Fuzzy argumentation system for decision support

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1 Fuzzy argumentation system for decision support Nouredine Tamani, Madalina Croitoru INRIA GraphIK, LIRMM, 161 rue Ada, Montpellier, France. Abstract. We introduce in this paper a quantitative preference based argumentation system relying on ASPIC argumentation framework and fuzzy set theory. The knowledge base is fuzzified to allow the experts to express their expertise (premises and rules) attached with grades of importance in the unit interval. Arguments are attached with a score aggregating the importance expressed on their premises and rules. Extensions are then computed and the strength of each of which can also be obtained based on its strong arguments. The strengths are used to rank fuzzy extensions from the strongest to the weakest one, upon which decisions can be made. The approach is finally used for decision making in a real world application within the EcoBioCap project. 1 Introduction Logically instantiated argumentation system (such as ASPIC/ASPIC+ framework [1, 2]) can be used to reason with arguments to provide means (i) to express argument as a combination of premises and inference rules, (ii) to define contradictions, attacks and defeat between arguments and (iii) to extract consistent subsets of arguments called extensions that also defend themselves against attacks. In practice, the decision-maker has often to deal with several extensions leading to conflicting decisions. In this context the argumentation process is not enough, since it cannot say what is the best extension to consider to make a decision. To address this area, several approaches for defining preference-based argumentation systems have been proposed during the last years [2 4]. These approaches extend the Dung abstract argumentation model by introducing a preorder or a total order on arguments through a preference relation, which states for each couple (or even for two subsets) of arguments either they are incomparable or which is the most preferred. In a logically instantiated argumentation framework ([2 4]), this preference order is used, among other, for a qualitative ranking of extensions. However, in real world application, the user may be unable to specify all the relative importance of arguments, because in the worst case complexity of computation of the ranking is exponential. Contribution. We introduce in this paper a quantitative approach to define a total order between extensions in logically instantiated argumentation framework. We propose to model the premises and rules as fuzzy sets such that the grade of membership of a premise (resp. a rule) expresses its importance from expert standpoints. The notions of strictness and defeasibility are then unified

2 2 Nouredine Tamani, Madalina Croitoru under the notion of importance. The rules of the system are then more expressive in the sense that the closer to 1 the importance of a premise (resp. a rule) is, the more strict it is, and conversely the closer to 0 a premise (resp. a rule) is, the more defeasible it is. Then, arguments are attached with a strength score based on the degrees of membership of the involved premises and rules, which can be computed with a t-norm. Based on the argument strengths, we can compute a force of each extension E, by using the decomposition based approach [5], corresponding to the truth value of the fuzzy quantified proposition almost all arguments in E are strong. We obtain a total order between extensions used to rank them from the strongest to the weakest one in a polynomial time. The proposed approach is finally applied within the EcoBioCap project to show its usefulness in decision making for packaging selection. In section 2, we recall the principles of argumentation framework. In section 3, the rationale behind the fuzzy ASPIC argumentation system is introduced. Its application in a real world application is detailed in Section 4. Section 5 summarizes related work in the field and finally Section 6 concludes the paper. 2 Argumentation framework 2.1 Dung argumentation principles Dung argumentation framework (AF ) [6] is a tuple (A, C), where C A A is a binary attack relation on the set of arguments A, having a meaning of defeat. For each argument X A, X is acceptable w.r.t. a set of arguments S A iff any argument attacking X, is attacked by an argument of S. A set of arguments S A is conflict free iff X, Y S, (X, Y ) / C. For any conflict free set of arguments S, S is an admissible extension iff X S implies X is acceptable w.r.t. S; S is a complete extension iff X S whenever X is acceptable w.r.t. S; S is a preferred extension iff it is a set inclusion maximal complete extension; S is the grounded extension iff it is the set inclusion minimal complete extension; S is a stable extension iff it is preferred and Y / S, X S such that (X, Y ) C. For T {complete, preferred, grounded, stable}, X is skeptically (resp. credulously) justified under the T semantics if X belongs to all (resp. at least one) T extension. Output of an extension E is Output(E) = {Conc(A), A E}, where Conc(A) is the conclusion of argument A. The skeptical output of AF is Output(AF ) = i=1,...,n Output(E i) such that E i are its T extensions. 2.2 ASPIC argumentation system (ASPIC AS) In this paper we consider ASPIC AS as a subset of ASPIC+ [7] and compatible with the one presented in [8]. An ASPIC AS is a tuple (L, cf, R, ), where L is a logical language, cf is a contrariness function which associates to each formula of L a set of its incompatible formulas (in 2 L ), R = R s R d is the set of strict (R s ) and defeasible (R d ) rules of the form ϕ 1,..., ϕ m ϕ and ϕ 1,..., ϕ m ϕ respectively, where ϕ i,i=1,...,m, ϕ are well-formed formulas in L, and R s R d =, and is a preference ordering over defeasible rules.

3 Fuzzy argumentation system for decision support 3 A knowledge base in AS = (L, cf, R, ) is K L such that K = K a K p and K a K p =, K a contains axioms and K p contains ordinary premises. 3 Fuzzy ASPIC argumentation system (F-ASPIC AS) 3.1 Fuzzy Set Theory Fuzzy set theory introduced by Zadeh [9] to express the gradual membership of an element to a set. Formally, a fuzzy set F is defined on a referential U by a membership function µ F : U [0, 1] such that µ F (x) denotes the membership grade of x in F. In particular, µ F (x) = 1 denotes the full membership of x in F, µ F (x) = 0 expresses the absolute non-membership and when 0 < µ F (x) < 1, it reflects a partial membership (the closer to 1 µ F (x), the more x belongs to F ). The core of F is Core(F ) = {x F : µ F (x) = 1} and its support is Support(F ) = {x F : µ F (x) > 0}. If a fuzzy set is a discrete set then it is denoted F = {(x 1, µ F (x 1 )),..., (x n, µ F (x n ))}, otherwise, it is characterized by its membership function, in practice often a trapezoidal function. The union and the intersection operators are defined by a couple of a t-norm and a t-conorm, such as (min, max). Let F, G be two fuzzy sets, µ F G (x) = max(µ F (x), µ G (x)), µ F G (x) = min(µ F (x), µ G (x)), and the complement of F, denoted F c, is µ F c(x) = 1 µ F (x). The logical counterparts of, and the complement are respectively, and. Other operators have also been defined such as fuzzy implications [10]. 3.2 Fuzzy argumentation system: F-ASPIC A fuzzy argumentation system F AS is equipped with a fuzzy membership function imp expressing for each premise and rule its importance. It is worth noticing that the rules do not model fuzzy implications but regular material implications attached with a score expressing a preference order between them reflecting the importance given by the experts. A fuzzy ASPIC argumentation system is a F AS = (L, cf, R, imp) such that L is a logical language, cf is a contrariness function (we consider the negation as its basic form), R is the fuzzy set of important rules of the form (φ 1,..., φ m φ, s) with φ i,i=1,...,m L are the premises of the rule, φ L is its conclusion and s [0, 1] is its importance, provided by the experts of the domain. For a given rule r, if µ imp (r) = 1 then r is a strict rule, if µ imp (r) = 0 then r is an insignificant rule, discarded by the system. If µ imp (r) ]0, 1[ then the closer to 1 µ imp (r) is, the more important r is, and conversely the closer to 0 µ imp (r) is, the more defeasible r is. In the same way, a knowledge base K in a F AS is a fuzzy set of important premises. For a given premise p, if µ imp (p) = 1 then it is an axiom, if µ imp (p) = 0 then p is insignificant (then discarded by the system), and If µ imp (p) ]0, 1[ then the closer to 1 µ imp (p) is the more important p is.

4 4 Nouredine Tamani, Madalina Croitoru F-ASPIC arguments. An F-ASPIC argument A can be of the following forms: 1. s c, with c K, P rem(a) = {c}, Conc(A) = c, Sub(A) = {c}, Rules(A) =, where P rem returns premises of A, Conc returns its conclusion, Sub returns its the sub-arguments, and Rules returns the rules involved in A, s ]0, 1] expresses the strength of A (defined below), 2. A 1,..., A m s c, such that there exists a rule r R of the form (Conc(A 1 ),..., Conc(A m ) c, s r ), and P rem(a) = P rem(a 1 )... P rem(a m ), Conc(A) = c, Sub(A) = Sub(A 1 )... Sub(A m ) {A}, Rules(A) = Rules(A 1 )... Rules(A m ) {r}, s ]0, 1] expresses the strength of A (defined below). Remark 1. An ASPIC argument has a nested form. A sub-argument is also an argument. To improve the readability, by abuse of notation, we associate to each argument a label made of a capital letter followed by a subscript number. The labels are then used in an argument to refer to its sub-arguments. In this notation, a label followed by colon is not a part of the argument. We make use of the same notation in F-ASPIC. Example 1. Let AS be an ASPIC argumentation system defining the rules R s = {a, b c} and the ordinary premises K p = {a, b}. The following are arguments in AS: A 1 : a A 2 : b A 3 : A 1, A 2 c. Definition 1 (Strength of an argument). The strength of argument A, denoted str(a) { computed efficiently as follows: µ imp (c), c K if A is of the form c, str(a) = T e Rules(A) P rem(a) (µ imp (e)) if A is of the form A 1,..., A m c, where T is a triangular norm such as min,, etc. Definition 2 (Strict/defeasible argument). An argument A is said strict iff str(a) = 1. Otherwise, it is called defeasible. Definition 3 (Consistency of R). Let A be the set of arguments of an F- ASPIC AS, the set of rule R is said consistent iff A, B A, such that str(a) = str(b) = 1 and Conc(A) = Conc(B). 3.3 Attacks and defeat between arguments We define the fuzzy rebut and fuzzy undercut attacks, denoted F-rebut and F-undercut respectively. Definition 4 (F-Rebut attack). An argument A rebuts argument B if A i Sub(A) and B j Sub(B) such that (i) Conc(A i ) = Conc(B j ), (ii) str(b j ) < 1 and (iii) str(a i ) str(b j ). Definition 5 (F-undercut attack). Argument A undercuts argument B iff B Sub(B) of the form B 1,..., B m s ϕ, with s < 1 and A Sub(A) : Conc(A ) = Conc(B 1),..., Conc(B m) s ϕ, with str(a ) s, and operator. converts a defeasible rule into a literal. Definition 6 (F-Defeat). Argument A defeats arguments B iff A F-rebuts or F-undercuts B.

5 Fuzzy argumentation system for decision support F-ASPIC AS extensions and rationality postulates Rationality postulates, defined in [11], ensure the completeness and the consistency of the output of a logical-based argumentation system. We consider here the following rationality postulates: 1. Closure under sub-arguments: for every argument in an extension, also all its sub-arguments are in the extension, 2. Closure under strict rules R s = {r R µ imp (r) = 1} of the output of an extension: all possible conclusions from applicable strict rules are derived in each extension, 3. Direct consistency: the output of each extension is consistent, so it is not allowed to derive a conclusion and its contradiction in an extension, 4. Indirect consistency: the closure under strict rules of the output of each extension is consistent. Proposition 1. F-ASPIC AS is closed under sub-arguments and under strict rules, direct and indirect consistent, iff the set of strict rules R s is closed under transposition [11] and R s is consistent. The proof of proposition 1 is detailed in the technical report [12] (pages 5-7). Proposition 1 shows that extensions can be built upon our defeat relation without falling in the inconsistency described in [3, 4, 2] when preferences are involved in the defeat relation and not in the attacks. So, our logical-based argumentation system is then compatible with Dung abstract model for argumentation. 3.5 Ordering extensions Several aggregating operators can be used (min, max, weighted mean, etc.) to compute the force of an extension combining the strength of its arguments. Although they are simple to implement they can lead to non-justifiable results, as in the case of max operator, for instance, an extension containing only strict arguments gets the same force as an extension having only one strict argument and numerous very weak arguments. A similar result can be obtained from a weighted mean operator. So, we need a hybrid operator able to deliver a force taking into account the strength of arguments and their number. Therefore, we define a force of an extension based on fuzzy quantified proposition as follows. Definition 7 (Force of an extension). The force of an extension E (denoted f orce(e)), under one Dung s semantics, is the truth value of the fuzzy quantified proposition P : almost all arguments in E are strong, denoted δ P. δ P can be computed efficiently in a polynomial time by the decomposition based approach [5]. So arguments in E are ranked from the most to the least strong: str(a 1 ) str(a 2 )... str(a n ) with n = E, and δ P = max i=1,...,n (min(str(a i ), µ almost all ( i )) (1) n

6 6 Nouredine Tamani, Madalina Croitoru Definition 8 (Ordering relation between extensions). Let E 1,..., E 2 be extensions, under one Dung semantics, of a F-ASPIC AS. The extension E i is preferred than the extension E j, denoted E i e E j iff force(e i ) force(e j ). Its strict counterpart e is: E i e E j iff force(e i ) > force(e j ). 4 Application to packaging selection 4.1 Selection of packaging according the aspect end of life The following text arguments about the end of life of packagings have been collected during an interview with experts in the domain of waste management: 1. Packaging materials, which are biodegradable, compostable, or recyclable having a low environmental impact are preferred, 2. Life Cycle Analysis (LCA) results are not in favor of biodegradable and compostable materials, 3. Consumers are in favor of biodegradable material because they help to protect the environment, 4. Biodegradable materials could encourage people to throw their packaging in nature, causing visual pollution. We model these arguments by a simple propositional language as follows: BP, CP and RP are symbols referring respectively to biodegradable, compostable and recyclable packagings, P EV, V P, HIP, LIP are symbols referring to packagings which respectively protect the environment, can cause visual pollution problem, have a high or low environmental impact (according to LCA), ACC, REJ are symbols referring to the decisions (accepted, rejected). The fuzzy set of important rules contains the following weighted implications (membership degrees are provided by the domain experts): R = {(BP HIP, 0.9), (CP HIP, 0.9), (RP LIP, 0.9), (BP P EV, 0.4), (BP V P, 0.8), (CP V P, 0.8), (HIP REJ, 0.9), (HIP ACC, 0.1), (LIP ACC, 0.9), (LIP REJ, 0.1), (V P REJ, 0.8), (V P ACC, 0.2), (P EV ACC, 0.8), (P EV REJ, 0.2)}. The premises of the system are defined by the fuzzy set corresponding to the important packaging choices K p = {(BP, 0.9), (CP, 0.9), (RP, 0.9)}, having the equal importance 0.9. The arguments derived from K p and R are the following (the strength of arguments are computed by the t-norm min): A 0 : 0.9 BP B 0 : 0.9 CP C 0 : 0.9 RP A 1 : A HIP A 2 : A REJ A 3 : A ACC A 4 : A P EV A 5 : A ACC A 6 : A REJ A 7 : A V P A 8 : A REJ A 9 : A ACC B 1 : B HIP B 2 : B REJ B 3 : B ACC B 4 : B V P B 5 : B REJ B 6 : B ACC C 1 : C LIP C 2 : C ACC C 3 : C REJ

7 ! Fuzzy argumentation system for decision support 7 μ almost_all(q)$ 1$ μ almost_all(q)$=$4q$4$2$ 0.5$ 0.75$ 1$ q$ Fig. 1. Example of definition of the quantifier almost all. We can compute the following two fuzzy preferred and stable extensions: E 1 = {(A 2, 0.9), (A 8, 0.8), (A 6, 0.2), (B 5, 0.8), (B 2, 0.9), (C 3, 0.1), (B 0, 0.9), (B 1, 0.9), (A 0, 0.9), (A 1, 0.9), (A 4, 0.4), (A 7, 0.8), (B 4, 0.8), (C 0, 0.9), (C 1, 0.9)}, and Output(E 1 )={REJ,CP,BP,RP,LIP,PEV,HIP,VP}. E 2 = {(A 3, 0.8), (A 5, 0.4), (B 6, 0.2), (C 2, 0.9), (B 3, 0.1), (B 0, 0.9), (B 1, 0.9), (A 0, 0.9), (A 1, 0.9), (A 4, 0.4), (A 7, 0.8), (B 4, 0.8), (C 0, 0.9), (C 1, 0.9)}, and Output(E 2 )={ACC,CP,BP,RP,LIP,PEV,HIP,VP}. Based on formula (1) and subjective definition of the fuzzy quantifier almost all (Figure 1), the force of each extension is δ E1 = 0.8 and δ E2 = 0.57, which means that arguments for rejection are stronger than arguments for acceptance. We can here make a decision but it is not justifiable because the rejection of the recyclable packagings is counter-intuitive, since they are supported by strong pros arguments and no arguments are against them. Independently of the aggregation operator used to compute the force of an extension, the strength of arguments against one type of packaging has an indirect influence on the rejection of the others. To fix the system, we split arguments according to the type of packaging on which they are expressed. So, arguments A i,i=0,...,9, B j,j=0,...,6 and C k,k=0,...,3 are about the acceptance or the rejection of respectively biodegradable (denoted Bio), compostable (denoted Com) and recyclable (denoted Rec) packagings. Then, we draw three argumentation graphs on which we compute preferred and stable extensions, and we obtain: E1 Bio = {(A 0, 0.9), (A 1, 0.9), (A 2, 0.9), (A 7, 0.8), (A 8, 0.8), (A 4, 0.4), (A 6, 0.2)}, then Output(E1 Bio )= {REJ, BP, HIP, PEV, VP} resulting to the rejection of biodegradable packagings, E1 Com = {(B 0, 0.9), (B 1, 0.9), (B 2, 0.9), (B 4, 0.8), (B 5, 0.8)}, so Output(E1 Com )= {REJ, CP, HIP, VP} leading to the rejection of compostable packagings, E1 Rec = {(C 0, 0.9), (C 1, 0.9), (C 2, 0.9)}, and Output(E1 Rec )={ACC, RP, LIP}, supporting the acceptance of recyclable packagings. This way to deal with arguments allows to make decisions about each packaging choice based on the force of the delivered extensions. So, δ E Bio = 0.5 (average 1 rejection), δ E Com = 0.8 (strong rejection) and δ 1 E Rec = 0.9 (strong acceptance). 1 It is worth noticing that in classical argumentation approach, the user gets the same extensions but no means to compare between them. Using the delivered extensions in the querying process. From the obtained extensions, we know the reasons why recyclable packagings are recom-

8 8 Nouredine Tamani, Madalina Croitoru mended and why biodegradable and compostable packagings are not. This result can be used in the querying process by adding to the query the predicate recycle = true to retrieve the recyclable packaging materials from the database. 5 Related work Comparison with preference-based ASPIC+ [2]. The approach of [2] relies on a preference-based Structured Argumentation Framework (SAF) built upon ASPIC+. An argumentation theory AT = (AS, K) is then a triple A, C, where A is the set of all finite arguments of AS, is an ordering on A, and (X, Y ) C iff X attacks Y. The instantiation of this SAF needs the definition of the operator, which is tricky from the application point of view, since must satisfy several conditions to be called reasonable [2]. Moreover, ASPIC+ requires the definition of two negations (the contrary and the contradiction), which is not intuitive for the experts providing the arguments. Comparison with fuzzy argumentation systems. Several fuzzy-based argumentation approaches have been proposed during the last decade. In [14 16] possibilistic defeasible logic programming [17] approaches have been developed. They suffer from the difficulties related to the definition of the necessity and possibility measures, since each argument is attached with a necessity degree, and possibilities are attached to the different models of the logical language. In the same context, a possibilistic label-based argumentation approach has been also proposed in [18, 19], expressing the reliability of argument sources. Then, arguments of a single source have a same reliability degree. In [20], a fuzzy argumentation system for trust has been proposed. The approach is similar to ours in the spirit but the context is quite different since in this approach arguments are not structured. In [21, 22] a fuzzy attack relation is defined to express the strength of attacks between arguments. This approach is also different than ours since we make use of crisp attacks. In [21] the strength of attacks depends on both fuzzy set of arguments supporting the attacker and the strength of the attack. In [22], attack and defeat relations involve a binary preference relation over arguments, in such a way that non-symmetric attacks can fail, resulting to non-intuitive extensions. [23] introduces an argumentation approach based on fuzzy description logic (fuzzy SHIF DL). Arguments are a combination of fuzzy linguistic variables and ontological knowledge, and involved in fuzzy attack and support relations. A preference relation is also used. This approach requires users to manually specify the attack and support relations between arguments. 6 Conclusion We have introduced a quantitative preference-based argumentation system relying on the fuzzy sets theory and ASPIC structured argumentation framework. Arguments are built upon the fuzzy set of important rules and premises, allowing the computation of their strength. Extensions are attached with a score in

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