condition that the  authors modify the part mentioned by the reviewer.  Now, if not wholly, the reviewer is stillattacking the authors. Despite  that, the decisions by the reviewer as well as by the handling editor  are overall favourable ones to them. They could take the expert opinion of the reviewer into account, weaken their claim; and then their  paper will be accepted. In this example, (1) that the authors have addressed some problem about LMN, (2) that a similar method was  used by XYZ et al. to solve problems ABC, and (3) that the paper  is accepted, are basically intrinsic arguments of the three arguments  given. The authors’ contribution is not completely defeated by the  reviewer, and the modif ication is a necessary compromise for the acceptance of their argument (that is, their paper). Intrinsic arguments  are the arguments that result from resolving, in a conflict-eliminable  set, internal non-defeating attacks.  S S  Definition 5 (Intrinsic arguments) Let α : 2 * 2 be such that  it is defined for S1 ⊆ S iff S1 is conflict-eliminable. If α is de- fined for S1 ⊆ S, then we define that α(S1 ) = {(π(1,s),n) | s ∈  max  S1 and n = π(2,s)−V (S1 ,s)}. We say that α(S1 ) are intrinsic  arguments of S1 .  Proposition 2 (Well-definedness) For any S1 ⊆ S, if α is defined  for S1 ,then every member of α(S1 ) is a member of S: in particular,  there exists no a ∈ A and no n ∈ N such that (a,−n) ∈ α(S1 ).  Intrinsic arguments of a conflict-eliminable set must be substantively  conflict-free.  Definition 6 (The view of intrinsic arguments) Let DelR (S,Sx )  bbee{SyyS ,,ss)||ss ∈∈ SSxx and Sy ⊆⊆ SSxx and R(Sy ,,ss)iiss ddeeffiinneedd..}},,  which is the set of attack relations within Sx . Now, let S1 be a  subset of S. If α is defined for S1 , then we say that ((S\\S1 ) ∪  α(S1 ),R\\Del(S,S1 )) is the view that S1 has about S, or simply  S1 ’s view of S. We denote S1 ’s view of S by ViewR (S,S1 ).  Proposition 3 Let S1 be a subset of S. If α(S1 ) is defined, then  α(S1 ) is conflict-free in ViewR (S,S1 ).  3.3 Coalition attacks, c-admissible sets and  c-preferred sets  A coalition may attack external arguments only by its intrinsic arguments.  Definition 7 (C-attacks and c-defeats) We say that S1 ⊆⊆ S c-attacks s ∈ S iff α is defined for S1 and there exists some  S2 ⊆ α(S1 ) such that π(2,ViewR (S,S1 )) is defined for (S2 ,s).  We say that S1 c-defeats s ∈ S iff S1 c-attacks s and  π(2,ViewR (S,S1 )))(α(S1 ),s) ≥ π(2,s).  There are no self c-attacks; Cf. Proposition 3.  Proposition 4 The following are equivalent. 1. S1 ⊆ S c-attacks s ∈ S. 2. S1 ⊆ S c-attacks s ∈ π(1,ViewR (S,S1 )).  Also, the following are equivalent. 1. S1 ⊆ S c-defeats s ∈ S. 2. S1 ⊆ S c-defeats s ∈ π(1,ViewR (S,S1 )).  Proof. By definition, π(2,ViewR (S,S1 ))(Sx ,s) is undef ined for  any Sx ⊆ S1 ∪ α(S1 ) and for any s ∈ S1 ∪ α(S1 ). Hence if  S1 ⊆ S c-attacks s ∈ S, then s 6∈ S1 and s 6∈ α(S1 ). Meanwhile,  (S\\S1 ) = (π(1,ViewR (S,S1 ))\\α(S1 )). 2  We now def ine the notions of c-admissible and c-preferred sets,  which are analogous to admissible and preferred sets we touched  upon in Section 2, but which are for conflict-eliminable sets. One  part in the def inition could appear diff icult at first. We italicise the  part, and elaborate it later.  Definition 8 (C-admissible/c-preferred sets) We say that S1 ⊆⊆  S is c-admissible iff α is defined for S1 and if S2 ⊆  π(1,ViewR (S,S1 )) attacks s ∈ S1 and if Sx ⊆ S2 is such that  R(Sx ,s) is defined, then there exists some S3 ⊆ α(S1 ) such that S3  c-defeat some sx ∈ Sx . We say that S1 ⊆ S is c-preferred iff S1  is c-admissible and there exists no S1 ⊂⊂ SSyy ⊆⊆ S such haat Syy is  c-admissible.  We explain why in the italicised part in the above definition it is  S2 ⊆ π(1,ViewR (S,S1 )) and not S2 ⊆ S; and why it is s ∈ S1  and not s ∈ α(S1 ). The notion of c-admissibility is intuitively the  sameas that of admissibility in Section 2: subsets of S2 are attacking  a conflict-eliminable set S1 ; and so S1 is not admissible unless some  member of S1 defeat all the attacking subsets of S2 . Now, because  we are presuming that S1 is conflict-eliminable and not necessarily  conflict-free, S2 ⊆ S would include any partial conflict in S1 as  an attack. However, c-admissibility, which is the admissibility of a  conflict-eliminable set in the view of the set, should not be defined to  defeat it. This explains why S2 ⊆ π(1,ViewR (S,S1 )) which compiles away all those purely internal partial conflicts. On the other  hand, the reason that it should be s ∈ S1 and not s ∈ α(S1 ) is because of the asymmetry in attacks to and from a conflict-eliminable  set. Recall Asymmetry in attacks to and from a coalition in Section 1.  3.4 Reduction to Nielsen-Parsons’ argumentation  frameworks  ↓  Theorem 1 (Restricted (S,R) as Nielsen-Parsons’) Let R be  such that it satisfies all but ",
    "[Quasi-closure by subset relation]. Let  S be such that for any S1 ⊆⊆ S aannddffooranyy s ∈ SS, iiRf R is defined  for (S1 ,s), then S1 defeats s. Then (S,R ↓) is Nielsen-Parsons’  argumentation framework, and the following all hold good. 1. Any conflict-eliminable set in S is a conflict-free set. 2. If α is defined for S1 ⊆ S, then α(S1 ) = S1 and  π(1,ViewR (S,S1 )) = S. 3. A c-attack by S1 ⊆ S on s ∈ S is an attack by S1 on s. 4. A c-attack is a c-defeat. 5. A c-admissible set is an admissible set, and a c-preferred set is a  preferred set.  Proof.  ↓ 1. Bydef inition,ifR isdefinedfor(S1 ,s),thenS1 defeatss.Hence  it isnecessary that a conflict-eliminable set be a conflict-free set. 2. By 1., it is vacuous that α(S1 ) = S1 . Further, Del(S,S1 ) = ∅,  and so π(1,ViewR (S,S1 )) = S. 3. By 2., both are trivial. 4. By definitionof (S,R ↓). 5. By 2.,both are trivial.  With these, itis straightforward to see that (S,R ↓) is NielsenParsons’ argumentation framework. 2  4 Semantics for coalition profitability and  formability  Itisof interestto learnmeaningful conflict-eliminablesets ofarguments.Inthissectionweshowsemanticcharacterisationsofcoalition  formabilityout of conflict-eliminablesets.Since coalition formation  presupposes at leasttwo groups of arguments, what we are characterisingisnot whether a set ofarguments is admissible and how good  anadmissible setover otheradmissible sets is, but whetheraconflicteliminable setcan form a coalitionwith another conflict-eliminable  set,and how good a coalitionover othercoalitions is.We presume  some conflict-eliminable set at fronand will talkof coalitionformabilitysemantics relativeto the conflict-eliminable set.We will first  discussprof itabilitrelationof a coalition,will show itstheoretical  properties,and willthen present four coalitionformability semanticseach of which formalises certain utilitypostulate(s)taking the  prof itabiliintoaccount. We assume the following notations.  Definition 9 (One-directional attacks) Let S1 ⊆ S be such that α(S1 ) is defined.We say that S1 is one-directionally attacked iff  thereexistsSx ⊆ π(1,ViewR (S,S1 ))such thatSx attackss ∈ S1  and S1 does not c-attackany sx ∈ Sx .  Definition 10 (States ofa conflict-eliminable set) Let ?: 2 S × 2S  be a binary relationsuch that (S1 ,S2 ) ∈?, written also S1 ? S2 ,  iffα is defined both for S1 and S2 and any of the three conditions  below is satisf ied: 1. S2 is c-admissible. 2. S1 is one-directionally attacked. 3. neither S1 nor S2 is c-admissible or one-directionally attacked.  Informally if Sx ⊆ S is c-admissible, then it is fully defended from  external attacks and is good. If Sx is one-directionally attacked, then  that means that Sx does not have any answer to external attacks,  which is bad. Any conflict-eliminable set that does not belong to either of them is better than being one-directionally attacked but is  worse than being c-admissible. Consequently, if S1 ? S2 , then S2 is  in a better state or in at least as good a state as S1 .  Definition 11 (Coalition permission) We say that coalition is permitted between S1 ⊆ S and S2 ⊆ S iiff S1 ∩ S2 = ∅ and α is  defined for S1 ∪ S2 .  The following results tell that this definition of coalition permission  isnot underspecif ied.  Lemma 1 (Defeats are unresolvable) Let S1 and s be such that  S1 ⊆ S and s ∈ S1 . If S1 defeats s, then for all S2 such that  S1 ⊆ S2 ⊆ S it holds that S2 defeats s.  Proof. By ",
    "[Attack strength monotonicity] of R. 2  Proposition 5 (Coalition and conflict-eliminability) If coalition is  permitted between S1 and S2 , then α is necessarily defined for S1  and S2 .  Proof. Suppose otherwise, then by conflict-eliminability of a set,  there must be an argument in Si ,i ∈ {1,2}, such that Si defeats  s ∈ Si .The required result obtains via Lemma 1. 2  We make one notion formally explicit for convenience, and then de- fine coalition prof itability.  S S  Definition 12 (Attackers) Let Attacker : 2 → 2 be such that  Attacker(S1 ) = {s ∈ S | there exists some s1 ∈ S1 such that s  attacks s1 .}. We say that Attacker(S1 ) is the set of attackers to S1 .  S S  Definition 13 (Coalition profitability) Let ? : 2 ×2 be such that  S1 ? S2 (or (S1 ,S2 ) ∈ ?) satisf iesthree axioms below. 1. S1 ⊆ S2 (larger set). 2. S1 ? S2 (better state). 3. |{s ∈ Attacker(S1 ) | S1 does not c-defeat s and s 6∈ S1 }| ≥  |{s ∈ Attacker(S1 ) | S2 does not c-defeat s and s 6∈ S2 }|  (fewer attackers).  Due to (better state), S1 ?S2 implies that α is definedboth for S1 and  S2 .By (larger set), a set that contains more arguments is a better set.  By (better state), a set that is in a better state is a better set. Finally,  by (fewer attackers), a set that is attacked by a smaller number of  arguments is a better set. The S1 in Attacker(S1 ) on the second line  is not a typo: this criterion is for measuring the prof its of coalition  formation for S1 .  Proposition 6 A (S,R) can be chosen in such a way that a pair of  S1 ⊆ S and S2 ⊆ S do not satisfy more than one axioms.  Proof. (larger set)  s3 s3  (a1,2) (a2,2) (a1,1) (a2,1)  Let S be {(a1 ,2),(a2 ,2),s3 }, and let R be such that it is defined only for any combination that matches the attack arrows  in the two drawings above. Def ine that R({(a1 ,2)},(a2 ,2)) =  R({(a2 ,2)},(a1 ,2)) = R({s3 },(a1 ,2)) = R({s3 },(a1 ,1)) =  R({(a2 ,2),s3 },(a1 ,2)) = 1, and that R({(a1 ,2)},s3 ) ≥  π(2,s3 ) (, among others implicit by the conditions of R). Then  the left and the right drawings represent (S,R) and respectively  ViewR (S,{(a1 ,2),(a2 ,2)}). Now, let S1 be {(a1 ,2)} and let  S2 be {(a1 ,2),(a2 ,2)}. Then clearly S1 ⊆ S2 . However, it  does not satisfy (better state): S1 is neither c-admissible nor onedirectionally attacked but S2 is one-directionally attacked. It does  not satisfy (fewer attackers), either: Attacker(S1 )) = {{ss3}},S11 cdefeats s3 ,and S2 c-defeats none. (better state)  s1 s2 s3  Let S be {s1 ,s2 ,s3 }, and let R be such that it is defined only mutually for s1 and s2 as shown above, and such that R({s1 },s2 ) ≥  π(2,s2 ) and R({s2 },s1 ) ≥ π(2,s1 ). Now, let S1 be {s1 } and  let S2 be {s3 }. Then S1 ? S2 because S2 is c-admissible. However, clearly it is not the case that S1 ⊆ S2 .Also, Attacker(S1 ) =  {s2 }, S1 c-defeats s2 , but S2 does not c-defeat it. (fewer attackers) Let S be {s1 ,s2 ,s3 }, and let R be such that it  is defined as is previously, and such that R({s1 },s2 ) < π(2,s2 )  aanndR(({{ss2}},,s)11< π(2,s1 ). Now, let S1 bbee {{ss3aanndlleeS2 be  {s2 }.ThenAttacker(S1 ) = ∅,andso(fewerattackers)issatisf ied  trivially.However, clearly it is not the case that S1 ⊆ S2 . And it  is not the case that S1 ? S2 ,because S1 is c-admissible, while S2  is neither c-admissible nor one-directionally attacked. 2  4.1 Theoretical results around profitability  It is triviato show that there exists some (S,R) and some S1 ,S2 ⊆  S such that S1 ? S1 ∪ S2 .We show other results.  Theorem 2 (Profitable coalition for subsets of a c-admissible set)  Let S1 ⊆ S be such that α(S1 ) is defined, and let Sx ⊆ S be a  c-admissible set.If S1 ⊂ Sx , then the following all hold good. 1. α is defined for S2 = Sx \\S1 . 2. Coalition is permitted between S1 and S2 . 3. S1 ? Sx .  Proof. 1. Suppose otherwise, then S2 would defeat at least one s ∈  S2 suchthat S2 woulddefeat s.Bylemma 1, Sx wouldthendefeat  s, for S2 ⊆ Sx . But Sx ,being c-admissible, does not defeat s. 2. α is defined for Sx by definition; and S1 ∩∩ SS22= ∅∅,aso by def-nition. 3. S1 ⊆ Sx ; Sx is a c-admissible set; and  |{s ∈ AAttacker(S1 ) | Sx does not c-defeat s and s 6∈ Sx }| = 0.  2  Theorem 3 (Existence theorem) If,for any S1 ⊆ S, there exists some S2 ⊆⊆ S su hh that coalion is perm ted between S11 and S2 and S1 ? S1 ∪ S2 and S1 ∪ S2 is c-admissible, then there exists some S3 ⊆ S such that coalition is permitted between S1 and S3 and S1 ? S1 ∪ S3 and S1 ∪ S3 is c-preferred and S2 ⊆ S3 . Proof. As S is of a finite size, it is straightforward to see that if S1 ∪ S2 is c-admissible, then there must exist some Sx such that it is c-preferred and that S1 ∪ S2 ⊆ Sx . We show that the choice of Sx \\S1 for S3 ensures the requirement to be satisf ied. By Theorem 2 coalition is permitted between S1 and S3 and S1 ? S1 ∪ S3 . By assumption, S1 ∪ S3 = Sx is c-preferred and S2 ⊆ S3 . 2 Theorem 4 (A c-preferred set as a mutually maximal coalition) Let S1 ⊆ S be such that α(S1 ) is defined, and let Pref(S1 ) be the set of all c-preferred sets that contain S1 as their sub- set. If Pref(S1 ) 6= ∅, then the following holds good: for any Sx ∈ Pref(S1 ), S1 ? Sx and (Sx \\S1 ) ? Sx and there exists no Sx ⊂ Sy ⊆ S such that S1 ? Sy or such that (Sx \\S1 ) ? Sy . Proof. By Theorem 2 and by definition of a set being c-preferrable. 2In  general, though, mutual prof itability is not guaranteed. Theorem 5 (Asymmetry of profitabilities) There exists an argu- mentation framework (S,R) such that there are disjoint subsets: S1 and S2 ,of S such that S1 ? S1 ∪ S2 , but such that it is not the case that S2 ? S1 ∪ S2 . Proof.  s3 s1 s3 s1  (a2,2) s4 (a2,1) s4  s5 s5  Let S be {s1 ,(a2 ,2),s3 ,s4 ,s5 }, and let R be such that it is defined only for any combination that matches the attack arrows in the drawings above, and such that: R({s1 },(a2 ,2)) = 1; R({s1 ,(a2 ,1)},s3 ) ≥ π(2,s3 ); R({s1 },s3 ) < π(2,s3 ); R({(a2 ,2)},s3 ) < π(2,s3 ); R({(a2 ,1)},s4 ) < π(2,s4 ); R({(a2 ,1)},s5 ) < π(2,s5 ); R({(a2 ,2)},s4 ) ≥ π(2,s4 ); R({(a2 ,2)},s5 ) ≥ π(2,s5 ); R({s3 },s1 ) ≥ 2; R({s4 },(a2 ,1)) ≥ 2; and R({s5 },(a2 ,1)) ≥ 2, among others. Now, let S1 be {s1 }, and let S2 be {(a2 ,2)}. Then S1 ? S1 ∪ S2 , for S1 and S1 ∪ S2 are both neither c-admissible nor one-directionally attacked. The axiom (fewer attackers) is also satisf ied. However, it is not the case that S2 ? S1 ∪ S2 , for it does not satisfy (fewer attackers). 2 Moreover, ? does not satisfy what we may at first expect hold good. Definition 14 (Continuation property of ?) Let S1 ⊆ S be such that α(S1 ) is defined, and let Max(S1 ) be the set of all Sx ⊆ S such that S1 ? Sx and such that if Sx ⊂ Sy ⊆ S, then it is not the case that S1 ? Sy . Then we say that ? is weakly continuous for S1 iff there exists some Sz ∈ Max(S1 ) such that, for any Sw ⊆ Sz , if coalition is permitted between S1 and Sw \\S1 , then S1 ?Sw .We say that ? is continuous for S1 iff it is weakly continuous for S1 for any Sz ∈ Max(S1 ). Theorem 6 (Coalition profitability discontinuation theorem) There exist S1 ,S2 ,Sx ⊆ S such that: (1) α is defined for S1 ,S2 and Sx ; (2) Sx ∈ Max(S1 ); and (3) S2 ⊆ Sx ,but such that S1 ? S2 does not hold good. Moreover, Max(S1 ) = Pref(S1 ) would not change this result. Proof. It suffices to consider the case where Max(S1 ) = Pref(S1 ).  s3  s3  s5 s5  s4 s4  (a1, 2) (a2,2) (a1,1) (a2,1) Let S be {(a1 ,2),(a2 ,2),s3 ,s4 ,s5 }, and let R be such that it is defined only for any combination that matches the attack arrows in the drawings above, and such that: R({(a1 ,2)},s4 ) ≥ π(2,s4 ); R({(a2 ,2)},s5 ) ≥ π(2,s5 ); R({(a1 ,2)},(a2 ,2)) = 1; R({(a2 ,2)},(a1 ,2)) = 1; R({s3 },s4 ) ≥ π(2,s4 ); R({(a2 ,1),s3 },s5 ) ≥ π(2,s5 ); R({s4 },(a1 ,1)) ≥ 2; and R({s5 },(a2 ,1)) ≥ 2, among oth- ers. Clearly α is defined for S1 = {(a1 ,2)}, Sa = {(a2 ,2)} S2 = S1 ∪ Sa , and Sx = {(a1 ,2),(a2 ,2),s3 }, among others.  Further, Sx is c-preferrable. However, it is not the case that  S1 ? S2 . In fact, it is also not the case that Sa ? S2 , since S2 is  one-directionally attacked (see the right drawing for the attacks  in ViewR (S,{(a1 ,2),(a2 ,2)})), whereas S1 and Sa are neither  c-admissible nor one-directionally attacked (see the left drawing). 2  Coalition prof itability continuation property satisf ies in certain special cases, however.  Theorem 7 (Coalition profitability continuation theorem) Let  Sx ⊆ S be a c-preferred set. Then ? is weakly continuous for  any S1 ⊂ Sx iff any disjoint pair Sy ,Sz of subsets of Sx satisfy  Sy ? Sy ∪ Sz .  Proof. If: Suppose, by way of showing contradiction, there are  three disjoint subsets of Sx : Sa ,Sb and Sc , such that Sa ?  Sa ∪ Sb 6?(Sa ∪ Sb ) ∪ Sc .By assumption we have (Sa ∪ Sb ) ?  (Sa ∪ Sb ) ∪ Sc ,contradiction.  Only if: By definition of weak continuation property of ?. 2  4.2 Coalition formability semantics  We use the prof itability relation to express our coalition formability  semantics. We set forth three rational utility postulates:  I Coalition is good when it is prof itable at least to one party.  II Coalition is good when it is prof itable to both parties. III Coalition is good when maximal potential future prof its are expected from it.  Of these, the first two can be understood immediately with the  prof itability relation. Our interpretation for the last postulate is as  follows. Suppose a party, some conflict-eliminable set S1 ⊆⊆ SS in our context, considers coalitionformation with another conflict- eliminable setS2 .We know thatS2 issome subset of S1 ⊆ S\\S1 . Before S1 forms a coalition with S2 , we have Max(S1 ) as the set of maximal coalitionspossible for S1. Once the coalition is formed, we have Max(S1 ∪ S2 )as theset ofmaximal coalitionspossible for the coalition. Here clearly Max(S1 ∪ S2 ) ⊆ Max(S1 ). What this means is thata particular choice of S2 blocks any possibilitiesin Max(S1 )MaxSS1 ∪∪ SS22 they become unrealisable from S1 ∪∪ SS.2 Hence S1 has an incentive not to form a coalition with such a S2 if all the members of Max(S1 ∪S2 ) are strictland comparatively less prof itablethan some member of Max(S1 ). We reflectthisintuition. Definition 15 (Maximal profitabilityrelation) Let ≤ll≤bb ≤ff : 2 S × 2 S be such thatthey satisfyall thefollowing: 1. S1 ≤l S2 iff|S1 | ≤|S2 |. 2. S1 ≤b S2 iffS2 is at leasas good by (better state)asS1 . 3. S1 ≤f S2 iffS2 is atleastas good by (fewer attackers)as S1 . We writeS1 <β S2 for each β ∈ {l,b,f} justwhen it is thecase that S1 ≤ββ S2 and it is not thcase that S2 ≤ββ S1 .Then we deine ?m :2S ×2 S to be such thatifS1 ?m S2 ,then both of thefollowing conditions satisfy: 1. S1 ? S2 . 2. Some Sx ∈ Max(S2 ) is such that, for all Sy ∈ Max(S1 ),  if Sx <β Sy for some β ∈ {l,b,f}, then there exists γ ∈  ({l,b,f}\\β) such that Sy <γ Sx . Intuitively,ifS1 ?m S2 ,then at leasone set iMax(S1 ) maximal in the three criteriathe setsize,the statequality and the number of ex- ternal attackers,is reachablefrom S2.The maximality here is judged by the principle that if S2 ,S3 ∈ Max(S1 ) are equal by two criteria, then S3 is better ifit is better than S2 by the remaining criterion.  We define four formability semantics: W which respects I, M which respects II (and implicitly also I), WS which respects I and III, and S which respects II and III. Definition 16 (Coalition formability semantics)  W(S1 ) = {S2 ⊆ S | S1 ? S1 ∪ S2 or S2 ? S1 ∪ S2 }.  M(S1 ) = {S2 ⊆ S | S1 ? S1 ∪ S2 and S2 ? S1 ∪ S2 }.  WS(S1 ) = {S2 ⊆ S | S1 ?m S1 ∪ S2 or S2 ?m S1 ∪ S2 }.  S(S1 ) = {S2 ⊆ S | S1 ?m S1 ∪ S2 and S2 ?m S1 ∪ S2 }. Here, or has the semantics of classical logic disjunction. Intuitively, ρ(S1 ) for ρ ∈ {W,M,WS,S} means that S1 is comfortable with forming a coalition with S2 ∈ ρ(S1 ) under the given criteria. Theorem 8 (The relation among coalition formability semantics) Let S1 ⊆ S be such that α(S1 ) is defined. The following all hold good. (1) M(S1 ) ⊆ W(S1 ). (2) WS(S1 ) ⊆ W(S1 ). (3) S(S1 ) ⊆ M(S1 ). (4) S(S1 ) ⊆ WS(S1 ). Meanwhile, neither WS(S1 ) ⊆ M(S1 ) nor M(S1 ) ⊆ WS(S1 ) is necessary. We provide one example to illustrate the semantic differences.  (a3, 2) s1 (a3, 2) s1 (a3, 1) s1  s4 s4 s4  (a2, 2) (a2, 1) (a2, 2)  s5 s5 s5  s6 s6 s6  s7 s7 s7 Let S = {{ss11,(a2 ,2),(a3 ,2),s4 ,s5 ,s6 }},, aanndd lleett R bbee ssuucchh that it is defined only for any combination that matches the attack arrows in the above drawings, and such that: RR(({{((aa22 ,,22))}},,((aa33,2)) = 1; RR(({{((aa22 ,,22))}},,ss44 )) ≥≥ ππ((22,,ss44 ); RR(({{((aa22 ,,22))}},,ss55)) ≥≥ ππ((22,,ss55 ); RR(({{((aa22 ,1),s1 }},,((aa33,,22)))) ≥≥ 22;; RR(({{((aa22 ,,11))}},,ss44) < π(2,s4 ); RR(({{((aa22 ,,11))}},,ss55 ) < π(2,s5 ); RR(({{ss11 }},,((aa33,2)) = 1; RR(({{((aa33 ,,11))}},,ss11 )) ≥≥ ππ((22,,ss11 ); RR(({{((aa33 ,,11))}},,ss66)) ≥≥ ππ((22,,ss66 ); RR(({{ss66 }},,((aa33,1)) = 2; R({s6 },s7 ) ≥ π(2,s7 ); R({s7 },s6 ) ≥ π(2,s6 ); RR(({{ss77 }},,ss55)) ≥≥ ππ((22,,ss55 ); RR(({{ss77 }},,ss44 )) ≥≥ ππ((22,,ss44 ); RR(({{ss44 }},,ss77) < π(2,s7 ); RR(({{ss44 }},,((aa22,1)) = 2; RR(({{ss55 }},,((aa22,1)) = 2; RR(({{ss55 }},,ss77 ) < π(2,s7 ); RR(({{ss44 ,s5 }},,ss77)) ≥≥ ππ((22,,ss77 ), among others that are implicit by the conditions of R. The left drawing, the middle drawing, and the right drawing represent (S,R), ViewR ((SS,,{{ss11 ,(a2 ,,22))}})),, and respectively ViewR (S,{(a2 ,2),(a3 ,2)}). We have:  W({(a2 ,2)}) = {{(a3 ,2)},{s1 },{s6 },{s7 },{s1 ,s6 },{s1 ,s7 },  {(a3 ,2),s7 }}.  M({(a2 ,2)}) = {{(a3 ,2)},{s7 },{s1 ,s6 },{s1 ,s7 },{(a3 ,2),s7 }}. WS({(a2 ,2)}) = {{(a3 ,2)},{s1 },{s7 },{s1 ,s7 },{(a3 ,2),s7 }}.  S({(a2 ,2)}) = WS({(a2 ,2)}).  The first two equalities can be checked one by one. For WS({(a2 ,2)}), we note that {s1 ,s6 } is excluded because: (1) for {(a2 ,2)}, {(a2 ,2),(a3 ,2),s7 } and {s1 ,(a2 ,2),s7 } are both strictly better in ?m than {s1 ,(a2 ,2),s6 }; and (2) for {s1 ,s6 }, {s1 ,s6 ,s4 ,s5 } is strictly better in ?m than {s1 ,(a2 ,2),s6 }. Similarlyl {s6 } is excluded. Finally, in this particular example, both {s1 ,(a2 ,2),s7 } and {(a2 ,2),(a3 ,2),s7 } are c-preferred, and, moreover, we have: {(a3 ,2)} ?m {(a2 ,2),(a3 ,2),s7 }; {s1 } ?m {s1 ,(a2 ,2),s7 }; {s7 } ?m {s1 ,(a2 ,2)s7 }; and {s7 } ?m {(a2 ,2),(a3 ,2),s7 }. By these, together with the subsumption of S({(a2 ,2)}) in WS({(a2 ,2)}) (Theorem 8), the last equality for S({(a2 ,2)}) follows. 5 Conclusion We proposed abstract-argumentation-theoretic coalition prof itability and formability semantics for conflict-eliminable sets. Further, we showed theoretical results around the semantics. We also showed how our framework can be reduced to Nielsen-Parsons’ argumen- tation framework. Our work has a connection to several important subf ields of abstract argumentation theory such as postulate-based abstract argumentation, attack-tolerant abstract argumentation and dynamic abstract argumentation. It is our hope that this study will further aid in linking the rich knowledge that is being accumulated in the literature. REFERENCES  ",
  ]

@sotetsuk Thank you for indication. I fix it. Please try this
http://<IP_ADDRESS>/arxiv-references-api/html/index.html


Adding the API Key and Secret

Sorry, I would love some help on how to add the API key and the Secret Key?
I cant figure out where to add these and what syntax we use to add these as.
Thanks

Add the sensor configuration as described to the configuration.yaml file of your Home Assistant. You can install Visual Studio Code to edit. You only need the API key in the config.

Add the sensor configuration as described to the configuration.yaml file of your Home Assistant. You can install Visual Studio Code to edit. You only need the API key in the config.

Yes, thats the KEY
"you only need the API key in the config"
Thanks


daprd startup logs are not useful for debugging

In what area(s)?

/area runtime

/area operator

/area placement

/area docs

/area test-and-release

What version of Dapr?

1.1.x

1.0.x
edge: output of git describe --dirty

Expected Behavior

daprd should emit a log which indicates what it is waiting for.
Actual Behavior
It is not easy and it may not be possible to tell why dapr is failing its readiness checks
Steps to Reproduce the Problem
Unknown
Example
These are the last lines I can see in the log before the pod fails its readiness check for the last time.
The only way to possibly diagnose what's wrong is to backtrack through the initRuntime function see what logging message is last emitted. This hasn't yielded anything helpful yet.
time="2021-08-09T20:52:53.750329918Z" level=debug msg="Starting loop to consume." app_id=indexer instance=pod-id-1234 scope=dapr.contrib type=log ver=1.1.2
time="2021-08-09T20:52:57.737967449Z" level=debug msg="{ERR_HEALTH_NOT_READY dapr is not ready}" app_id=indexer instance=pod-id-1234 scope=dapr.runtime.http type=log ver=1.1.2
time="2021-08-09T20:52:59.703507456Z" level=debug msg="{ERR_HEALTH_NOT_READY dapr is not ready}" app_id=indexer instance=pod-id-1234 scope=dapr.runtime.http type=log ver=1.1.2
time="2021-08-09T20:53:03.737871015Z" level=debug msg="{ERR_HEALTH_NOT_READY dapr is not ready}" app_id=indexer instance=pod-id-1234 scope=dapr.runtime.http type=log ver=1.1.2
time="2021-08-09T20:53:05.703592581Z" level=debug msg="{ERR_HEALTH_NOT_READY dapr is not ready}" app_id=indexer instance=pod-id-1234 scope=dapr.runtime.http type=log ver=1.1.2
time="2021-08-09T20:53:09.737935583Z" level=debug msg="{ERR_HEALTH_NOT_READY dapr is not ready}" app_id=indexer instance=pod-id-1234 scope=dapr.runtime.http type=log ver=1.1.2

Release Note

RELEASE NOTE:

@wcs1only Were you looking at a related question of improving daprd logging for startup failures?

When the kafka pubsub component can't start a session with kafka because the server is in error, it fails silently and causes the sidecar to fail readiness with no logging.


Information in incorrect location within Properties window for SELECT operator

Azure Data Studio Version:
Version: 1.35.0-insider (user setup)
Commit: 3cfd1d23da0efba022104635b01b22cfa9a08524
Date: 2022-02-14T06:49:59.062Z
VS Code: 1.59.0
Electron: 13.5.0
Chrome: 91.0.4472.164
Node.js: 14.16.0
V8: 9.1.269.39-electron.0
OS: Windows_NT x64 10.0.19044

Steps to Reproduce:

Open a query window and run a SELECT statement, with the plan (CTRL +M)
In the plan window, open the properties for the SELECT operator.
The values for SET OPTIONS appear above the SET OPTIONS text:

Memory grant information is located above and below the Memory Grant Info tag:

The Wait Stats and Query Time stats information is also misaligned:

Does this issue occur when all extensions are disabled?: Yes/No

Fix available.


Search: site-wide search doesn't filter by location

When searching the site using the search bar (/search), you don't have a filter-by-location.

However, when searching from the articles (/articles/search), the location filter is available.

This is an issue because according to Brennan, the spec should've been "searching from the homepage should let me search articles related to my location".

Hey, did you post this (and #947) on the wrong project?

embarassing :| sorry!

embarassing :| sorry!

Haha! No worries at all.


DARK MODE

Description
we can add dark mode : to switsh the modes in top right most corner  : in nav bar
Screenshots

Checklist

[X] I have checked the existing issues.
[X] I follow Contributing Guidelines of this project.
[ ] I am a IWOC'24 contributor.
[ ] I am a JWOC'24 contributor.
[X] I am a SWOC'24 contributor.

Code of Conduct

[X] I agree to follow this project's "Code of Conduct".

No need for this as UI enhancement issue is open.
Pls work on your assigned issues @student-aditipaliwal


Large image support

Describe the issue
There is an error when i try to generate a world using provider type: image
The image is 25000x25000
Error: https://mclo.gs/MIY81FW
The files were .yml but i can't upload them, so i converted in txt
image.txt
pack.txt

Send the entire pack folder, including the image you used. Zip all the files, so you can upload them as a single file.

I believe the issue is with how the image is being loaded via Java's BufferedImage class, specifically that exception seems to be thrown when images are too large to be stored in memory (since each image sample point is stored in an array limited by the max integer size)
A solution would be to implement image stitching (which would be invisible to users) in the backend for images that are too large, such that they can be loaded into memory as multiple smaller BufferedImages.

I would recommend implementing on top of https://github.com/PolyhedralDev/Terra/pull/356 if anyone else wishes to work on this

Thanks so much for the support! Still need the full pack? In case I'd prefer to send it in private.
If I may ask, how soon will this bug be fixed?

hope #356 doesnt get stale. rn, I think the image provider is not usable

Stitching implemented in 6efff02c19ef089d5d74df5ce4f30067afcab9f8


Nuget package missing platform

When adding the project to a net47 target framework project via paket using dotnet restore I get
 : error NU1202: Package GeonBit.UI 3.4.0 is not compatible with net47 (.NETFramework,Version=v4.7). Package GeonBit.UI 3.4.0 supports: geonbitui (geonbitui,Version=v0.0) []
Since portable is deprecated, I think GeonBit.UI should be netstandard2.0 compatible to ensure maximum compatibility with other platforms & monogame. However, I'm not entirely sure. Because of this i need to manually add the reference as opposed to using nuget.

Thanks @ollipekka,
I'll keep this issue opened until my next release, then I'll see what's the best way to be cross platform and version.
Cheers :)

I have created a pull request for .NetStandard2.0 and .NetFramework4.6.1. Please have a look at the description of the pull request though.


Ascii table broken with some cities

Hi,
Please check attached screenshot:

I find this happening with many other cities as well.
Let me know if this is a wego issue instead to report the bug on their repo.
Thanks!

It's not about cities, it's about Thundery part. I guess thunder emoji(?) breaks the terminal output or font because it is animated.

@slump64 yes, see here https://github.com/chubin/wttr.in/issues/100
But just saw that @cig0 screenshot only shows rain symbols so the drops are wide symbols as well

@MAGICCC now it is not rain, it is thunder (or better to say lighting)

@slump64 it is not an emoji, it is just a normal wide unicode character; I will fix it. Actually after I've added ja/ko support, that us wide characters, the fix is trivial

@chubin OFFTOPIC: pretty sure the fact that this symbol is a normal wide unicode character doesn't make it an not emoji (https://emojipedia.org/high-voltage-sign/)

@slump64 yes, you are right, strictly speaking it is an emoji, it would be better to write "even if it is an emoji, it does not matter, because it is just a normal wide unicode symbol and so it can be handled as such". Actually many of normal symbols are emojis, as far as I know, even such symbols as +, - and ! and so on (©, ®, ☯, ™, etc. I'm not sure, but  I think that all of them are emojis). Thus, you are surely right

The problem was reported again: https://twitter.com/neproshennie/status/1001575302647242753

I love this tool 🙂 but unfortunately have the same display error

also problems http://prntscr.com/kpdl56

@07pepa What was that? Chrome?

nope and i know what is causing it... it is debian in WSL

Howdy! I'm using KiTTY as my SSH program, which is a fork of Putty to access my Raspberry Pi 3 which I've set up to use 256 colors and zsh.
I decided to browse around to see what other icons are used to represent the weather when I took notice of an area that was having thunderstorms both as part of the weather report and in the forecast.
One of the things I've noticed is how the lightning bolts' appearance.

I was thinking as a substitute for the ASCII/Emoji lightning icons, try ASCII Art, because this is what that forecast looks like here on Github.
Weather report: Jackson, MS

      .-.      Thunderstorm In Vicinity
     (   ).    82..86 °F
    (___(__)   ↑ 16 mph
   ‚‘⚡‘‚⚡‚‘  9 mi
   ‚’‚’⚡’‚’   0.0 in

I thought about some alternatives. After a google search to I saw this (The last example down at the bottom) and tried to minimize it. I don't think I could make it any smaller without losing some detail, but do you think you can work with something like this?
    ,/		 Lightning bolt (Similar to Corcoran's but smaller. Still too big.)
  .'/___,
.'__  ,'
   /,'
  /'

    ___       Lightning bolt (but this one would be a little too big)
   /  /_,
  /_  ,'
   /,'
  /'

 So I came up with this. For all the following case, the lightning bolt would be colored yellow to differentiate that from rain. which would use similar character.

    __       Lightning bolt (raw, no clouds or precip.)
   / /_,
  /_ ,'
   /'

   .-.      Thunderstorm (small cloud, no precip.)
  ( __).
 (_/ /_,)
  /_ ,'
   /'

_`/"".-.      Thunderstorm (small cloud, no precip. with sun)
 ,\_( __).
  /(_/ /_,)
    /_ ,'
     /'

_`/"".-.      Thunderstorm (small cloud, light rain. with sun)
 ,\_( __).
  /(_/ /_,)
   ‘/_ ,' ‘
  ‘ ‘/' ‘ ‘

_`/"".-.      Thunderstorm (small cloud, light moderate/heavy rain. with sun)
 ,\_( __).
  /(_/ /_,)
  ‚‘/_ ,'‚‘
  ‚‘,/'‚‘‚‘

_`/"".-.      Thundersnow (small cloud, snow. with sun)
 ,\_( __).
  /(_/ /_,)
   */_ ,'*
  * */'*

    .--.		Thunderstorm (large cloud, no precip.)
 .-(  __).
(___./ /__)
    /_ ,'
     /'

    .--.		Thunderstorm (large cloud, light rain.)
 .-(  __).
(___./ /__)
 ‘ ‘/_ ,'‘
‘ ‘  /' ‘

     .--.		Thunderstorm (large cloud, with moderate/heavy rain.)
  .-(  __).
 (___./ /__)
‚‘‚‘‚/_ ,'‚‘
‚‘‚‘, /'‚‘‚‘

     .--.		Thundersnow (large cloud, snow.)
  .-(  __).
 (___./ /__)
 * * /_ ,'*
* * * /'*

Might want to move the lightning bolt to the left a little bit to make it more clear. (I wish Github would change the comment textarea to use a monospace font instead of a sans-serif. I had to whip these up in my text editor.

OK, let's try to make those no-sun icons look better.
    .--.		Thunderstorm (large cloud, no precip., lightning more centered)
 .-( __ ).
(__./ /___)
   /_ ,'
    /'

     .--.		Thunderstorm (large cloud, no precip., lightning more centered)
  .-( __ ).
 (__./ /___)
 ‘ ‘/_ ,' ‘
‘ ‘ ‘/'‘ ‘

     .--.		Thunderstorm (large cloud, heavy/moderate rain, lightning more centered)
  .-( __ ).
 (__./ /___)
‚‘‚‘/_ ,'‚‘
‚‘‚‘,/'‚‘‚‘

     .--.		Thundersnow (large cloud, snow, lightning more centered)
  .-( __ ).
 (__./ /___)
 * */_ ,'* *
* * */' * *

Now that I think of it, I should have submitted these comments to schachmat/wego

@jrcharney Sure, you can do it, and so improve the both projects; we can use your improvements independently anyway; wego and wttr.in diverged many years ago (in 2016).
Maybe we should have create a non-unicode/non-emoji version, where only real ASCII symbols would be used? There we could the proposed by @jrcharney fixes.
Another option: we could make an option by means of which actual width of the lightning emoji could be chosen. The problem is (as you probably understand) that the with of the lighting emoji/sign varies in different terminals/browsers, so when we fix it for ones it does not work the others and other way around. But such an option could be a workaround

The problem is currently visible in Firefox 73 (on Fedora 31). No display issue on the XFCE Terminal.

@nixpanic Niels, thank you for reporting. We should fix it some day


Rewriting some parts

Description
I don't think I plan to add any new feature at this PR, rather refactoring what exists, and hope to improve the simplification algorithm
Tasks

[x] Classes in Core/Continuous refactored
[x] Numbers refactored
[x] SymPy's constants fixed
[x] SymPy's complex numbers fixed
[ ] Refactor the sorting algorithm
[ ] Reconsider domains (???)
[x] Reals, Integers, etc. are now subclasses of SpecialSet

Codecov Report

Merging #271 into master will decrease coverage by 0.22%.
The diff coverage is 82.14%.

@@            Coverage Diff             @@
##           master     #271      +/-   ##
==========================================
- Coverage   76.12%   75.90%   -0.23%
==========================================
  Files          97      104       +7
  Lines        8629     8641      +12
  Branches     1453     1459       +6
==========================================
- Hits         6569     6559      -10
- Misses       1578     1604      +26
+ Partials      482      478       -4

Impacted Files
Coverage Δ

AngouriMath/Core/Domains.Classes.cs
8.16% <0.00%> (-2.95%)
:arrow_down:

...Continuous/Entity.Continuous.Complex.Definition.cs
77.38% <ø> (ø)

...ty/Continuous/Entity.Continuous.Real.Definition.cs
79.66% <ø> (ø)

AngouriMath/Functions/Substitute.cs
62.31% <ø> (-0.54%)
:arrow_down:

.../Continuous/Entity.Continuous.Number.Definition.cs
17.64% <17.64%> (ø)

...ontinuous/Entity.Continuous.Rational.Definition.cs
47.69% <33.33%> (ø)

AngouriMath/Core/Antlr/AngouriMathParser.cs
78.55% <50.00%> (ø)

AngouriMath/Functions/Output/ToSympy.Classes.cs
80.28% <63.15%> (-5.21%)
:arrow_down:

AngouriMath/Functions/Output/ToString.Classes.cs
55.24% <68.57%> (+4.33%)
:arrow_up:

...s/Simplification/Patterns/Patterns.Trigonometry.cs
77.45% <75.00%> (-0.21%)
:arrow_down:

... and 27 more

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Possibility of changes in App settings

Hi Team,
Please let us know if the below are feasible.

Can we change the default message of bot “You cannot send messages to this bot”?

Can we add a hyperlink in App description like make email address and clickable link.

Hi @ArunkumarTMEYGDS
Thanks for reaching out to us
we will get back to you with proper responce

Hi @ArunkumarTMEYGDS
Can we add a hyperlink in App description like make email address and clickable link.
Yes, we can make email clickable. For instance, we have implemented this feature for one of our demo tenants.
In manifest file, we have a description field where we added the following information to make the email address clickable.

Can we change the default message of bot “You cannot send messages to this bot”?
we are checking on it we will get back to you ASAP with proper responce

Thanks @peddivyshnavi. We wait for the second point.

Hi @ArunkumarTMEYGDS
Can we change the default message of bot “You cannot send messages to this bot”?
Could you please let us know for which bot you are seeing this message for author bot?

@peddivyshnavi User bot

@peddivyshnavi User bot

@peddivyshnavi User bot

Hi @ArunkumarTMEYGDS
Basically, it is a one way communication where user bot will receive message whenever author bot sends message So it is not possible to send a message to the bot and this is expected.and we cant change default message

Thanks @peddivyshnavi for the updates

Could you please update us on the status of the issue?

Hi @peddivyshnavi
We have deployed for testing. Awaiting to confirm if the changes are reflecting.

Could you please update us on the status of the issue?

Hi @peddivyshnavi
The concern with hyperlink in point worked with given steps.
Since we cannot change the default bot message, we have left as it is.
You can close this ticket. Thanks.

Marking this ticket as closed


Ksenia Gateway HTTP

Hi,
please could you confirm if it's necessary the Gateway HTTP of Ksenia for this integration?
Thanks a lot in advance!
Giovanni

The integration works with pre 4.0 version with a direct connection. If you conenct to http://LOCAL-IP:port, you should get a browser window (not a webpage) with a login. If so, this integration is compatible