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Encoding Argumentation Semantics by Boolean Algebra

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Encoding Argumentation Semantics by Boolean Algebra IEICE TRANS. INF. & SYST., VOL.E100–D, NO.4 APRIL 2017 838 PAPER Encoding Argumentation Semantics by Boolean Algebra Fuan PUya), Guiming LUOy, Nonmembers, and Zhou JIANGy, Student Member SUMMARY In this paper, a Boolean algebra approach is proposed to tations and some interesting properties, each semantics can encode various acceptability semantics for abstract argumentation frame- be equivalently encoded into more than one Boolean con- works, where each semantics can be equivalently encoded into several straint model. Then, finding the extensions of a semantics Boolean constraint models based on Boolean matrices and a family of Boolean operations between them. Then, we show that these models can is equivalently to finding the Boolean vectors that satisfy all be easily translated into logic programs, and can be solved by a constraint constraints in one of its Boolean constraint models. To solve solver over Boolean variables. In addition, we propose some querying these constraint models, in this paper, we employ a Con- strategies to accelerate the calculation of the grounded, stable and complete straint Logic Programming over Boolean variables (CLPB) extensions. Finally, we describe an experimental study on the performance of our encodings according to different semantics and querying strategies. system, which is an algebraically oriented Constraint Pro- key words: argumentation framework, acceptability semantics, Boolean gramming solver and has abilities to handle any Boolean algebra, Boolean constraint solver, logic programming, querying strategies expressions with little conversation. In addition, we propose some querying strategies to accelerate the calculation of sta- 1. Introduction ble and complete semantics by initializing some Boolean variables based on the grounded extension, which is op- Argumentation is a reasoning model based on construct- tionally calculated by iterating the Boolean-algebra-based ing and comparing arguments about conflicting options and characteristic function using bit-vector operations. At last, opinions. The most popularly used framework to talk about we describe an experimental study on the performance of ff general issues of argumentation is that of Dung’s abstract ar- the proposed encodings according to di erent semantics and gumentation frameworks (AFs) [1], which consists of a set querying strategies. of arguments and a binary relation that represents the con- The rest of this paper is organized as follows. Section flicting arguments. It offers a series of extension-based se- 2 presents some basic concepts about AF. Section 3 intro- mantics for solving inconsistent knowledge by selecting ac- duces a Boolean algebra approach to represent these basic ceptable subsets. Intuitively, an acceptable set of arguments concepts. Section 4 encodes Dung’s semantics into Boolean must be in some sense coherent (no attacks between its ar- constraint models, which are translated into logic programs guments, i.e., conflict-free) and strong enough (e.g., able to based on CLPB in Sect. 5. Section 6 gives some experimen- defend itself against all attacking arguments, i.e., admissi- tal results, and Sect. 7 concludes. ble). Based on these, various semantics have been estab- lished, such as grounded, complete, and stable (see [2] for 2. Abstract Argumentation Framework an overview). However, finding extensions can be a com- plex procedure when done without any computational help, In this section, we briefly outline key elements of Dung’s when the AF contains numerous arguments and attacks [3]. abstract argumentation frameworks [1]: The main goal of this paper is to demonstrate how Definition 1. An abstract argumentation framework (AF) is Boolean algebra can be used for representing and solving a tuple ∆ def= hX; Ri where X is a finite set of arguments and such complex problems in the research of argumentation. R ⊆ X × X is a binary relation, called attack relation. We propose to represent a subset of arguments by a Boolean For two arguments x; y 2 X,(x; y) 2 R means that vector, attack relations by a Boolean matrix. Moreover, we x attacks y, or x is an attacker of y. Often, we write introduce a series of Boolean operations, such as element- (x; y) 2 R as xRy. An AF can be represented by a di- wise operations on Boolean vectors, and the Boolean ma- rected graph, whose nodes are arguments and edges rep- trix multiplication operation, which can be used to repre- resent attack relations. We denote by R−(x) (respectively, sent the neutrality, innocuousity and characteristic functions R+(x)) the subset of X containing those arguments that at- of AFs by Boolean functions, three cornerstones of Dung’s tack (respectively, are attacked by) x 2 X, extending this acceptability semantics. We show that with these represen- notation in the natural way to sets of arguments, so that Manuscript received July 24, 2016. for S ⊆ X, R−(S ) def= fx 2 X : 9y 2 S such that xRyg and Manuscript revised December 19, 2016. R+(S ) def= fx 2 X : 9y 2 S such that yRxg. Manuscript publicized January 18, 2017. yThe authors are with the Tsinghua University, Beijing, China. The justified arguments are evaluated based on subsets a) E-mail: [email protected] (Corresponding author) of X (called extensions) defined under a range of seman- DOI: 10.1587/transinf.2016EDP7313 tics. The arguments in an extension are required to not at- Copyright ⃝c 2017 The Institute of Electronics, Information and Communication Engineers PU et al.: ENCODING ARGUMENTATION SEMANTICS BY BOOLEAN ALGEBRA 839 tack each other (extensions are conflict-free), and attack any we denote the collection of all σ-extensions of ∆ by Eσ(∆). argument that in turn attacks an argument in the extension A stable extension is a maximal (w.r.t. ⊆) conflict-free (extensions reinstate or defend their contained arguments). extension. The existence of stable extensions is not guar- Now, let us formally characterise these two fundamental no- anteed. If there exists a stable extension, it must be a com- tions of conflict-free and defence as below: plete extension (but not vice versa), and thus it is admissible. Definition 2. Let ∆ = hX; Ri be an AF, S ⊆ X and x 2 X. Each admissible set is contained in a complete extension. • S is conflict-free iff @x; y 2 S such that xRy (i.e., S \ Every AF possesses at least one complete extension. The R+(S ) = ;, or, equivalently, S \R−(S ) = ;). grounded extension is conflict-free and unique, and always • S defends x iff 8y 2 X if yRx then 9z 2 S such that zRy, exists, for proofs presented in [1, Thm. 25] . It can be com- (i.e., R−(x) ⊆ R+(S )). puted by iteratively applying the characteristic function F Dung introduces a function for an AF, called charac- from the empty set until we reach a fixed point. We summa- teristic function. It applies to some set S ⊆ X and returns rize some properties of these semantics from [1], [2], [5] as all arguments that are defended by S . below: hX; Ri ; ⊆ X Definition 3. Let ∆ = hX; Ri be an AF. The characteristic Proposition 2. Let be an AF, and S E , then, X X ff ⊆ N ∆ F∆ 7! ⊆ X (i) S is an admissible extension i S (S ) and function of is : 2 2 such that, given S , − + def R (S ) ⊆ R (S ). F∆(S ) = fx 2 X : S defends xg. (ii) S is an admissible extension iff S ⊆ F (S ) \N(S ). 2 X ⊆ X An argument x is not attacked by a set S if no (iii) S is an admissible extension iff S ⊆ F (S \N(S )). @ 2 R argument in S attacks x (i.e., y S such that y x, alterna- (iv) S is a complete extension iff S = F (S ) \N(S ). < R+ tively, x (S )). Then, we can define a function, which is (v) S is a complete extension iff S = F (S \N(S )). ⊆ X applied to some set S , returns all arguments that are not (vi) If S is the (unique) grounded extension, then S is in- attacked by S . This function, called neutrality function, was cluded in each complete extensions, i.e., introduced in [4]. Similarly, we also define another func- innocuousity function tion, called , which is applied to some S ⊆ E; 8E 2 ECO(∆) (1) set S ⊆ X and returns all arguments that do not attack S . Definition 4. Let ∆ = hX; Ri be an AF. The neutrality func- and arguments that attacked by S are not included in X X tion of ∆ is N∆ : 2 7! 2 such that, given S ⊆ X, any complete extension, formally, N def= f 2 X < R+ g ∆(S ) x : x (S ) . Equivalently, we can write + def + R (S ) * E; 8E 2 ECO(∆) (2) the neutrality function as N∆(S ) = R (S ), where the bar on a set means the complement of the set relative to X. The in- X X They are also hold for stable semantics if ∆ exists sta- nocuousity function of ∆ is I∆ : 2 7! 2 such that, given E ∆ , ; def − ble extensions, i.e., ST( ) . S ⊆ X, I∆(S ) = fx 2 X : x < R (S )g, or alternatively, def Note that most semantics are multi-extension seman- I = R− ∆(S ) (S ). tics. That is, there is not always an unique extension in- From now on, we will omit the subscript ∆ from F∆, duced by the semantics. In order to reason with multi- N∆ and I∆ if there is no danger of ambiguity. extension semantics, usually, one takes either a credulous Proposition 1. Let ∆ = hX; Ri be an AF. For any S ⊆ X, or skeptical perspective, i.e., an argument x 2 X is cred- the following holds: (i) S is conflict-free iff S ⊆ N(S ); (ii) S ulously (respectively, skeptically) justified under some se- is conflict-free iff S ⊆ I(S ); (iii) F (S ) = N(N(S )).
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