Decision Support for Agri-Food Chains: A Reverse Engineering Argumentation-Based Approach

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To cite this version:

Rallou Thomopoulos, Madalina Croitoru, Nouredine Tamani. Decision Support for Agri-Food Chains: A Reverse Engineering Argumentation-Based Approach. Ecological Informatics, Elsevier, 2015, 26, pp.182-191. �10.1016/j.ecoinf.2014.05.010�. �lirmm-01242932�

DOI: doi:10.1016/j.ecoinf.2014.05.010

Reference: ECOINF 482

To appear in: Ecological Informatics

Received date: 24 December 2013 Revised date: 25 May 2014 Accepted date: 27 May 2014

Please cite this article as: Thomopoulos, Rallou, Croitoru, Madalina, Tamani, Nouredine, Decision Support for Agri-Food Chains: A Reverse Engineering Argumentation-Based Approach, Ecological Informatics (2014), doi: 10.1016/j.ecoinf.2014.05.010

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 !""#$%"&"''"(

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$% R7+"!08*'0-. 23 F +3 #'"(" +- $ -"6)"%0" 23 -"#- 70$**"& $ +"!08*'0-. 3"7 9&".4"8 (F0, F1, . . . , Fn) -)0' #'$# F0 ⊆ F9 F0 +- R:02%-+-#"%#9 32( ","(.

i ∈ {1, . . . , n − 1}9 +# '2*&- #'$# Fi +- $% +;;"&+$#" &"(+,$#+2% 23 Fi−1

<+,"% $ -"# {F0, . . . , Fk} ⊆ F $%& $ -"# 23 ()*"- R9 #'" 0*2-)(" 23 {F0, . . . , Fk} = ( # R9 &"%2#"& ClR({F0, . . . , Fk})9 +- &">%"& $- #'" -;$**: "-# -"# 7=+#' ("-/"0# #2 ⊆8 ='+0' 02%#$+%- {F0, . . . , Fk}9 $%& +- 0*2-"& 32( R:&"(+,$#+2% 7#'$# +-9 32( ","(. R:&"(+,$#+2% Fn 23 {F0, . . . , Fk}9 =" '$," Fn ⊆ ClR({F0, . . . , Fk})) ?+%$**.9 =" -$. #'$# $ -"# F $%& $ -"# 23 ()*"- R ".'*0% $ 3$0# G 7$%& =" =(+#" F, R |= G8 +@ #'" 0*2-)(" 23 #'" 3$0#- 1. $** #'" ()*"- "%#$+*- F 7+ " +3 ClR(F) |= G8 !"#$%&'"$#( 5 02%-#($+%# +- $ 32(;)*$ ∀x1 . . . ∀xn (H(x1, . . . , xn) → ⊥)9 ='"(" H +- $% $#2; 2( $ 02%A)%0#+2% 23 $#2;- $%& n ∈ {0, 1, 2, . . .} B6)+,$*"%#*.9 $ 02%-#($+%# 0$% 1" =(+##"% $- ¬(∃x1, ..., ∃xnH(x1, ...xn)) 5-$% "C$;/*" 23 $ 02%-#($+%#9 02%-+&"( N = ¬(∃ 1 #:!-;'<#1( ∧ ="4!"*3"#1((( <+,"% $ D%2=*"&E" 1$-" K = (F, R, N )9 $ -"# {F1, . . . , Fk} ⊆ F +- -$+& #2 1" 0.4-.303'".' +3 $%& 2%*. +3 #'"(" "C+-#- $ 02%-#($+%# N ∈ N -)0' #'$# {F1, . . . , Fk} |= HN9 ='"(" HN &"%2#"- #'" "C+-#"%#+$* 0*2-)(" 23 #'" './2#': "-+- 23 N 5 -"# +- 02%-+-#"%# +3 $%& 2%*. +3 +# +- %2# +%02%-+-#"%# 5 -"# {F1, . . . , Fk} ⊆ F +- R70.4-.303'".' +3 $%& 2%*. +3 #'"(" "C+-#- $ 02%-#($+%# FG

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 N ∈ N !"# $#%$ ClR({F1, . . . , Fk}) |= HN& '#()( HN *(+,$( $#( (-. $(+$.%/ "/, !)( ,0 $#( #12,$#( . ,0 N3 4($ K = (F, R, N ) '#()(5 • F ",+$%.+ $#( 0,//,'.+6 0%"$ 5 − F1= !"#$%&'"("))"* ∧ +,-)#./-#-)0!""%&'"("))"* − F2= ∃ " 12)!#3)/,-4#)"%"5&'"("))"* − F3= ∃ 6 %0/&"!+,-)"-)%65&'"("))"* ∧ 7/89%6** • R ",+ . $ ,0 $#( 0,//,'.+6 )!/( 5 − R1= ∀ 25: % !"#$%2* ∧ 12)!#3)/,-4#)"%:52* ∧ ;"<)/3/$"0!""%2* →="3!"#<"%:** − R2 = ∀25:5> % !"#$%2* ∧ 12)!#3)/,-4#)"%:52* ∧ 0/&"!+,-)"-)%>52* ∧ 7/89%>* → ?!,@)9%:** − R3= ∀ 2 % !"#$%2* ∧ +,-)#./-#-)0!""%2* →;"<)/3/$"0!""%2* ∧ A:3,),2/-0!""%2** • N ",+$%.+ $#( 0,//,'.+6 +(6%$.7( ",+ $)%.+$5 − N = ¬(∃2 %?!,@)9%2* ∧ ="3!"#<"%2*** K _{. .+",+ . $(+$ .+"( (F, R) |= N3 8+*((*& F}1 %+* R3 %//,' $, *(*!"( ;"<)/3/$"0!""%&'"("))"*3 9,:;.+(* $, F2 %+* R1 '( ,;$%.+ ="3!"#<"%"*3 F3 %+* R2 *(*!"( ?!,@)9%"*& 7.,/%$.+6 $#( +(6%$.7( ",+ $)%.+$ N3 <.7(+ % =+,'/(*6( ;% (& ,+( "%+ % = % ",+>!+"$.7( ?!()1 .+ ,)*() $, =+,' '#($#() ,:($#.+6 #,/* ,) +,$3 @.$#,!$ /, ,0 6(+()%/.$1 '( ",+ .*() AB

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

!!"#$% &!%'(%&)*+# ,(#-*#. /01*&1 $-# 2$&).34 5. $% #6$78"# !2 $ ,(#-9: )$;# ∃x1cat(x1)4 <1# $%.0#- )! ,(#-9 α *. 8!.*)*+# *2 $%= !%"9 *2 F, R |= α4

5%.0#-*%> Q: )-$=*)*!%$""9: 1$. )0! =*?#-#%) $">!-*)17*& $88-!$&1#.@ #*A )1#- 2!-0$-= &1$*%*%> !- $&;0$-=. &1$*%*%>4 <1# )0! $88-!$&1#. &!7# )! #*)1#- /B3 C%=*%> $% $%.0#- !2 Q *% )1# RA=#-*+$)*!%. !2 )1# 2$&). *% )1# ;%!0"#=># $.# !- /D3 &!78()*%> )1# *%+#-.# RA=#-*+$)*!%. !2 )1# ,(#-9 $%= C%=*%> *2 )1#-# *. $ 7$)&1 *% )1# 2$&).4 E# 0*"" 2!&(. !% )1# "$))#- $88-!$&1 *% )1# 2!""!0*%>4

!"! #$%&'()*+ ,)- #**,./+

<1*. .#&)*!% .1!0. )1$) *) *. 8!..* "# )! =#C%# $% *%.)$%)*$)*!% !2 F(%>G. $ .)-$&) $->(7#%)$)*!% )1#!-9 /F(%>: BHHI3 )1$) &$% # (.#= )! -#$.!% 0*)1 $% *%&!%.*.)#%) !%)!"!>*&$" JK4 E# C-.) =#C%# )1# %!)*!% !2 $% $->(7#%)4 L!- $ .#) !2 2!-7("$# G = {G1, . . . , Gn}: %!)$)*!% V G *. (.#= $. $% $ -#+*$)*!% 2!- G1∧ . . . ∧ Gn4 !"#$%$&# '( 012() , /)345(-%( 6,+( K = (F, R, N )7 ,) ,$%&'()* a 1+ , *&85( a = (F0, F1, . . . , Fn) 49($(: • (F0, . . . , Fn−1) 1+ , -($12,*13) +(;&().( 41*9 $(+8(.* *3 K • Fn 1+ ,) ,*3'7 , .3)<&).*13) 3= ,*3'+7 *9( (>1+*()*1,5 .53+&$( 3= ,)

,*3' 3$ *9( (>1+*()*1,5 .53+&$( 3= , .3)<&).*13) 3= ,*3'+ +&.9 *9,* Fn−1|= Fn!

<1*. =#C%*)*!%: 2!""!0*%> )1# =#C%*)*!% !2 /M-!*)!-( $%= N#.*&: DOBP3 *. $ .)-$*>1)2!-0$-= 0$9 )! =#C%# $% $->(7#%): .*%&# $% ,$%&'()* &!--#.8!%=. )! $ -($12,*13)4

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 ! "#$%&#'( )*+ ,!))#!,. '/!$ ,!0 !,. 0+ "1%%!"+ )*) 0+ /+ 2#3+, -45+6 7,!0&+62+ 8-"+ K = (F, R, N ) -,6 6! ,!) +5%&#9#)&( $+,)#!, F. R ,!/ N _{#' ,!) ,+9+""-/(: ;+) a = (F}0, ..., Fn) 8+ -, -/21$+,): *+,. 0+ 6+,!)+ Supp(a) = F0 -,6 Conc(a) = Fn: </21$+,)" $-( -))-97 +-9* !)*+/. 0*#9* #" 9-%)1/+6 8( - 8#,-/( -))-97 /+&-)#!, Att ⊆ Arg(F) × Arg(F):

!"#$%$&# '( !" K = (F, R, N ) #! $ %&'()!*+! #$,! $&* )!" a $&* b #! "(' $-+./!&",0 12! $-+./!&" a $""$3%, $-+./!&" b4 *!&'"!* (a, b) ∈ Att4 56 $&* '&)7 56 "2!-! !85,", ϕ ∈ Supp(b) ,.32 "2$" "2! ,!" {Conc(a), ϕ} 5, R_{95&3'&,5,"!&"0}

*#" -))-97 /+&-)#!, #" ,!) "($$+)/#9: ! "++ 0*(. 9!,"#6+/ )*+ '!&&!0#,2 +5-$%&+: ;+) F = {p(m), q(m), r(m)}. R = ∅. N = {∀x1(p(x1) ∧ q(x1) ∧

r(x1) → ⊥)}: ;+) a = ({p(m), q(m)}, p(m) ∧ q(m)). b = ({r(m)}, r(m)): =+

*-3+ (a, b) ∈ Att -,6 (b, a) /∈ Att: *#" 0#&& +,"1/+ )*-) )*+ ,-#3+ +5)+,"#!, #" 6#>+/+,). -) &+-") #, )*+!/(. '/!$ )*+ %/+'+//+6. ")-8&+. +)9: "+$-,)#9": ?!0+3+/. #, !1/ -%%&#9-)#!, )*+( -&& +,)-#& )*+ "-$+ #,'!/$-)#!, -" "*!0, &-)+/ !,:

!"#$%$&# )( :5;!& $ %&'()!*+! #$,! K = (F, R, N )4 "2! 9!//+"%!,6#,2 -/21$+,)-)#!, '/-$+0!/7 AFK 5, $ <$5- (A = Arg(F), Att) (2!-! Arg(F)

5, "2! ,!" '6 $)) $-+./!&", "2$" 3$& #! 3'&,"-.3"!* 6-'/ F $&* Att 5, "2! 3'--!,<'&*5&+ $""$3% -!)$"5'& $, ,<!35=!* 5& >!=&5"5'& ?0

!" E ⊆ A $&* a ∈ A0 @! ,$7 "2$" E 5, 9!,@#9) '/++ 5A "2!-! !85,", &' $-+./!&", a, b ∈ E ,.32 "2$" (a, b) ∈ Att0 E 6+'+,6" a 5A 6'- !;!-7 $-+./!&" b ∈ A4 56 (! 2$;! (b, a) ∈ Att "2!& "2!-! !85,", c ∈ E ,.32 "2$" (c, b) ∈ Att0

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

E ! !"#$$#%&' " # ! $%&' $# ()** +&, ,*(*&,! +-- #! +)./0*&#!1 E ! + ()"*&'+' ',+'-$#)- " E ! +& +,0 !! 2-* !*# 34 $4 $%&#+ &! +-- #4* +)./0*&#! # ,*(*&,!1 E ! + *.'/'..'! ',+'-$#)- " # ! 0+5 0+- 63 #4 )*!7*$# #% !*# &$-/! %&8 +,0 !! 2-* !*#1 E ! + $+ %&' ',+'-$#)- " # ! $%&' $#9()** +&, (%) +-- a ∈ A \ E: #4*)* *5 !#! +& +)./0*&# b ∈ E !/$4 #4+# (b, a) ∈ Att1

E ! + 0.)1-!'! ',+'-$#)- " E ! + 0 & 0+- 6(%) !*# &$-/! %&8 $%07-*#* *5#*&! %&1

;%) +& +)./0*&#+# %& ()+0*3%)< AS = (A, Att) 3* ,*&%#* 2= Extx(AS)

6%) 2= Extx(A, Att)8 #4* !*# %( #! *5#*&! %&! 3 #4 )*!7*$# #% !*0+&# $! x1

>* /!* #4* +22)*? +# %&! c: p: s: +&, g (%) )*!7*$# ?*-= $%07-*#*: 7)*(*))*,: !#+2-* +&, .)%/&,*, !*0+&# $!1

@& +)./0*&# ! !$*7# $+--= +$$*7#*, ( # ! & +-- *5#*&! %&!: $)*,/-%/!-= +$$*7#*, ( # ! & +# -*+!# %&* *5#*&! %& +&, )*A*$#*, ( # ! &%# & +&= *5#*&! %&1

2 $'! )- +3#$ !'4-#+#)- )/ .01"'-+$ -! ++ (5$ #- 67.)#+).1 -! 8'$#(9 :;<=> ? $ &$) $3)?- +3 + +3' . +#)- &#+@ *)$+1& +'$ )/ 67 "#- ! -! A"B 0)1!9 :;;C> .' .'$*'(+'!D E3#$ #-$+ -+# +#)- .'$*'(+$ +3' !#.'(+9 #-!#.'(+ ()-$#$+'-(@ $ ?'&& $ +3' (&)$1.'D

B1C1 ;%)0+- ! &. #4* /!* $+!*

F- +3#$ $1%$'(+#)- ?' /)." &#$' +3' -)+#)-$ *.'$'-+'! #- $'(+#)- GD H'+ K = (F, R, N ) %' ()-$#$+'-+ 5-)?&'!0' % $'D E3#$ #$ +3' 5-)?&'!0' % $' +3 + && (+).$ $3 .' -! 0.'' 1*)-D F- +3#$ * *'. ?' $$1"' +3 + +3' .1&'$ -! -'0 +#I' ()-$+. #-+$ .' ()"")- +) 'I'.@%)!@D

E3' 0) &$ )/ +3' !#J'.'-+ (+).$ ( - %' $''- $ $'+ )/ ',#$+'-+# &&@ (&)$'! ()-K1-(+$D L' !'-)+' +3'" %@ G1, G2, ..., GnD

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

!" Gi #! $ %&$' $() K "*! +(&,'!)%! #$-!. K /- 0&(-/-"!(" $() K

)&!-(&" !("$/' Gi. 1! 0&234"! "*! /(5!6-! R7)!6/5$"/&(- &8 Gi 9,*!6! R /- "*!

-!" &8 64'!- &8 "*! +(&,'!)%! #$-!:. 1! $)) $'' &8 "*! R−1

(Gi) "& "*! 8$0"-.

1! "*4- &#"$/( $ (!, +(&,'!)%! #$-! Ki ,*/0* )/;!6- 86&2 K -&'!'< #<

/"-8$0"- -!" 9,*/0* (&, $'-& /(0'4)!- R−1

(Gi):= K = (F ∪ R−1(Gi), R, N ). 1!

$'-& /23&-! "*$" Ki /- 0&(-/-"!(".

>/5!( G = {G1, G2, ..., Gn}? "*! %&$'- 0&66!-3&() "& $ -!" &8 5/!,3&/("- V

9"*!6! !@/-"- $ 84(0"/&( κ : G → 2V:. A*/- 84(0"/&( 0$( $--/%( $ %&$' "& &(!

&6 2&6! 5/!,3&/("- $() !$0* 5/!,3&/(" 0$( #! $--&0/$"!) ,/"* &(! &6 2&6! %&$'-. >/5!( $ %&$' Gi? "*! 9-!" &8: 5/!,3&/("9-: $--&0/$"!) ,/"* "* %&$'

/-)!(&"!) #< κ(Gi). B/2/'$6'<? %/5!( $ 5/!,3&/(" vi? "*! -!" &8 %&$'- $--&0/$"!)

,/"* /" /- )!(&"!) #< κ−1 (vi). !"#$%& '( !" "#! $!" %& '(!)*%(+"$ V = ,+-".("(%+/ $0+("0.1/ %.20+%3!*4 "(56 0+7 G 5%+$($"(+2 %& "#! &%33%)(+2 2%03$8 G1= ∃9 :;.!07:9< ∧ %)=03":9<</ G2 = ∃9 :;.!07:9< ∧ >%+"0?(+0+"@.!!:9<</ G3 = ∃9 :;.!07:9< ∧ >.-$"1:9<</ G4 = ∃ 9 :;.!07:9< ∧ A.05!B3!?!+"C(5#:9<<D E! #0'! κ(G1) F κ(G4) F nutrition/ κ(G2) F sanitary 0+7 κ(G3) F organolepticD >%+'!.$!31 κ−1 (nutrition)F {G1, G4}/ κ−1(sanitary)F {G2} 0+7 κ−1 (organoleptic)F {G3}D

A*! 64'!- ,/'' 0&66!-3&() "& "*! -!" &8 -4C0/!(" 0&()/"/&(- (!!)!) 8&6 "*! %&$' Gi. D( "*! 0&("!@" &8 &46 36$0"/0$' $33'/0$"/&( "*/- /- /''4-"6$"!) /(

E/%46! F 9,/"* 6!-3!0" "& (4"6/"/&( %&$'-:.

!"#$%& )( A% .!05# "#! 2%03 G1 = ∃ 9 :;.!07:9< ∧ %)=03":9<</ "#!

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 !"#$% &' ()*+ ,- $%)./ 0#,$!,!-0)1 "-)1+

!"#$%&'% ()*% K +"!,)-!* ,.% /"$$"#-!' 01$%2 ∀ 345 670%)&638 ∧ 9)$,:&; <1!+,-"!65438 ∧ =%+0%)*%658 → >"#9)$,6388

!" #$ %&' (&%$)*!+ ",! $!" &- .&/0$ G = {G1, G2, ..., Gn} /%* ",! )%)")/0

1%&'0!*.! 2/$! K = (F, R, N )3 4$ *!$(+)2!* /2&5! '! (&67#"! ",! n 1%&'0!*.! 2/$!$8 (&++!$7&%*)%. "& !/(, .&/09 Ki = (F ∪ R−1(Gi), R, N ) -&+

!/(, i = 1, ..., n3 :! (&%$)*!+ ",! #%)&% &- /00 ",!$! 1%&'0!*.! 2/$!$9 Kagg = (F [ i=1,...,n R−1 (Gi), R, N ) !"#$%& '( >%, K = (F, R, N ) #.%0% 2 • F = {F1} =?@100%!,A3,0)+,-"!B),%6CDE8F • R +"!,)-!* ,.% /"$$"#-!' 01$%*2 ;<

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

− R1 = ∀ !" #$%&'(# ) ∧ * +%',+-./0'+&#"! ) ∧ 1&,%&'2&#")

→1-3&2+-45&# )) − R2 = ∀ !6 #$%&'(# ) ∧ 7'5+8(9:/,+-./#6! ) ∧ 1&,%&'2&#6)

→ ;.<7'5+# )) − R3 = ∀ !" #$%&'(# ) ∧ * +%',+-./0'+&#"! ) ∧ =%.<+>#")

→ ?%',&*5&@&/+0-,># )) − R4 = ∀ !" #$%&'(# ) ∧ * +%',+-./0'+&#"! ) ∧ 1&,%&'2&#")

→ A&2+-,-(&B%&&# )) • N ,./+'-/2 +>& C.55.<-/3 /&3'+-D& ,./2+%'-/+E

− N = ¬(∃ #=%.<+># ) ∧ 1&,%&'2&# ))) ;&+ +>& 3.'5 2&+ G '2 C.55.<2E

• G1= ∃ F #$%&'(#F) ∧ 1-3&2+-45&#F))! <>&%& κ(G1) = nutrition

• G2= ∃ F #$%&'(#F) ∧ ;.<7'5+#F))! <>&%& κ(G2) = nutrition

• G3= ∃F #$%&'(#F) ∧ ?%',&*5&@&/+0-,>#F))! <>&%& κ(G3) = nutrition

• G4= ∃ F #$%&'(#F) ∧ A&2+-,-(&B%&&#F))! <>&%& κ(G4) = sanitaryG

?>&/E

• K1= (F1, R, N )<>&%& F1 = F ∪ R −1

(G1) ,./+'-/2 +>& C.55.<-/3 C',+2E

− F1= H:%%&/+* +%',+-./0'+&#?IJ)

− F2= $%&'(#F) ∧ * +%',+-./0'+&#τ!F) ∧ 1&,%&'2&#τ)

• K2= (F2, R, N )<>&%& F2 = F ∪ R −1 (G2) ,./+'-/2 +>& C.55.<-/3 C',+2E − F1= H:%%&/+* +%',+-./0'+&#?IJ) − F3= $%&'(#F) ∧ 7'5+8(9:/,+-./#2!F) ∧ 1&,%&'2&#2) • K3= (F3, R, N )<>&%& F3 = F ∪ R −1 (G3) ,./+'-/2 +>& C.55.<-/3 C',+2E

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 − F1= !""#$%&'%"()%*+$,(%#-./01 − F4= 2"#(3-41 ∧ &'%"()%*+$,(%#-τ541 ∧ 6"+7%8-τ1 • K4= (F4, R, N )78#"# F4 = F ∪ R−1(G4) )+$%(*$9 %8# :+;;+7*$< :()%9= − F1= !""#$%&'%"()%*+$,(%#-./01 − F2= 2"#(3-41 ∧ &'%"()%*+$,(%#-τ541 ∧ >#)"#(9#-τ1 ?*$(;;@ Kagg = (FSi=1,...,nR −1 (Gi), R, N ) 78#"# FS_i=1,...,nR−1 (Gi) = {F1, F2, F3, F4}A

! "#!$%&$' () *+$ ,%$&("-! $./0,1$2 (* 0/3 +/,,$) *+/* Kagg (! ()4")5

!(!*$)* 6/)' (* '"$! !" $&$) 7"% 8"/1! #$1")8()8 *" *+$ !/0$ &($9,"()*:; <$ *+$) -!$ /%8-0$)*/*(")2 9+(4+2 #3 *+$ 0$/)! "7 $.*$)!(")! 9(11 (!"1/*$ !-#5 !$*! "7 7/4*! 9$ 4/) /44$,* *"8$*+$% 64/11$' $.*$)!(")!:; =-%*+$%0"%$2 *+$ $.*$)!(")! 9(11 /11"9 -! *" !$$ 9+(4+ /%$ *+$ &($9,"()*! /!!"4(/*$' *" $/4+ 0/.(0/1 4")!(!*$)* !-#!$* "7 >)"91$'8$ 6#3 *+$ 0$/)! "7 *+$ 7-)4*(") κ:; 4+"(4$ ,%"4$'-%$ *+$) +/! *" #$ -!$' 6!$$ $./0,1$ #$1"9:; ?+$ /%8-0$)* 7%/0$9"%> 9$ 4/) 4")!*%-4* 7%"0 *+$ /#"&$ >)"91$'8$ #/!$ (! (A, Att) 9+$%$ A 4")*/()! *+$ 7"11"9()8@ • a = ({F2}, F2, R1(F2))9+$%$ R1(F2) =2"#(3-41 ∧ &'%"()%*+$,(%#-τ541 ∧ >#)"#(9#-τ1 ∧ >*<#9%*B;#-41; • b = ({F4}, F4, R3(F4)) 9+$%$ R3(F4) = 2"#(3-41 ∧ &'%"()%*+$,(%#-τ541 ∧ 6"+7%8-τ1 ∧ ."()#&;#C#$%,*)8-41; • c = ({F2}, F2, R4(F2)) 9+$%$ R4(F2) = 2"#(3-41 ∧ &'%"()%*+$,(%#-τ541 ∧ >#)"#(9#-τ1 ∧ D#9%*)*3#?"##-41; • d = ({F3}, F3, R2(F3))9+$%$ R2(F3) = 2"#(3-41 ∧ E(;%F3G!$)%*+$-9541 ∧ >#)"#(9#-91 ∧ H+7E(;%-41 /)' Att = {(a, b), (b, a), (b, c), (c, b)};

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 ! "#$% &'()*+!"&"$,! %-%"+* .+/!+. 0+ !,0 ,1"&$!2

• Extstable(A, Att) = Extsemi−stable(A, Att) = Extpref ered(A, Att) = {{a, c, d},

{b, d}}₃

4"&'"$!( 5',* "#+ +6"+!%$,!% Extx(A, Att)7 "#+ 8',8,%+. .+9$%$,! %)88,'"

%-%"+* 5)!9"$,!% &% 5,::,0%2 5,' +;+'- +6"+!%$,! ε ∈ Extx(A, Att)2

• <,!%$.+' "#+ 5&9"% ,99)''$!( $! "#+ &'()*+!"% ,5 ε = • .+!"$5- "#+ >!,0:+.(+ 1&%+% Ki 0#+'+ "#+%+ 5&9"% ,99)'=

• ?1"&$! "#+ (,&:% Gi 0#$9# &'+ %&"$%/+. 1- "#+ +6"+!%$,!=

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• @%+ 8'+5+'+!9+ '+:&"$,!% ,5 +68+'"% ,! (,&:% &!. F ,' ;$+08,$!"%3 ! "#+ 8'+;$,)% +6&*8:+ G8:+&%+ '+9&:: "#&" "#+ (,&:% G1 &!. G2 &'+ &%%,A

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%&!$"&'-;$+08,$!"H 0+ #&;+2

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

• !" #$%& "'&"(%)*( {a, c, d} )% +,%"- *( &!" .,/&% F2 ,(- F3

*+&,)("-.$*0 K11 K2 ,(- K4 &!,& %,&)%.2 &!" 3*,4% G11 G2 ,(- G45

• !" %"/*(- "'&"(%)*( {b, d} )% +,%"- *( F3,(- F4*+&,)("- .$*0 K2

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,(- 8)"69*)(&%5

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)% 9$"."$$"- &* &!" *&!"$%5 ! "#$%&$'()*

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!" #$%& 0""&)(3 -",4& 6)&! &!" -"4)0)&,&)*( *. &!" 9$*L"/& *+L"/&)8"% ,(-,--$"%%"- .7(-,0"(&,4 M7"%&)*(% %7/! ,%E =% )& 9*%%)+4" &* 7()M7"42 -"#(" ,

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

!""#$ %&'(#) *+,-+ .-'/(&,, "0 !""# %&'(#$ .+"12# %' -"/.,#'&'#) 3"4 -"12# 5+'6 %' #'7/'# 0&"8 ( /15&,5,"/(29 .(/,5(&69 .'/."&,(2 (/# '-"/"8,-:",/5 "0 ;,'4) *+,-+ (&' 5+' 8(,/ </"4/ 4(6. 5" (-+,';' 5+'8)

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A/ 5+' 0"22"4,/! :2'/(&6 8''5,/! 5+' &'(2 :"5'/5,(2 "0 5+' (::&"(-+ 4(. .+"4/> =+' '@:'&5. 4'&' 0"&812(5,/! !"(2. (/# ;,'4:",/5. 5+'6 4'&' ,/5'&B '.5'# ,/ (/# 5+' C"!1, .6.5'8 5"!'5+'& 4,5+ 5+' (&!18'/5(5,"/ '@5'/.,"/ 4(. 6,'2#,/! 5+' (.."-,(5'# :"..,%2' :&":".,5,"/.> D,!1&' E .+"4. ( .-&''/.+"5 "0 5+' #'8"/.5&(5"& (/.4'&. 0"& ( 54"B!"(2 F1'&6G ( /15&,5,"/(2 !"(2 H+,!+ 7%'& -"/5'/5I (/# (/ "&!(/"2':5,- !"(2 H-&1.56 %&'(#I> =4" .'5. "0 -"8:(5,%2' (-5,"/. (&' :&":".'#9 ."8' -+",-'. H.1-+ (. ,/-&'(.,/! "& #'-&'(.,/! 5+' '@B 5&(-5,"/ &(5'I %',/! ,/-"8:(5,%2' 0"& %"5+ !"(2.9 (/# 5+1. .':(&(5'# ,/ 5+' 54" (25'&/(5,;' .'5.>

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ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 !"#$% &' (%)*+,-$.-*$ ,/$%%+,0*- ,0*1!+" -1* ,%-, *2 3*,,!45% ./-!*+,

!"# $%& '&(!)*( +&'& ,"-(./&'&/ 0( &12)0-03)& 34 &12&'*(5 3!* -"* "36."!(5 (.-,& 70-4 ,"-(./&'0*."-( %0/ *" 3& *08&- .-*" 0,,"!-*#

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ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 !"#$ !%&!'#()! *'!# (' $!+,# *- $(,!. / *'!01"2 &!+(*1 " *3# 4*$5 ! (6($"$('7 8'*3 !17! $5+*975 "' ('$!+)(!3 "'1 -*+," (:('7 ($ (' $5! #*-$3"+! #2#$!, ; 35(65 6*'#$($9$!# $5! *'7!#$ &"+$ *- $5! 3*+8. <*3!)!+= $5(# +! "$()! 2 #5*+$ $(,! 5(1!# " #$+*'7 &+!+!>9(#($!? 5")('7 " +!"12 " 6 !"+ )(!3 *- $5! 6"#! #$912= " #2'*&#(# *- $5! >9!#$(*'# $* "#8 $5! !%&!+$ "'1 "' (,& !,!'$!1 8'*3 !17! ,*1! . @' $5! *$5!+ 5"'1= 3!4#($!#= $!65'(6" +!&*+$# "'1 #6(!'$(A6 "+$(6 !# "+! ,*+! 6*#$ 2 $* "'" 2:!. B*+ ('#$"'6!= $5! 6+($(6" +!"1('7 " #6(!'$(A6 &"&!+ *- $5! 1*,"(' ,"2 +!>9(+! " *'!01"2 &!+(*1 *' ($# *3'= -*+ " 1(#6!+'('7 +!"1!+. <*3!)!+ $5!2 " *3 *'! $* $* 7+"#& $5! ('# "'1 *9$# *- $5! >9!#$(*'. C9+('7 $5! !)" 9"$(*' #$!&= $5! !%&!+$# +"(#!1 $5! >9!#$(*' *- $5! (,&*+0 $"'6! "$$"65!1 $* $5! 1(D!+!'$ &(!6!# *- 8'*3 !17! ,*1! !1 (' $5! #2#$!,. E*+!*)!+= (' #*,! 6"#!# !%&!+$# ,"2 5!#($"$! *' $5! +! !)"'6! *- #*,! -"6$# *+ +9 !#. / &*##(4( ($2 3*9 1 $59# 4! $* "1*&$ " &+!-!+!'6!04"#!1 "+79,!'0 $"$(*' #2#$!,= "# &+*&*#!1 (' #!)!+" 3*+8# #965 "# F/,7*91 "'1 G"2+* = HIIHJ K!'650G"&*'= HIIL4J M"6( "'1 )"' 1!+ N*++!= HIIOJ /,7*91 !$ " .= HIIIJ K*9+79!$ !$ " .= HIPL"Q= "4 ! $* $"8! ('$* "66*9'$ 1(D!+!'$ !)! # *-(,&*+$"'6! ",*'7 "+79,!'$#.

! "#$%&'()#$

R)!' (- "+79,!'$"$(*' 4"#!1 1!6(#(*' ,"8('7 ,!$5*1# "&& (!1 $* $5! -**1 ('19#$+2 3!+! " #* &+*&*#!1 42 FK*9+79!$= HIPIJ K*9+79!$ !$ " .= HIPL4Q= $5(# &"&!+ "11+!##!# " 8!2 &*('$ (' $5! 6*'$!%$ *- 69++!'$ $!65'(>9!# 9#!1 42 $5! -**1 #!6$*+ "'1 '",! 2 "11+!##('7 +!)!+#! !'7('!!+('7. / #*= (' $5(# "&&+*"65= "' "+79,!'$ (# 9#!1 5!+! "# " ,!$5*1 6*,&9$('7 6*,&"$(4 ! *40 S!6$()!# (' $5! #!6$*+. N5(# 6"#! #$912 +!&+!#!'$# "' *+(7('" "&& (6"$(*' "'1

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 ! "!#$%&'()#"*( ''$% )+ "! #+( ,$%!%-. /(01 2. '$%*"1"!, ! $,3-(!# 4 #"%! 2 &(1 1()"&"%!4&3''%$# &.&#(- 5%$ #+( * $"%3& 5%%1 &()#%$&6 7# $(83"$(& !(*($#+(0(&& #+( *($. (9'(!&"*( # &: %5 :!%;0(1,( -%1(0"!,6 <3)+ # &:= "! "#& )3$$(!# &# #( ) !!%# 2( 3#%- #(16 7# &#$%!,0. 1('(!1& %! #+( 83 0"#. %5 (9'($# %'"!"%! !1 (0")"# #"%! >(9+ 3&#"*(!(&&= )($# "!#.= (#)?6 @+( )3$$(!# #$(!1 5%$ 1()"&"%!4- :"!, #%%0& "!)031(& -%$( !1 -%$( -(#+%1& %5 $,34 -(!# #"%! & -( !& %5 "!)031"!, (9'($#& "! #+( # &: %5 -%1(0"!, !1 #+( 1()"&"%!4- :"!, '$%)(&&(&6 A!%#+($ (0(-(!# #% # :( "!#% ))%3!#= !%# 1"&4 )3&&(1 "! #+"& ' '($= "& #+( 1"B)30#. %5 #()+!%0%,") 00. >5$%- ! ,$%!%-. *"(;'%"!#? '3##"!, "! '0 )( #+( 5 )#& %5 ( )+ %'#"%!6 C%1(0"!, #+"& &'()# "! #+( 5%$- 0"&- "& &#"00 #% 2( &#31"(16

A-,%31= D6= E .$%0= E6= FGGF6 A $( &%!"!, -%1(0 2 &(1 %! #+( '$%13)#"%! %5 ))('# 20( $,3-(!#&6 A!! 0& %5 C #+(- #")& !1 A$#"/)" 0 7!#(00",(!)( HI= JKLMFJN6

A-,%31= D6= O $&%!&= <6= O($$3&&(0= D6= FGGG6 A! $,3-(!# #"%! 5$ -(;%$: 2 &(1 %! )%!#(9#3 0 '$(5($(!)(&6 7!P O$%)((1"!,& %5 #+( 7!#($! #"%! 0 E%!4 5($(!)( %! Q%$- 0 !1 A''0"(1 !1 O$ )#") 0 R( &%!"!,6 ''6 SKMNL6 A-,%31= D6= O$ 1(= T6= FGGK6 U&"!, $,3-(!#& 5%$ - :"!, !1 (9'0 "!"!,

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A&& ,+"$= X6= W '%0"= A6= Y .#%3(= C6= Z32%"&= Z6= O$ 1(= T6= FGJJ6 W3-($"4

ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

!" #$%&'(!)#&$ %*+#&$, -!))# . &% !$+/.'+ /#)0 +*11&')#$2 !'2*(.$)+3 4$, 456743 113 89:;89<3

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ACCEPTED MANUSCRIPT

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!!"#$% &' (''% )* "#&+ #, !*-"#. /$ "&/ !'"#.+0 12!$3& 4+5&$65 7#&/ 8!!"#. &#',5 9: ;<<=> 9?@AB9?9C0

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ACCEPTED MANUSCRIPT

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

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