'! Prudent Semantic for Argumentation ramework Sylvie Cote-Marqui, Caroline Devred and Pierre Marqui CRL/CNRS Univerité d Artoi Len RANCE cote,devred,marqui @cril.univ-artoi.fr Abtract We preent new prudent emantic within Dung theory of argumentation. Under uch prudent emantic, two argument cannot belong to the ame extenion whenever one of them attack indirectly the other one. We argue that our emantic lead to a better handling of controverial argument than Dung one. We compare the prudent inference relation induced by our emantic w.r.t. cautioune; we alo compare them with the inference relation induced by Dung emantic. 1 ntroduction Argumentation i a general approach to model defeaible reaoning, in which the two main iue are the generation of argument and their exploitation o a to draw ome concluion baed on the way argument interact (ee e.g., [17, 15]). Several theorie of argumentation have been propoed o far (ee among other [10, 14, 16, 12, 4, 1, 7]). Among them i Dung theory 1 [10], which i quite influential ince it encompae many approache to nonmonotonic reaoning and logic programming a pecial cae. n Dung approach, no aumption i made about the nature of an argument (it can be a tatement upported by ome aumption like in the theory introduced by Elvang-Gøranon et al. [13] but thi i not mandatory). What really mean i the way argument interact w.r.t. the attack relation. Argument and the way they interact w.r.t. the attack relation are conidered a initial data of any argumentation framework, which can thu be viewed a a labeled digraph. Several inference relation can be defined within Dung theory. Uually, inference i defined at the argument level: an argument i conidered derivable from an argumentation framework when it belong to one (credulou accept- The author have been partly upported by the UT de Len, the Région Nord/Pa-de-Calai through the RCCA Conortium and by the European Community EDER Program. 1 Refined and extended by everal author, including [3, 2]. ability) (rep. all (keptical acceptability)) extenion() of under ome emantic, where an extenion of i a conflict-free and elf-defending et of argument, maximal for a given criterion (made precie by the emantic under conideration). While keptical acceptability can be afely extended to the level of et of argument, thi i not the cae for credulou acceptability. ndeed, it can be the cae that both argument and are (individually) derivable from while the et i not included into any extenion of. Now, defining acceptability for et of argument a incluion into ome (rep. all) extenion under Dung emantic doe not alway lead to expected concluion. Conider Example 1: Example 1 Let with!"#$&%' and ()+* -,./*!" -,0/*1%!2,.*1' 2,.*#$!2,0/*1' #,.. The digraph for i depicted on the following figure. # % On thi example, whatever the emantic and the inference relation (keptical or credulou) among Dung one, i derivable from. However, i attacked by and the unique defeater of i controverial w.r.t. (an argument ' i aid to be controverial w.r.t. an argument if ' indirectly attack and ' indirectly defend ). So it i not cautiou to infer. One way to cope with thi problem i to uppre indirect conflict 2 from the different extenion (under all emantic). orbidding indirect conflict in extenion doe not prevent all argument which are indirectly attacked from belonging to ome extenion. Thi only prevent pair of 2 There i an indirect conflict between two argument when one of them attack indirectly the other one.
argument which conflict indirectly from belonging to the ame extenion. n our opinion, uch a cautiou approach i enible whenever we want to infer et of argument: infering together two argument which conflict indirectly i not prudent. Thi problem i not avoided by Dung emantic. Thu, ha a ingle extenion /'&% 3 whatever the emantic among Dung one. Conequently, and ' are alway jointly derivable from while ' i controverial w.r.t.. n thi paper, we define and tudy new emantic for Dung framework, baed on a more demanding notion of abence of conflict, ince indirect attack are not allowed within a prudent, admiible et. Epecially, a prudent, admiible et never include pair of controverial argument. We compare the inference relation induced by uch new emantic with Dung one and how that in many cae one obtain more cautiou notion of derivability. 2 Dung Theory of Argumentation Let u preent ome baic definition at work in Dung theory of argumentation [10]. We retrict them to finite argumentation framework. Definition 1 (argumentation framework) A finite argumentation framework i a pair 45 6 where i a finite et of o-called argument and i a binary relation over (a ubet of 879 ), the attack relation. A firt important notion i the notion of acceptability: an argument i acceptable w.r.t. a et of argument whenever it i defended by the et, i.e., every argument which attack i attacked by an element of the et. Definition 2 (acceptable et) Let :;5 6 be a finite argumentation framework. An argument =<> i acceptable w.r.t. a ubet? of if and only if for every <@.t. * " -,<A, there exit!b<c?.t. *!" 0,<A. A et of argument i acceptable w.r.t.? when each of it element i acceptable w.r.t.?. A econd important notion i the notion of abence of conflict. ntuitively, two argument hould not be conidered together whenever one of them attack the other one. Definition 3 (conflict-free et) Let DE5 be a finite argumentation framework. A ubet? of i conflictfree if and only if for every <?, we have *5 0, <@. Requiring the abence of conflict and the form of autonomy captured by elf-acceptability lead to the notion of admiible et. Definition 4 (admiible et) Let :; be a finite argumentation framework. A ubet? of i admiible for if and only if? i conflict-free and acceptable w.r.t.?. The ignificance of the concept of admiible et i reflected by the fact that every extenion of an argumentation framework under the tandard emantic introduced by Dung (preferred, table, complete and grounded extenion) i an admiible et, atifying ome form of optimality: Definition 5 (extenion) Let GH5 6 be a finite argumentation framework. A ubet? of i a preferred extenion of if and only if it i maximal w.r.t. among the et of admiible et for. A ubet? of i a table extenion of if and only if it i conflict-free for and for every argument from CKL?, there exit <?.t. * " -,M<@. A ubet? of i a complete extenion of if and only if it i admiible for and it coincide with the et of argument acceptable w.r.t. itelf. A ubet? of i the grounded extenion of if and only if it i the leat element w.r.t. among the complete extenion of. Example 1 (cont ed) Let NO8 '&%P. N i the grounded extenion of Q, the unique preferred extenion of R, the unique table extenion of Q and the unique complete extenion of. ormally, the complete extenion of can be characterized a the fixed point of it characteritic function S TVU : Definition 6 (characteritic function) The characteritic function, denoted S TVU, of an argumentation framework WO X i defined a follow: S TVUZY$[ T]\ [ T S TVU *?Q,P^ `_" i acceptable w.r.t.?m. Among the fixed point of S TVU, the grounded extenion of i the leat element w.r.t. [10]. inally, everal notion of derivability of an argument (or more generally a et of argument) from an argumentation framework can be defined by requiring that the (et of) argument() i included into an extenion (credulou acceptability) or every extenion (keptical acceptability) of of a pecific kind. Obviouly enough, credulou derivability and keptical derivability w.r.t. the grounded extenion coincide, ince there cannot be more than one grounded extenion for any argumentation framework.
3 Prudent Extenion n order to addre cenario like in Example 1 in a more atifying way, we need to refine Dung notion of admiibility, by requiring that no indirect conflict occur within an admiible et of argument; thi lead to the notion of p-admiible et: Definition 7 (p-admiible et) Let ab be a finite argumentation framework.?cd i p(rudent)- admiible for if and only if every <Z? i acceptable w.r.t.? and? i without indirect conflict, i.e., there i no pair of argument and of?.t. there i an odd-length path from to in. Example 1 (cont ed) '&%P and it ubet are the p- admiible et for. rom thi definition, the next lemma follow immmediately: Lemma 1 Let, be two argument of a finite argumentation framework. f i controverial w.r.t., then cannot be included into any p-admiible et for. Note that thi lemma doe not prevent or from belonging to a p-admiible et for, but not to the ame one. Actually, the abence of controverial argument i only neceary. n particular, no argument belonging to an oddlength cycle of can alo belong to a p-admiible et. Thu, our approach depart from [2] who conider that oddlength and even-length cycle in an argumentation framework hould be conidered in the ame way. Epecially, it i not cautiou to conider within a ingle extenion the argument of an odd-length cycle ince they attack themelve indirectly. urthermore, any argument from an odd-length cycle i controverial w.r.t. an argument of the cycle. On thi ground, one can define everal notion of prudent extenion, echoing Dung one. Let tart with preferred p- extenion: Definition 8 (preferred p-extenion) Let ee X be a finite argumentation framework. A p-admiible et?af for i a preferred p-extenion of if and only if gh?ij>.t.?>kl?qi and?qi i p-admiible for. Example 1 (cont ed) /'&%P i the unique preferred p- extenion of. We have the following propoition: Propoition 1 Let mh X be a finite argumentation framework. 1. The et of all p-admiible ubet of for i a complete et of * [ T 2,. 2. or every p-admible et?nn for, there exit at leat one preferred p-extenion N4D of.t.?>>n. Since o i p-admiible for any, we obtain: Corollary 1 Every finite argumentation framework WO X ha a preferred p-extenion. What can be found in preferred p-extenion? Though every argument which i not attacked belong at leat to one preferred p-extenion of, it i not the cae in general that it belong to every preferred p-extenion of. On the other hand, whenever an argument belong to a preferred p- extenion of, all it mandatory defender alo belong to it (but the condition i not ufficient): Propoition 2 Let mg5 be a finite argumentation framework. Let N be a preferred p-extenion of and ^<O. ^<mn only if all the mandatory defender of belong to N a well, where <] i a mandatory defender of iff there exit!f<d.t. *5!p -,=<D and i not defended againt! by an element of qr<z E_-q i a direct defender of and q doe not attack indirectly MK ". Example 1 (cont ed) i not derivable (even credulouly) w.r.t. our prudent emantic while it i keptically derivable w.r.t. Dung one (which coincide here). The rationale for it i a follow: while ' i a mandatory defender of, it alo attack it indirectly, i.e. ' i controverial w.r.t.. So cannot be accepted. Let u now conider the notion of table p-extenion: Definition 9 (table p-extenion) Let D( X be a finite argumentation framework. A ubet? of without indirect conflict i a table p-extenion of iff? attack (in a direct way) every argument from CKL?. Example 1 (cont ed) ha no table p-extenion. A for Dung extenion, we have: Lemma 2 Every table p-extenion of a finite argumentation framework alo i a preferred p-extenion of. The convere doe not hold. Let u now explain how p-extenion can be characterized uing ome fixed point contruction: Definition 10 (p-characteritic function) The p-characteritic function of a finite argumentation framework WO X i defined a follow: S TVU Y$[ T=t\ [ T S TVU *?Q,u v_ a i acceptable w.r.t.? and?wwh i without indirect conflict.
Contrariwie to SxTVU, S6 TVU i in general nonmonotonic w.r.t. (and thi i alo the cae of it retriction to the et of all p-admiible ubet of ). Thi prevent u from defining a notion of grounded p-extenion a the leat fixed point of S6 TVU. Neverthele: Lemma 3 Let yz5 6 be a finite argumentation framework. The equence *5S{ TVU * o},&,&~1" i monotonic w.r.t., and each element of it i a p-admiible et of. Since i finite, the equence *1S{ TVU * o},&, ~1" i tationary from ome rank, o the following definition of the grounded p-extenion of i well-founded: Definition 11 (grounded p-extenion) Let GH X be a finite argumentation framework. Let be the lowet integer uch that the equence *5S{ TVU * o},, ~5" i tationary from rank. S6"ƒ TVU *o}, i the grounded p-extenion of. Example 1 (cont ed) /'&%P i the grounded p-extenion of. Like the grounded extenion, the grounded p-extenion of an argumentation framework include the et of the element of which are not attacked. Hence: Lemma 4 Let yz5 6 be a finite argumentation framework. f i acyclic, then the grounded p-extenion of i nonempty. Thu, every finite argumentation framework ha at leat one preferred p-extenion, a unique grounded p-extenion and zero, one or many table p-extenion. Let u now introduce a notion of complete p-extenion. Definition 12 (complete p-extenion) Let X be a finite argumentation framework and let? be a p- admiible et for.? i a complete p-extenion of if and ony if every argument which i acceptable w.r.t.? and without indirect conflict with? belong to?. rom the definition, it come immediately that: Lemma 5 A et of argument? without indirect conflict i a complete p-extenion of if and only if S{ TVU *?Q,Q?. We alo have: Lemma 6 The grounded p-extenion of a finite argumentation framework i a complete p-extenion of. While the grounded extenion of an argumentation framework i included into the interection of all the complete extenion of, it i not the cae in general that the grounded p-extenion of i included into every preferred p-extenion of. Let u now define everal inference relation baed on our prudent emantic for argumentation framework: Definition 13 (prudent inference relation) _ˆX ƒ Š denote the inference relation obtained by conidering a prudent emantic (where AŒ *5Ž"# #ŽŽ"#q$,, A4? *1 -. #, or ] *1Ž" %jq #q+, ) and i an inference principle, either credulou ( Xn ) or keptical ( Xfš ). or intance,? v i a conequence of w.r.t. _ˆ œ ƒ, noted B_ˆ œ ƒ?, indicate that? i included into every preferred p-extenion of. We have compared all the prudent inference relation w.r.t. cautioune, auming that every under conideration ha a table p-extenion. 3 Lemma 7 Let be a finite argumentation framework. f ha a table p-extenion, then the interection of all preferred p-extenion of i included into the grounded p-extenion of. Baed on the previou lemmata, we have obtained the following reult: Propoition 3 The cautioune relation reported in the following table hold for every finite argumentation framework which ha a table p-extenion (Each time a cell contain a, it mean that for every (u5 and every?dž, if? i a conequence of w.r.t. the inference relation indexing the row, then? i a conequence of w.r.t. the inference relation indexing the column.) _ˆXŸ ƒ _ˆXŸ ƒ _ˆ œ ƒ _ˆXŸ ƒ _ˆ œ ƒ One can note that the cautioune picture for prudent inference relation i imilar to the one for the inference relation induced from Dung emantic (auming that the argumentation framework under conideration have table extenion): k^_ˆ œ ƒ k8_ˆxÿ ƒ k8_ˆxÿ ƒ d 4 Comparion with Dung ramework Let _ˆ ƒ Š denote the inference relation obtained by conidering Dung emantic (where ObŒ *5Ž#" #ŽŽ"#q+,,? *1 -. #", or *5Ž %jq$#q+, ) and i an inference principle, either credulou ( ) or keptical ( fš ). We have obtained the following reult: 3 When it i not the cae, both inference relation R /ª «and "ª «trivialize.
Propoition 4 The cautioune relation reported in the following table hold for every finite argumentation framework which ha a table p-extenion. _ˆ Ÿ ƒ _ˆ œ ƒ _ˆ Ÿ ƒ,,,,,,, _ˆ œ ƒ _ˆ œ ƒ,,, _ˆ ƒ n the light of the two table, one can oberve that the mot cautiou inference relation among thoe conidered here i _ˆxœ ƒ. A expected, credulou prudent inference relation are trictly more cautiou than credulou non- i trictly more cau- prudent one. More urpriingly, tiou than _ˆxœ ƒ. Before concluding the paper, let u briefly conider ome complexity iue related to our prudent inference relation. irt of all, it i eay to how that, given a finite argumentation framework, deciding whether an argument indirectly attack another argument and deciding whether a et of argument i free of indirect conflict (rep. i p- admiible for, i a table p-extenion of, i the grounded p-extenion of ) are in P. We howed in a previou paper [6] that conidering et of argument (intead of ingle argument) a input querie for the inference problem doe not lead to a complexity hift (the purpoe i to determine whether uch et are derivable from a given finite argumentation framework ). A to the prudent inference relation, it come that deciding whether a et of argument i a preferred p-extenion of or whether it i included into all table p-extenion of are in conp, and that deciding whether a et of argument i included into a preferred p-extenion (rep. a table p- extenion) of i in NP. inally, deciding whether a et of argument i included into all preferred p-extenion of i in L±. Accordingly, our prudent inference relation are not computationally more complex that the correponding one baed on Dung emantic (ee [8, 11]). 5 Concluion and Perpective We have preented new prudent emantic within Dung theory of argumentation. Under uch prudent emantic, two argument cannot belong to the ame extenion whenever one of them attack indirectly the other one. Thi lead to a better handling of controverial argument than in Dung approach. Our work call for ome perpective. One of them conit in developing pecific algorithm for computing prudent extenion, baed on algorithm for computing extenion like thoe decribed in [5, 9]. Reference [1] L. Amgoud and C. Cayrol. nferring from inconitency in preference-baed argumentation framework.. of Automated Reaoning, 29:125 169, 2002. [2] P. Baroni and M. Giacomin. Solving emantic problem with odd-length cycle in argumentation. n ECSQARU 03, volume 2711 of LNA, page 440 451, 2003. [3] P. Baroni, M. Giacomin, and G.Guida. Extending abtract argumentation ytem theory. Art. ntelligence, 120(2):251 270, 2000. [4] P. Benard and Anthony Hunter. A logic-baed theory of deductive argument. Art. ntelligence, 128:203 235, 2001. [5] C. Cayrol, S. Doutre, and. Mengin. On deciion problem related to the preferred emantic for argumentation framework.. of Logic and Computation, 13(3):377 403, 2003. [6] S. Cote-Marqui, C. Devred, and P. Marqui. Symmetric argumentation framework. n ECSQARU 05, volume 3571 of LNA, page 317 328, 2005. [7] Y. Dimopoulo, B. Nebel, and. Toni. On the computional complexity of aumption-baed argumentation for default reaoning. Art. ntelligence, 141:57 78, 2002. [8] Y. Dimopoulo and A. Torre. Graph theoretical tructure in logic program and default theorie. Theoretical Computer Science, 170:209 244, 1996. [9] S. Doutre and. Mengin. On ceptical v credulou acceptance for abtract argument ytem. n NMR 04, page 134 139, 2004. [10] P. M. Dung. On the acceptability of argument and it fundamental role in nonmonotonic reaoning, logic programming and n-peron game. Art. ntelligence, 77(2):321 358, 1995. [11] P. Dunne and T. Bench-Capon. Coherence in finite argument ytem. Art. ntelligence, 141:187 203, 2002. [12] M. Elvang-Gøranon and A. Hunter. Argumentative logic: Reaoning with claically inconitent information. Data and Knowledge Engineering, 16:125 145, 1995. [13] M. Elvang-Gøranon,. ox, and P. Kraue. Acceptability of argument a logical uncertainty. n ECSQARU 93, volume 747 of LNCS, page 85 90, 1993. [14]. Pollock. How to reaon defeaibly. Art. ntelligence, 57(1):1 42, 1992. [15] A. Prakken and G. Vreewijk. Logic for defeaible argumentation. volume 4 of Handbook of Philoophical Logic, 2²/³ edition, page 219 318. Kluwer Academic Publiher, 2002. [16] G. Simari and R. Loui. A mathematical treatment of defeaible reaoning and it implementation. Art. ntelligence, 53(2 3):125 157, 1992. [17] S. Toulmin. The Ue of Argument. Cambridge Univerity Pre, 1958.