On the Revision of Argumentation Systems: Minimal Change of Arguments Status

Transcription

1 On the Revision of Argumenttion Systems: Miniml Chnge of Arguments Sttus Sylvie Coste-Mrquis, Séstien Koniezny, Jen-Guy Milly, n Pierre Mrquis CRIL Université Artois CNRS Lens, Frne {oste,koniezny,milly,mrquis}@ril.fr Astrt. In this pper, we investigte the revision issue for rgumenttion systems à l Dung. We fous on revision s miniml hnge of the rguments sttus. Contrrily to most of the previous works on the topi, the ition of new rguments is not llowe in the revision proess, so tht the revise system hs to e otine y moifying the ttk reltion, only. We introue lnguge of revision formule whih is expressive enough for enling the representtion of omplex onitions on the eptility of rguments in the revise system. We show how AGM elief revision postultes n e trnslte to the se of rgumenttion systems. We provie orresponing representtion theorem in terms of miniml hnge of the rguments sttus. Severl istne-se revision opertors stisfying the postultes re lso pointe out. Keywors: rgumenttion, elief revision 1 Introution In this pper, we investigte the revision issue for strt rgumenttion systems à l Dung [1]. Argumenttion systems re irete grphs, where noes orrespon to rguments n rs to ttks etween rguments. In suh systems, the sttus (eptne) of eh rgument epens on the hosen eptility semntis (groune, preferre, stle mong others). In his ook [2] Gärenfors introue strtly elief hnge s the opertion llowing to hnge the epistemi sttus of piee of informtion with respet to the epistemi stte of n gent. There re three possile sttus: epte, refuse or unefine. An revision, ontrtion n expnsion re efine s the possile trnsitions etween these sttus, s illustrte y Figure 1. Then Gärenfors instntites this generl efinition within logil frmework. But it is interesting to note tht efore efining elief hnge opertors, one hs to mke preise the logil setting into whih the sttus epte, refuse n unefine re efine. Similrly, if one wnts to instntite Gärenfors generl efinition of elief hnge within Dung s rgumenttion theory, it is first neessry to efine wht

2 2 Unefine Expnsion Contrtion Expnsion Contrtion Aepte Revision Revision Refuse Fig. 1. Gärenfors epistemi trnsitions re the ville piees of informtion n wht these sttus men. In Dung s rgumenttion theory, the si piees of informtion re the rguments of the system, n their sttus epens on the eptility semntis uner onsiertion. Thus, it oes not mke sense to stuy the revision of rgumenttion systems iretly on the ttk grph, inepenently of ny semntis. Stte otherwise, the revision of given rgumenttion system uner two ifferent semntis my esily le to two ifferent results. For instne, in the se of the rgumenttion system given on Figure 2, we note tht uner the stle n preferre semntis, elongs to every extension, wheres oes not elong to ny extension in the se of the groune semntis. So, revise to ept oes not nee ny hnge for stle n preferre semntis, ut hnge is require for the groune semntis. Fig. 2. The hnge neee to revise this system is not the sme oring to the semntis. In this pper, we fous on revision s miniml hnge of the rguments sttus. To e more preise, uner hosen semntis, n given n rgumenttion system n revision formul expressing how the sttus of some rguments hs to e hnge, we wnt to erive one or severl rgumenttion systems whih stisfy the revision formul, n re suh tht the orresponing extensions re s lose s possile to the extensions of the input system. Contrrily to most of the previous works on the topi, the ition of new rguments is not llowe in the revision proess, so tht the revise system hs to e otine y moifying the ttk reltion, only. Espeilly, the revision formul oes not inite why the sttus of rguments hve hnge.

3 3 Miniml hnge of the ttk grph n e onsiere s riterion for efining the revise systems, ut not s the min one, sine, s expline ove, the eptne sttus of rguments is more funmentl informtion. Aoringly, ensuring miniml hnge of these sttus is more importnt thn (n ifferent from) ensuring miniml hnge of the ttk grph. Suh revision opertors, where miniml hnge ers on the rguments sttus, n e very useful for pplitions of rgumenttion on soil network etes [3]. When n gent A initites ete out n rgument α, if nother gent B oes not gree with A out α ut onsiers tht A is trustworthy, B hs to revise her eliefs to inlue α in t lest one of her extensions. This kin of resoning uses reulous inferene, ut it n e reple y skeptil inferene n so the revise system n e ompute with one of the opertors efine in this pper. The rest of the pper is orgnize s follows. After short introution to the Dung s theory of strt rgumenttion in Setion 2, we introue lnguge of revision formule whih is expressive enough for enling the representtion of omplex onitions on the eptility of rguments in the revise system in Setion 3. In Setion 4 we show how AGM elief revision postultes n e trnslte to the se of rgumenttion systems. An we provie orresponing representtion theorem in terms of miniml hnge of the rguments sttus. In Setion 5 severl istne-se revision opertors stisfying the postultes re pointe out. Then Setion 6 isusses how to ssoite rgumenttion systems to the otine sets of extensions, n isuss the issue of miniml hnge of the ttk grph. Setion 7 isusses some relte work. Setion 8 onlues the pper. Proofs re omitte for spe resons. 2 Preliminries We strt with very short introution to Dung s theory of rgumenttion (see [1] for more etils). A (finite) rgumenttion system (lso referre to s n rgumenttion frmework) is pir AF = A, R where A is (finite) set of so-lle rguments n R is inry reltion over A ( suset of A A). In the following, A is suppose to ontin t lest two elements n the ttk reltion R is suppose to e irreflexive, i.e., self-ontriting rguments re rejete. AFs A enotes the set of ll suh systems on the set of rguments A. An rgument A is eptle with respet to set of rguments S A whenever it is efene y the set, i.e., for every A s.t. (, ) R, there exists S suh tht (, ) R. We sy tht suset S of A is onflit-free if n only if for every, S, we hve (, ) R. A suset S of A is missile for AF if n only if S is onflit-free n eptle with respet to S. Solutions of n rgumenttion systems re sets S of rguments tht n e, so to sy, epte together. Severl semntis σ (espeilly, the omplete semntis, the preferre semntis, the stle semntis, the groune semntis) n e onsiere for pturing formlly this notion, n eh of them gives rise to speifi notion of extensions. For instne:

4 4 S is omplete extension of AF if n only if it is n missile set n every rgument whih is eptle w.r.t. S elongs to S, S is preferre extension of AF if n only if it is mximl (with respet to set inlusion) in the set of missile sets for AF, S is stle extension of AF if n only if S is onflit-free n A \ S, S suh tht (, ) R, S is the (unique) groune extension of AF if n only if it is the smllest element (with respet to set inlusion) mong the omplete extensions. Ext σ (AF ) enotes the set of extensions of AF for the semntis σ. The epistemi sttus of ny rgument A with respet to the epistemi stte represente y AF is then given y: is epte if elongs to every ε Ext σ (AF ), is refuse if oes not elong to ny ε Ext σ (AF ), n is unefine in the remining se. Moreover, one introue nottion out miniml elements of set. First, < enotes the strit prt of n enotes the inifferene reltion ssoite. Given set E n pre-orer, the miniml elements of E w.r.t. this pre-orer re min(e, ) = {e E e E, e < e}. 3 On Revision Formule We wnt to efine revision setting for Dung s rgumenttion systems in whih sophistite revision formule n e tken into ount, n not only the ft tht given rgument shoul e epte or refuse. To this en, we onsier logil lnguge L A, where negtion is use to enote the ft tht given rgument shoul e refuse, n formule n e onnete using onjuntion n isjuntion. Definition 1 Given A = {α 1,..., α k } set of rguments, L A is the lnguge generte y the following ontext-free grmmr in BNF: rg ::= α 1... α k Φ ::= rg Φ (Φ Φ) (Φ Φ) For instne, ϕ 1 = ( (( ) ( ))) expresses tht in the revise epistemi stte one wnts to e epte n n to e oth epte or oth refuse. The epistemi sttus of suh formule ϕ 1 from L A in n rgumenttion system AF AFs A for given semntis σ is given y: Definition 2 Let ε A n ϕ L A. The onept of stisftion of ϕ y ε, note ε ϕ, is efine inutively s follows: If ϕ = A, then ε ϕ iff ε, If ϕ = (ϕ 1 ϕ 2 ), ε ϕ iff ε ϕ 1 n ε ϕ 2, If ϕ = (ϕ 1 ϕ 2 ), ε ϕ iff ε ϕ 1 or ε ϕ 2, If ϕ = ψ, ε ϕ iff ε ψ.

5 5 Then for ny AF in AFs A, n ny semntis σ, we sy tht: ϕ is epte w.r.t. AF, note AF σ ϕ, if ε ϕ for every ε Ext σ (AF ), ϕ is refuse w.r.t. AF, note AF σ ϕ, if ε ϕ for no ε Ext σ (AF ), ϕ is unefine w.r.t. AF in the remining se. From now on we ll nite 1 ny suset ε of A. Cnites n e interprete s moels or ounter-moels of revision formule. Continuing the previous exmple, if A = {,, }, ϕ 1 is stisfie y the nites from {{}, {,, }}. Thus, for the groune semntis, ϕ 1 is epte w.r.t. AF 1 with R 1 = {(, ), (, )} ut is refuse w.r.t. AF 2 with R 2 = {(, ), (, )}. Oviously enough, for ny set M = {ε 1,..., ε k } of nites, there exists formul ϕ L A so tht M = A ϕ, where A ϕ = {ε A ε ϕ} is the set of nites whih stisfy ϕ. However, in the generl se, A ϕ is not the set of ll σ extensions of n AF in AFs A. Consier for instne, A = {,, } n ϕ 1 = ( ) ( ) 2. {} n {,, } re the two nites stisfying ϕ 1, n there is no AF in AFs A suh tht Ext σ (AF ) = {{}, {,, }} for σ = groune, σ = preferre or σ = stle. A onept of σ-onsisteny n then e efine s follows: Definition 3 A formul ϕ L A is σ-onsistent 3 iff there exists set S of rgumenttion systems AF in AFs A suh tht A ϕ = AF S Ext σ(af ). When A = {,, }, ϕ 1 = ( (( ) ( ))) is σ onsistent for σ = groune, σ = preferre or σ = stle sine {{}, {,, }} = Ext σ (AF 3 ) Ext σ (AF 4 ) where R 3 = {(, ), (, )} n R 4 =. Contrstingly, ϕ 2 = is groune-onsistent n preferre-onsistent ut not stle-onsistent. ϕ 3 = neither is groune-onsistent nor is preferre-onsistent, ut is stle-onsistent (onsier AF 5 suh tht R 5 = {(, ), (, ), (, )}.) 4 On the Revision of Argumenttion Systems A revision opertor on rgumenttion systems is mpping ssoiting set of rgumenttion systems to the input rgumenttion system n the input revision formul: Definition 4 Given ny set of rguments A, revision opertor on rgumenttion systems is mpping from AFs A L A to 2 AFs A. Clerly, the result of the revision of n rgumenttion system is not unique rgumenttion system in the generl se, ut set of rgumenttion systems. The reson is quite simple: there n e severl possile results whih hve extly the sme mximum plusiility. So in this se there is no reson to 1 Tht is to sy nite to e n extension 2 Equivlent to ( (( ) ( ))). 3 or simply onsistent if the semntis is fixe.

6 6 hoose priori one of them (we will return to this point lter on.) If this is prolemti for prtiulr pplition, seletion funtion n e use s tie-rek rule for ensuring the uniity of the result (just like, for instne, the mxihoie seletion funtion is onsiere in AGM elief revision [2].) In orer to efine revision opertors, our pproh follows two-step proess. Intuitively, one first selets from the nites stisfying the revision formul ϕ those whih re s lose s possile to the σ-extensions of AF. Then, one genertes some rgumenttion systems suh tht the union of their σ extensions preisely oinies with the selete nites. Of ourse, it is not the se tht eh mpping from AFs A L A to 2 AFs A is resonle revision opertor. For instne, the onstnt, yet trivil opertor efine y AF ϕ = shoul e isre. In orer to ientify interesting revision opertors, one hs to ientify the logil properties tht gurntee rtionl ehviour. Suh n xiomti pproh is stnr in logi n the AGM postultes [4,5] hve een pointe out for hrterizing vlule revision opertors in logil setting. As in [6], we n revisit these postultes in set-theoreti frmework, here suite to the rgumenttion se. Let S e set of rgumenttion systems AF in AFs A. For eh semntis σ, we efine the set Ext σ (S) of σ-extensions of S s AF S Ext σ(af ). The ounterprt of AGM postultes in the rgumenttion se is given y: (AE1) Ext σ (AF ϕ) A ϕ (AE2) If Ext σ (AF ) A ϕ, then Ext σ (AF ϕ) = Ext σ (AF ) A ϕ (AE3) If ϕ is σ-onsistent, then Ext σ (AF ϕ) (AE4) If A ϕ = A ψ, then Ext σ (AF ϕ) = Ext σ (AF ψ) (AE5) Ext σ (AF ϕ) A ψ Ext σ (AF (ϕ ψ)) (AE6) If Ext σ (AF ϕ) A ψ, then Ext σ (AF (ϕ ψ)) Ext σ (AF ϕ) A ψ (AE1) sttes tht the σ-extensions of the resulting set of rgumenttion systems must e mong the nites stisfying ϕ. (AE2) emns tht if there re σ-extensions of the input system stisfying ϕ, then the resulting σ- extensions must oinie with them. (AE3) requires the resulting set of σ- extensions to e non-empty s soon ϕ is σ-onsistent. (AE4) sys tht the revision y equivlent formule must le to the sme results. The lst two postultes (AE5) n (AE6) express miniml hnge priniple with respet to the rguments sttus: one expets to hnge s few s possile the sttus of rguments in the input system. Interestingly, s in the logil se, we n erive representtion theorem whih hrterizes extly the revision opertors stisfying the postultes in more onstrutive wy. To this en, we first nee to exten the notion of fithful ssignment [5]: Definition 5 A fithful ssignment is mpping ssoiting ny rgumenttion system AF = A, R (uner semntis σ) with totl pre-orer σ AF on the set of nites suh tht: 1. if ε 1 Ext σ (AF ) n ε 2 Ext σ (AF ), then ε 1 σ AF ε 2, 2. if ε 1 Ext σ (AF ) n ε 2 Ext σ (AF ), then ε 1 < σ AF ε 2.

7 7 The representtion theorem n then e stte s follows: Proposition 1 Given semntis σ, revision opertor stisfies the rtionlity postultes (AE1) - (AE6) iff there exists fithful ssignment whih mthes every system AF = A, R to totl pre-orer σ AF so tht Ext σ (AF ϕ) = min(a ϕ, σ AF ) This theorem is useful for efining opertors stisfying the rtionlity postultes, s those presente in the next setion. 5 Distne-Bse Revision Let us now present some istne-se revision opertors stisfying the rtionlity postultes (AE1) - (AE6). Let e ny istne on 2 A, for instne, the Hmming istne given y H (ε 1, ε 2 ) = (ε 1 \ ε 2 ) (ε 2 \ ε 1 ). Given ε 2 A n E 2 A, n e extene to istne etween ε n E, y stting tht (ε, E) = min ε E (ε, ε ). For ny rgumenttion system AF AFs A, this istne inues totl pre-orer etween nites ε 1, ε 2 2 A given y ε 1 σ, AF ε 2 iff (ε 1, Ext σ (AF )) (ε 2, Ext σ (AF )). On this groun, revision opertors n e efine y: Definition 6 Let σ e ny given semntis. A istne-se revision opertor is ny revision opertor for whih there exists istne on 2 A suh tht for every AF n every ϕ, we hve Ext σ (AF ϕ) = min(a ϕ, σ, AF ). Proposition 2 Let σ e ny semntis. Any istne-se revision opertor stisfies the rtionlity postultes (AE1) - (AE6). Let us now efine nother fmily of istne-se opertors, whih tke vntge of lellings. Let us first rell tht, inste of using extensions, the solutions of n rgumenttion system n e expresse using the onept of lelling [7]. Formlly, lelling is mpping L ssoiting lel in, une or out with every rgument of the set A. The stronger notion of reinsttement lelling epens on the ttk reltion R: n rgument is lelle in iff every rgument ttking is out; n rgument is out iff there exists n rgument in ttking ; n rgument is une iff it is neither in nor out. These reinsttement lellings orrespon to Dung s omplete extensions in ijetive wy. Thus, for ny omplete extension ε, the ssoite reinsttement lelling is suh tht every rgument ε is in, every rgument ttke y n rgument in is out, every other rgument is une. Conversely, for every reinsttement lelling L, the orresponing extension E(L) is the set of rguments lelle in y the lelling. Other semntis n lso e enoe with lellings. We introue some nottions: L ϕ is the set of lellings L suh tht E(L) A ϕ. Given set of lellings L, E(L) = {E(L) L L}. Finlly, given system AF

8 8 n semntis σ, Ls σ (AF ) enotes the set of lellings orresponing to the σ-extensions of AF. Lellings, whih ring riher informtion thn extensions, n e use to efine interesting istne-se revision opertors. Inee, onsier the following notion of eition istne: Definition 7 Let m, n, o e three integers, let L 1 n L 2 e two lellings, n let X n Y e ny of in, out n une. An eition istne (m,n,o) etween lellings is efine s: (m,n,o) (L 1, L 2 ) = A (L 1 (), L 2 ()), where (in, out) = m (in, une) = n (out, une) = o (X, Y ) = (Y, X) (X, X) = 0 Proposition 3 Let m, n, o e three integers. (m,n,o) is istne. An interesting point to note is tht these eition istnes re not neessrily neutrl or symmetril. We ll neutrl istne suh tht (in, une) + (une, out) = (in, out) n symmetril istne suh tht (in, une) = (une, out). Defining non symmetril istnes is wy for instne to fvor eptne of rguments over rejet, y hoosing (in, une) = 1, (out, une) = 9, n (in, out) = 10. For ny istne L etween lellings, we n efine pre-orer σ, L AF etween lellings s we i it for nites: L 1 σ, L AF L 2 iff L (L 1, L σ (AF )) L (L 2, L σ (AF )). Definition 8 Let σ e ny given semntis. A lelling-istne-se revision opertor L is ny revision opertor for whih there exists istne L = (m,n,o) on 2 A suh tht for every AF n every ϕ, we hve Ls σ (AF L ϕ) = min(l ϕ, σ, L AF ). The following exmple illustrtes the impt of the hosen istne on the revise system: Exmple 1 Let σ e the stle semntis. We revise the system AF 6 elow y the formul ϕ = e. e Ext σ (AF 6 ) = {{,, e}, {,, e}}, the ssoite stle lellings re {(, in), (, out), (, out), (, in), (e, in)} n {(, out), (, in), (, out), (, in),

9 (e, in)}. When we revise AF 6 y ϕ using the istne-se opertor inue y the istne (1,9,10) on lellings, one gets s result system of whom the lellings re {(, in), (, out), (, out), (, out), (e, out)} n {(, out), (, in), (, out), (, out), (e, out)}. When the istne (9,1,10) is use, one otins {(, in), (, out), (, out), (, une), (e, une)} n {(, out), (, in), (, out), (, une), (e, une)} s lellings of the result systems. If the genertion of the systems tke ount of the lellings, the struture of the result grph will e ifferent : when the refuse rguments re out, it mens tht there exists n ttk from n epte rgument to refuse rgument. When the rguments re une, those ttks o not exist. With the first istne (1,9,10), it is less ostly to hnge n rgument from in to out thn to une. Suh istne llow to hoose nites whih refuses rguments y. Contrrily, the istne (9,1,10) llow to hoose nites whih ept more rguments. The hoie of prtiulr istne llow to influene result of the revision. Like opertors se on extensions, istne-se opertors using leling exhiit goo logil properties: Proposition 4 Let σ e ny semntis. Any lelling-istne-se revision opertor L stisfies the rtionlity postultes (AE1) - (AE6). 9 6 Revision t the System Level The opertors efine in the previous setion fous on the nites tht re s lose s possile to the extensions of the input system. However, they o not inite how to generte the orresponing rgumenttion systems, i.e., the rgumenttion systems suh tht the union of their extensions oinies with the selete nites 4. In orer to o this jo, we onsier mpping AF σ from 2 2A to 2 AFs A, lle genertion opertor, tht ssoites with ny set C of nites set of rgumenttion systems suh tht Ext σ (AF σ (C)) = C. Oserve tht, whtever the semntis σ, suh mpping AF σ exists. This omes esily from the ft tht: Proposition 5 Whtever the semntis σ, for every non empty set C of nites from 2 A, suh tht C, there exists finite set S AFs A suh tht C = AF S Ext σ(af ). The point is tht every nite C n e ssoite with n rgumenttion system AF suh tht C is the unique σ-extension of it. For instne, if A = {,, } n C = {, }, AF given y R = {(, ), (, )} oes the jo whtever the semntis σ. This metho llows to prove tht set of systems exists, 4 The onstrution of systems from lellings is still n open question, this setion only els with genertion of systems from nites.

10 10 ut it oes not stuy the genertion of miniml set of systems orresponing to the nites. We give first pproh to this prolem in the rest. On this groun, revision opertors n then e efine s follows: Definition 9 Given semntis σ, fithful ssignment tht mthes every rgumenttion system to totl pre-orer σ AF, n genertion opertor AF σ, the orresponing si revision opertor is efine y: AF ϕ = AF σ (min(a ϕ, σ AF )) One of the key results of the pper is tht: Proposition 6 Every si revision opertor stisfies the postultes (AE1)- (AE6). Bsi opertors el only with minimlity of hnge of rguments sttus. Inee, the rtionlity postultes sks for preserving s muh s possile the sttus of rguments in the input system: oing so while ensuring tht the revision formul is stisfie oes not usully imply miniml hnge of the ttk reltion, n vie-vers. As mtter of illustrtion, onsier the rgumenttion systems AF 7, AF 8, AF 9, n AF 10 represente respetively in Figure 3., 3., 3., 3.. e (3.) e (3.) e (3.) e (3.) Fig. 3. Miniml Chnge Suppose tht our gol is to rejet e, tht is to get system so tht e oes not pper in ny extension. We onsier the revision formul ϕ = e. A miniml hnge on the ttk reltion of AF 7 les to AF 8 or AF 9 : eh of them iffers with AF 7 on single ttk. This ontrsts with AF 10 sine the hnge on the ttk reltion require to go from AF 7 to AF 10 is stritly greter thn the hnge on the ttk reltion require to go from AF 7 to AF 9. Eh of these four systems hs unique extension whtever the semntis: {,, e} for AF 7, {,, } for AF 8, n {, } for AF 9 n AF 10. Hene, the hnge on the sttus of rguments hieve when going from AF 7 to AF 9 or AF 10 is stritly smller thn the hnge on the sttus of rguments hieve when going from AF 7 to AF 8. A simple genertion opertor AF σ onsists in generting sets S of inresing rinlity n until the onition C = AF S Ext σ(af ) is stisfie n then returns the union of ll the miniml-size S whih stisfies it. However, using this simple genertion opertor oes not ensure to otin n optiml set of rgumenttion systems, when optimlity is unerstoo either s the miniml

11 11 numer of rgumenttion systems require to generte the nites, or s miniml hnge of the ttk reltion. Minimizing the hnge of the ttk reltion n nevertheless e tken into ount in the efinition of the genertion opertor (hene s seon riterion, with smller priority thn the riterion pturing the miniml hnge of the rguments sttus.) This n e esily one y onsiering istne g on the rgumenttion grphs. One simple exmple is the Hmming istne, given y g H (AF 1, AF 2 ) = (R 1 \ R 2 ) (R 2 \ R 1 ). But one n lso hoose more elorte eition istnes suh s those given in [8]. The g H istne etween two rgumenttion systems orrespons to the numer of ttks tht must e e or remove to mke them ientil. Eh istne g inues pre-orer etween rgumenttion systems, efine y AF 1 g AF AF 2 iff g(af 1, AF ) g(af 2, AF ). This pre-orer n e use to filter out unneessry rgumenttion systems proue y the genertion opertor. However, simple minimiztion w.r.t. g AF of the output of the genertion opertor is not suffiient in generl, s illustrte y the following exmple: Exmple 2 Suppose tht the nites re C = {C 1, C 2 }, n the genertion opertor AF σ pplie on C gives AF σ (C) = {AF 11, AF 12 } so tht, for the onsiere semntis σ, Ext σ (AF 11 ) = {C 1 } n Ext σ (AF 12 ) = {C 2 }. If AF 11 n AF 12 re not t equl istne from the input system AF (tht is AF 11 < AF AF 12 or AF 12 < AF AF 11 ), one of the systems is not miniml n must e remove. So n extension is lost. To tkle this prolem, we onsier seletion funtions removing rgumenttion systems when this oes not hnge the set of extensions: Definition 10 Given semntis σ, set of nites C, pre-orer AF n genertion opertor AF σ, let us efine the C-minimum-over seletion funtion γ C y: γ C (AF σ (C)) AF σ (C) Ext σ (γ C (AF σ (C))) = Ext σ (AF σ (C)) AF AF σ (C) n AF γ C (AF σ (C)) only if AF AF σ (C) so tht AF < AF AF Bse on this notion, full revision opertors n e efine s follows: Definition 11 Given semntis σ, fithful ssignment σ, AF, genertion opertor AF σ, pre-orer g AF over AFs A n C-minimum-over seletion funtion γ, the orresponing full revision opertor σ, AF, g AF,γ is efine y: AF σ, AF, g AF,γ ϕ = γ min(aϕ, σ, AF )(AF σ(min(a ϕ, σ, AF ))) Clerly full revision opertors re si revision opertors. Sine the minimiztion of the ttk reltion is one without moifying the selete nites, we get tht:

12 12 Proposition 7 Every full revision opertor σ, AF, g AF,γ stisfies the postultes (AE1)-(AE6). For illustrtion purposes, we now provie some exmples of revision opertors. For these exmples we will use the following C-minimum-over seletion funtion γ g AF 1 : V is n orere list efine from AF σ (C) y sorting it in esening orer 5 w.r.t. g AF. For i from 1 to n o if V j V s.t. V j < g AF V i n Ext σ (V \V i ) = Ext σ (V ), then V := V \V i. γ g AF 1 := V. With this metho, the systems remove from V re those whih re the frthest from the input AF. Fig. 4. The system AF 13 Let us now illustrte those onepts y onsiering the revision of the rgumenttion system AF 13 represente t Figure 4, using the full revision opertor efine from the Hmming istne etween nites, the Hmming istne etween grphs n the previous seletion funtion γ g AF 1. The set of extensions of AF 13 is {{, }} for the preferre n stle semntis, n its groune extension is. Hene, whtever the hosen semntis, we hve AF 13. Suppose now tht we wnt to e epte, so we mke revision with ϕ =. For the preferre semntis n the stle semntis, we first uil the set of nites whih stisfy so tht the Hmming istne with {{, }} is miniml. This set is {{,, }}. Then we uil the rgumenttion systems whih over this nite, fousing on those whih re t miniml istne to AF 13. The result is AF 14, given t Figure 5.. (5.) (5.) Fig. 5. The result of the revision of AF 13 y 5 There exists severl orere lists, epening on the wy to sort elements whih re equls w.r.t. the pre-orer. We hoose ritrrily one of these lists.

13 13 We proee similrly for the groune semntis: the set of extensions {{}} is first ompute, n then the unique system, AF 15, represente on Figure 5. is generte. A lst point we woul like to isuss is the ft tht si revision opertor outputs set of rgumenttion systems, n not single rgumenttion system in the generl se. Atully, this is onsequene of the expressiveness of the lnguge of revision formule we wnt to onsier. In orer to illustrte it, onsier A = {,,, }, n AF 16 s represente in Figure 6. Fig. 6. The system AF 16 (7.) (7.) Fig. 7. Revision of AF 16 The extensions of AF 16 re the sme for the groune, stle n preferre semntis, Ext(AF 16 ) = {{, }}. Let ϕ =. Oserve tht n ply symmetri roles, oth in AF 16 n in ϕ. When omputing the result of the revision with the full revision opertor se on Hmming istne etween nites, Hmming istne on the grph n the seletion funtion γ g AF 1, we otin two nites {,, } n {,, }, n two orresponing rgumenttion systems AF 17 n AF 18 (given respetively t Figure 7. n Figure 7.). Choosing one of these systems woul require to ept some ritrriness given the symmetri roles of n. Finlly, few wors out the itertion issue for revision. Given tht the output of revision opertor is not single system in the generl se ut set of suh systems, revision nnot e iretly iterte. Nevertheless, it is esy to exten revision opertors so tht sets of rgumenttion systems n e epte s input. Bsilly, S = {AF 1,..., AF n } ϕ n e efine s AF i S AF AF i ϕ AF. 7 Relte Work Some previous works hve lrey onsiere the hnge issue for rgumenttion systems à l Dung.

14 14 Thus, Boell, Ki n Vn er Torre, [9,10] hve stuie strtion n refinement priniples. An strtion is reution of the ttk reltion or the set of rguments, wheres refinement is the ition of ttks or rguments to the system. The uthors fouse on the stuy of semntis whih ensure the existene of unique extension (for instne, the groune extension), n they formulte some priniples of the form if we o this prtiulr hnge, then the extension of the result is like this. They ientifie some priniples stisfie y the groune semntis. Cyrol, Dupin e Sint-Cyr n Lgsquie-Shiex [11] stuie the ition of n rgument to n rgumenttion system. They stte some properties tht n e stisfie when hnge ours in n rgumenttion system, n pointe out those whih re stisfie (n uner whih onitions) when n rgument (n the ttks onerning it) is e to the grph. With Bisquert, they i similr stuy out the eletion of n rgument [12]. Bumnn [13] lso stuie the miniml hnge prolem in strt rgumenttion. He reporte some ouns on the numer of moifitions to mke on n rgumenttion system so s to enfore given set of rguments. These ouns epen on the semntis n the type of hnge llowe. All these works re silly onerne y the moifition of the ttk reltion. As suh, they iffer in signifint wy from the hnge prolem s stuie in this pper, where hnge primrily lies on the sttus of rguments (n seonry only on the ttk reltion.) 8 Conlusion In this pper, we investigte the revision prolem for strt rgumenttion systems à l Dung. We fouse on revision s miniml hnge of the rguments sttus. We introue lnguge of revision formule whih is expressive enough for enling the representtion of omplex onitions on the eptility of rguments in the revise system. We showe how AGM elief revision postultes n e trnslte to the se of rgumenttion systems. We provie orresponing representtion theorem in terms of miniml hnge of the rguments sttus, n pointe out severl istne-se revision opertors stisfying the postultes. As future work we first pln to enoe our revision opertors y representing rgumenttion systems with logil onstrints (in similr wy to Besnr n Doutre [14]), so s to e le to enefit from the power of onstrint solvers to ompute revise systems. The stuy of other hnge opertions on rgumenttion systems is nother perspetive for further reserh. Moreover, the ssoition of miniml set of rgumenttion systems to set of nites is very interesting, inluing for other pplitions thn revision. So we n eepen this question. The sme question for the se of lellings (see Exmple 1) is still open n will e stuie.

15 15 Referenes 1. Dung, P.M.: On the eptility of rguments n its funmentl role in nonmonotoni resoning, logi progrmming, n n-person gmes. Artifiil Intelligene 77(2) (1995) Gärenfors, P.: Knowlege In Flux. Cmrige University Press, Cmrige, UK (1988) 3. Griellini, S., Torroni, P.: MS Dilogues: Persuing n getting persue, A moel of soil network etes tht reoniles rguments n trust. (2013) 4. Alhourrón, C.E., Gärenfors, P., Mkinson, D.: On the logi of theory hnge : Prtil meet ontrtion n revision funtions. Journl of Symoli Logi 50 (1985) Ktsuno, H., Menelzon, A.O.: Propositionl knowlege se revision n miniml hnge. Artifiil Intelligene 52 (1991) Qi, G., Liu, W., Bell, D.A.: Knowlege se revision in esription logis. In: Proeeings of the Europen Conferene on Logis in Artifiil Intelligene (JELIA 06). Volume 4160 of Leture Notes in Computer Siene., Springer (2006) Cmin, M.: On the issue of reinsttement in rgumenttion. In: Proeeings of the Europen Conferene on Logis in Artifiil Intelligene (JELIA 06), Springer (2006) Coste-Mrquis, S., Devre, C., Koniezny, S., Lgsquie-Shiex, M.C., Mrquis, P.: On the merging of Dung s rgumenttion systems. Artifiil Intelligene Journl 171 (2007) Boell, G., Ki, S., vn er Torre, L.: Dynmis in rgumenttion with single extensions: ttk refinement n the groune extension. In: Proeeings of the Interntionl Conferene on Autonomous Agents n Multigents Systems (AA- MAS 09). (2009) Boell, G., Ki, S., vn er Torre, L.: Dynmis in rgumenttion with single extensions: Astrtion priniples n the groune extension. In: Proeeings of the Europen Conferenes on Symoli n Quntittive Approhes to Resoning with Unertinty (ECSQARU 09). Volume 5590 of Leture Notes in Computer Siene., Springer (2009) Cyrol, C., e Sint-Cyr, F.D., Lgsquie-Shiex, M.C.: Chnge in strt rgumenttion frmeworks: Aing n rgument. Journl of Artifiil Intelligene Reserh 38 (2010) Bisquert, P., Cyrol, C., e Sint-Cyr, F.D., Lgsquie-Shiex, M.C.: Chnge in rgumenttion systems: Exploring the interest of removing n rgument. In: Proeeings of the Interntionl Conferene on Slle Unertinty Mngement (SUM 11). Volume 6929 of Leture Notes in Computer Siene., Springer (2011) Bumnn, R.: Wht oes it tke to enfore n rgument? miniml hnge in strt rgumenttion. In: Proeeings of the Europen Conferene on Artifiil Intelligene (ECAI 12). (2012) Besnr, P., Doutre, S.: Cheking the eptility of set of rguments. In: Proeeings of the 10th Interntionl Workshop on Non-Monotoni Resoning (NMR 04). (2004) 59 64