Abduction and Dialogical Proof in Argumentation and Logic Programming
Richard Booth, Dov Gabbay, Souhila Kaci, Tjitze Rienstra, Leendert van der Torre
aa r X i v : . [ c s . A I] J u l Abduction and Dialogical Proofin Argumentation and Logic Programming
Richard Booth and Dov Gabbay , and Souhila Kaci and Tjitze Rienstra , and Leendert van der Torre Abstract.
We develop a model of abduction in abstract argumenta-tion, where changes to an argumentation framework act as hypothe-ses to explain the support of an observation. We present dialogicalproof theories for the main decision problems (i.e., finding hypothe-ses that explain skeptical/credulous support) and we show that ourmodel can be instantiated on the basis of abductive logic programs.
In the context of abstract argumentation [12], abduction can be seenas the problem of finding changes to an argumentation framework(or AF for short) with the goal of explaining observations that canbe justified by making arguments accepted. The general problem ofwhether and how an AF can be changed with the goal of changing thestatus of arguments has been studied by Baumann and Brewka [3],who called it the enforcing problem, as well as Bisquert et al. [4],Perotti et al. [5] and Kontarinis et al. [15]. None of these works,however, made any explicit link with abduction. Sakama [20], on theother hand, explicitly focused on abduction, and presented a modelin which additions as well as removals of arguments from an abstractAF act as explanations for the observation that an argument is ac-cepted or rejected.While Sakama did address computation in his framework, hismethod was based on translating abstract AFs into logic programs.Proof theories in argumentation are, however, often formulated as dialogical proof theories, which aim at relating the problem they ad-dress with stereotypical patterns found in real world dialogue. Forexample, proof theories for skeptical/credulous acceptance have beenmodelled as dialogues in which a proponent persuades an opponentto accept the necessity/possibility of an argument [17], while credu-lous acceptance has also been related to Socratic style dialogue [9].Thus, the question of how decision problems in abduction in argu-mentation can similarly be modelled as dialogues remains open.Furthermore, argumentation is often used as an abstract model fornon-monotonic reasoning formalisms. For example, an instantiated AF can be generated on the basis of a logic program. Consequencescan then be computed by looking at the extensions of the instantiatedAF [12]. In the context of abduction, one may ask whether a modelof abduction in argumentation can similarly be seen as an abstractionof abductive logic programming. Sakama, however, did not explorethe instantiation of his model, meaning that this question too remainsopen. Computer Science and Communication, University of Luxembourg([email protected], [email protected], [email protected]) Dept. Computer Science, King’s College London ([email protected]) LIRMM, University of Montpellier 2 ([email protected])
This brings us to the contribution of this paper. We first present amodel of abduction in abstract argumentation, based on the notion ofan AAF (abductive argumentation framework) that encodes differentpossible changes to an AF, each of which may act as a hypothesis toexplain an observation that can be justified by making an argumentaccepted. We then do two things:1. We present sound and complete dialogical proof procedures forthe main decision problems, i.e., finding hypotheses that explainskeptical/credulous acceptance of arguments in support of an ob-servation. These proof procedures show that the problem of ab-duction is related to an extended form of persuasion, where theproponent uses hypothetical moves to persuade the opponent.2. We show that AAFs can be instantiated by ALPs (abductive logicprograms) in such a way that the hypotheses generated for an ob-servation by the ALP can be computed by translating the ALP intoan AAF. The type of ALPs we focus on are based on Sakama andInoue’s model of extended abduction [13, 14], in which hypothe-ses have a positive as well as a negative element (i.e., facts addedto the logic program as well as facts removed from it).In sum, our contribution is a model of abduction in argumentationwith dialogical proof theories for the main decision problems, whichcan be seen as an abstraction of abduction in logic programming.The overview of this paper is as follows. After introducing the nec-essary preliminaries in section 2 we present in section 3 our modelof abduction in argumentation. In section 4 we present dialogicalproof procedures for the main decision problems (explaining skep-tical/credulous acceptance). In section 5 we show that our model ofabduction can be used to instantiate abduction in logic programming.We discuss related work in section 6 and conclude in section 7.
An argumentation framework consists of a set A of arguments and abinary attack relation over A [12]. We assume in this paper that A is a finite subset of a fixed set U called the universe of arguments . Definition 1.
Given a countably infinite set U called the universeof arguments , an argumentation framework ( AF , for short) is a pair F = ( A, ) where A is a finite subset of U and a binary relationover A . If a b we say that a attacks b . F denotes the set of allAFs.Extensions are sets of arguments that represent different view-points on the acceptance of the arguments of an AF. A semantics is a method to select extensions that qualify as somehow justifiable.We focus on one of the most basic ones, namely the complete seman-tics [12]. efinition 2. Let F = ( A, ) . An extension of F is a set E ⊆ A . An extension E is conflict-free iff for no a, b ∈ E it holds that a b . An argument a ∈ A is defended by E iff for all b such that b a there is a c ∈ E such that c b . Given an extension E ,we define Def F ( E ) by Def F ( E ) = { a ∈ A | E defends a } . Anextension E is admissible iff E is conflict-free and E ⊆ Def F ( E ) ,and complete iff E is conflict-free and E = Def F ( E ) . The set ofcomplete extension of F will be denoted by Co ( F ) . Furthermore,the grounded extension (denoted by Gr ( F ) ) is the unique minimal(w.r.t. ⊆ ) complete extension of F . An argument is said to be skeptically (resp. credulously ) acceptediff it is a member of all (resp. some) complete extensions. Notethat the set of skeptically accepted arguments coincides with thegrounded extension. Furthermore, an argument is a member of acomplete extension iff it is a member of a preferred extension, whichis a maximal (w.r.t. ⊆ ) complete extension. Consequently, credulousacceptance under the preferred semantics (as studied e.g. in [17]) co-incides with credulous acceptance under the complete semantics. Abduction is a form of reasoning that goes from an observation to ahypothesis. We assume that an observation translates into a set X ⊆ A . Intuitively, X is a set of arguments that each individually supportthe observation. If at least one argument x ∈ X is skeptically (resp.credulously) accepted, we say that the observation X is skeptically(resp. credulously) supported . Definition 3.
Given an AF F = ( A, ) , an observation X ⊆ A is skeptically (resp. credulously) supported iff for all (resp. some) E ∈ Co ( F ) it holds that x ∈ E for some x ∈ X . The following proposition implies that checking whether an obser-vation X is skeptically supported can be done by checking whetheran individual argument x ∈ X is in the grounded extension. Proposition 1.
Let F = ( A, ) and X ⊆ A . It holds that F skep-tically supports X iff x ∈ Gr ( F ) for some x ∈ X .Proof of proposition 1. The if direction is immediate. For the onlyif direction, assume F = ( A, ) explains skeptical support for X .Then for every complete extension E of F , there is an x ∈ X s.t. x ∈ E . Define G by G = ( A ∪{ a, b } , ∪{ ( x, a ) | x ∈ X }∪{ ( a, b ) } ) ,where a, b A . Then for every complete extension E of G it holdsthat b ∈ E , hence b ∈ Gr ( G ) . Thus x ∈ Gr ( G ) for some x ∈ X .But Gr ( F ) = Gr ( G ) ∩ A , hence x ∈ Gr ( F ) for some x ∈ X .It may be that an AF F does not skeptically or credulously supportan observation X . Abduction then amounts to finding a change to F so that X is supported. We use the following definition of an AAF ( Abductive AF ) to capture the changes w.r.t. F (each change repre-sented by an AF G called an abducible AF) that an agent considers.We assume that F itself is also an abducible AF, namely one thatcaptures the case where no change is necessary. Other abducible AFsmay be formed by addition of arguments and attacks to F , removalof arguments and attacks from F , or a combination of both. Definition 4. An abductive AF is a pair M = ( F, I ) where F is anAF and I ⊆ F a set of AFs called abducible such that F ∈ I . Given an AAF ( F, I ) and observation X , skeptical/credulous sup-port for X can be explained by the change from F to some G ∈ I that skeptically/credulously supports X . In this case we say that G explains skeptical/credulous support for X . The arguments/attacksadded to and absent from G can be seen as the actual explanation. Definition 5.
Let M = (
F, I ) be an AAF. An abducible AF G ∈ I explains skeptical (resp. credulous) support for an observation X iff G skeptically (resp. credulously) supports X . One can focus on explanations satisfying additional criteria, suchas minimality w.r.t. the added or removed arguments/attacks. Weleave the formal treatment of such criteria for future work.
Example 1.
Let M = ( F, { F, G , G , G } ) , where F, G , G and G are as defined in figure 1. Let X = { b } be an observation. Itholds that G and G both explain skeptical support for X , while G only explains credulous support for X . F b cad G b cad e G b c G b ce Figure 1.
The AFs of the AAF ( F, { F, G , G , G } ) . Remark 1.
The main difference between Sakama’s [20] model ofabduction in abstract argumentation and the one presented here, isthat he takes an explanation to be a set of independently selectableabducible arguments, while we take it to be a change to the AF that isapplied as a whole. In section 5 we show that this is necessary whenapplying the abstract model in an instantiated setting.
In this section we present methods to determine, given an AAFM = (
F, I ) (for F = ( A, ) ) whether an abducible AF G ∈ I explains credulous or skeptical support for an observation X ⊆ A .We build on ideas behind the grounded and preferred games , whichare dialogical procedures that determine skeptical or credulous ac-ceptance of an argument [17]. To sketch the idea behind these games(for a detailed discussion cf. [17]): two imaginary players (PRO andOPP) take alternating turns in putting forward arguments accordingto a set of rules, PRO either as an initial claim or in defence againstOPP’s attacks, while OPP initiates different disputes by attacking thearguments put forward by PRO. Skeptical or credulous acceptance isproven if PRO can win the game by ending every dispute in its favouraccording to a “last-word” principle.Our method adapts this idea so that the moves made by PRO areessentially hypothetical moves. That is, to defend the initial claim(i.e., to explain an observation) PRO can put forward, by way of hy-pothesis, any attack x y present in some G ∈ I . This marks achoice of PRO to focus only on those abducible AFs in which the at-tack x y is present. Similarly, PRO can reply to an attack x y ,put forward by OPP, with the claim that this attack is invalid, markingthe choice of PRO to focus only on the abducible AFs in which theattack x y is not present. Thus, each move by PRO narrows downthe set of abducible AFs in which all of PRO’s moves are valid. Theobjective is to end the dialogue with a non-empty set of abducibleAFs. Such a dialogue represents a proof that these abducible AFsexplain skeptical or credulous support for the observation.Alternatively, such dialogues can be seen as games that deter-mine skeptical/credulous support of an observation by an AF thatre played simultaneously over all abducible AFs in the AAF. In thisview, the objective is to end the dialogue in such a way that it repre-sents a proof for at least one abducible AF. Indeed, in the case where M = ( F, { F } ) , our method reduces simply to a proof theory forskeptical or credulous support of an observation by F .Before we move on we need to introduce some notation. Definition 6.
Given a set I of AFs we define: • A I = ∪{ A | ( A, ) ∈ I } , • I = ∪{ | ( A, ) ∈ I } , • I x y = { ( A, ) ∈ I | x, y ∈ A, x y } , • I X = { ( A, ) ∈ I | X ⊆ A } . We model dialogues as sequences of moves , each move being of acertain type, and made either by PRO or OPP.
Definition 7.
Let M = (
F, I ) be an AAF. A dialogue based on M isa sequence S = ( m , . . . , m n ) , where each m i is either: • an OPP attack “ OPP: x y ”, where x I y , • a hypothetical PRO defence “ PRO: y + x ”, where y I x , • a hypothetical PRO negation “ PRO: y − x ”, where y I x , • a conceding move “ OPP: ok ”, • a success claim move “ PRO: win ”.We denote by S · S ′ the concatenation of S and S ′ . Intuitively, a move
OPP: y x represents an attack by OPP onthe argument x by putting forward the attacker y . A hypotheticalPRO defence PRO: y + x represents a defence by PRO who putsforward y to attack the argument x put forward by OPP. A hypo-thetical PRO negation PRO: y − x , on the other hand, representsa claim by PRO that the attack y x is not a valid attack. Theconceding move OPP: ok is made whenever OPP runs out of pos-sibilities to attack a given argument, while the move
PRO: win ismade when PRO is able to claim success.In the following sections we specify how dialogues are structured.Before doing so, we introduce some notation that we use to keeptrack of the abducible AFs on which PRO chooses to focus in a di-alogue D . We call this set the information state of D after a givenmove. While it initially contains all abducible AFs in M , it is re-stricted when PRO makes a move PRO: x + y or PRO: x − y . Definition 8.
Let M = (
F, I ) be an AAF. Let D = ( m , . . . , m n ) be a dialogue based on M. We denote the information state in D aftermove i by J ( D, i ) , which is defined recursively by: J ( D, i ) = I if i = 0 ,J ( D, i − ∩ I x y if m i = PRO: x + y,J ( D, i − \ I x y if m i = PRO: x − y,J ( D, i − otherwise.We denote by J ( D ) the information state J ( D, n ) . We define the rules of a dialogue using a set of production rules thatrecursively define the set of sequences constituting dialogues. (Thesame methodology was used by Booth et al. [7] in defining a dia-logical proof theory related to preference-based argumentation.) Ina skeptical explanation dialogue for an observation X , an initial ar-gument x ∈ X is challenged by the opponent, who puts forward allpossible attacks OPP: y x present in any of the abducible AFspresent in the AAF, followed by OPP: ok . We call this a skepticalOPP reply to x . For each move OPP: y x , PRO responds with a skeptical PRO reply to y x , which is either a hypothetical defence PRO: z + y (in turn followed by a skeptical OPP reply to z ) or ahypothetical negation PRO: y − x . Formally: Definition 9 (Skeptical explanation dialogue) . Let F = ( A, ) ,M = ( F, I ) and x ∈ A . • A skeptical OPP reply to x is a finite sequence ( OPP: y x ) · S · . . . · ( OPP: y n x ) · S n · ( OPP: ok ) where { y , . . . , y n } = { y | y I x } and each S i is a skeptical PRO reply to y i x . • A skeptical PRO reply to y x is either: (1) A sequence ( PRO: z + y ) · S where z I y and where S is a skepticalOPP reply to z , or (2) The sequence ( PRO: y − x ) . Given an observation X ⊆ A we say that M generates the skepticalexplanation dialogue D for X iff D = S · ( PRO: win ) , where S is askeptical OPP reply to some x ∈ X . The following theorem establishes soundness and completeness.
Theorem 1.
Let M = (
F, I ) be an AAF where F = ( A, ) . Let X ⊆ A and G ∈ I . It holds that G explains skeptical support for X iff M generates a skeptical explanation dialogue D for X such that G ∈ J ( D ) . Due to space constraints we only provide a sketch of the proof.
Sketch of proof.
Let M = (( A, ) , I ) , X ⊆ A and G ∈ I . (Onlyif:) Assume x ∈ Gr ( G ) for some x ∈ X . By induction on thenumber of times the characteristic function [12] is applied so as toestablish that x ∈ Gr ( G ) , it can be shown that a credulous OPPreply D to x exists (and hence a dialogue D · ( PRO: win ) for X )s.t. G ∈ J ( D · ( PRO: win )) . (If:) Assume M generates a skepticalexplanation dialogue D for X s.t. G ∈ J ( D ) . By induction on thestructure of D it can be shown that x ∈ Gr ( G ) for some x ∈ X . Example 2.
The listing below shows a skeptical explanation dia-logue D = ( m , . . . , m ) for the observation { b } that is generatedby the AAF defined in example 1. i m i J ( D, i ) OPP: c b { F, G , G , G } PRO: e + c { G , G } OPP: ok { G , G } OPP: a b { G , G } PRO: e + a { G } OPP: ok { G } OPP: ok { G } PRO: win { G } The sequence ( m , . . . , m ) is a skeptical OPP reply to b , inwhich OPP puts forward the two attacks c b and a b . PROdefends b from both c and a by putting forward the attacker e (move2 and 5). This leads to the focus first on the abducible AFs G , G (inwhich the attack e c exists) and then on G (in which the attack e a exists). This proves that G explains skeptical support for theobservation { b } . Another dialogue is shown below. i m i J ( D, i ) OPP: c b { F, G , G , G } PRO: e + c { G , G } OPP: ok { G , G } OPP: a b { G , G } PRO: a − b { G } OPP: ok { G } PRO: win { G } ere, PRO defends b from c by using the argument e , but defends b from a by claiming that the attack a b is invalid. This leads to thefocus first on the abducible AFs G , G (in which the attack e c exists) and then on G (in which the attack a b does not exist).This dialogue proves that G explains skeptical support for { b } . The definition of a credulous explanation dialogue is similar to that ofa skeptical one. The difference lies in what constitutes an acceptabledefence. To show that an argument x is skeptically accepted, x mustbe defended from its attackers by arguments other than x itself. Forcredulous acceptance, however, it suffices to show that x is a memberof an admissible set, and hence x may be defended from its attackersby any argument, including x itself. To achieve this we need to keeptrack of the arguments that are, according to the moves made by PRO,accepted. Once an argument x is accepted, PRO does not need todefend x again, if this argument is put forward a second time.Formally a credulous OPP reply to ( x, Z ) (for some x ∈ A I andset Z ⊆ A I used to keep track of accepted arguments) consists of allpossible attacks OPP: y x on x , followed by OPP: ok when allattacks have been put forward. For each move
OPP: y x , PRO re-sponds either by putting forward a hypothetical defence PRO: z + y which (this time only if z Z ) is followed by a credulous OPPreply to ( z, Z ∪ { z } ) , or by putting forward a hypothetical nega-tion PRO: y − x . We call this response a credulous PRO reply to ( y x, Z ) . A credulous explanation dialogue for a set X consistsof a credulous OPP reply to ( x, { x } ) for some x ∈ X , followed bya success claim PRO: win .In addition, arguments put forward by PRO in defence of the ob-servation may not conflict. Such a conflict occurs when OPP putsforward
OPP: x y and OPP: y z (indicating that both y and z are accepted) while PRO does not put forward PRO: y − z . If thissituation does not occur we say that the dialogue is conflict-free . Definition 10 (Credulous explanation dialogue) . Let F = ( A, ) ,M = ( F, I ) , x ∈ A and Z ⊆ A . • A credulous OPP reply to ( x, Z ) is a finite sequence ( OPP: y x ) · S · . . . · ( OPP: y n x ) · S n · ( OPP: ok ) where { y , . . . , y n } = { y | y I x } and each S i is a credulous PRO reply to ( y i x, Z ) . • A credulous PRO reply to ( y x, Z ) is either: (1) a sequence ( PRO: z + y ) · S such that z I y , z Z and S is a credulousOPP reply to ( z, Z ∪ { z } ) , (2) a sequence ( PRO: z + y ) suchthat z I y and z ∈ Z , or (3) the sequence ( PRO: y − x ) . Given a set X ⊆ A we say that M generates the credulous expla-nation dialogue D for X iff D = S · ( PRO: win ) , where S is acredulous OPP reply to ( x, { x } ) for some x ∈ X . We say that D is conflict-free iff for all x, y, z ∈ A I it holds that if D containsthe moves OPP: x y and OPP: y z then it contains the move PRO: y − z . The following theorem establishes soundness and completeness.
Theorem 2.
Let M = (
F, I ) be an AAF where F = ( A, ) . Let X ⊆ A and G ∈ I . It holds that G explains credulous support for X iff M generates a conflict-free credulous explanation dialogue D for X such that G ∈ J ( D ) .Sketch of proof.. Let M = (( A, ) , I ) , X ⊆ A and G ∈ I . (Onlyif:) Assume for some x ∈ X and E ∈ Co ( G ) that x ∈ E . Usingthe fact that E ⊆ Def G ( E ) one can recursively define a credulous OPP reply D to ( x, Z ) for some Z ⊆ A and hence a credulous ex-planation dialogue D · ( PRO: win ) . Conflict-freeness of E impliesconflict-freeness of D . (If:) Assume M generates a credulous expla-nation dialogue D · ( PRO: win ) for X such that G ∈ J ( D ) . Then D is a credulous OPP reply to ( a, { a } ) for some a ∈ X . It can beshown that the set E = { a } ∪ { x | PRO: x + z ∈ D } satisfies E ⊆ Def G ( E ) . Conflict-freeness of D implies conflict-freeness of E . Hence a ∈ E for some E ∈ Co ( G ) . Example 3.
The listing below shows a conflict-free credulous expla-nation dialogue D = ( m , . . . , m ) for the observation { b } gener-ated by the AAF defined in example 1. i m i J ( D, i ) OPP: c b { F, G , G , G } PRO: b + c { F, G , G , G } OPP: a b { F, G , G , G } PRO: a − b { G , G } OPP: ok { G , G } PRO: win { G , G } Here, the sequence ( m , . . . , m ) is a credulous OPP reply to ( b, { b } ) . PRO defends b from OPP’s attack c b by putting forwardthe attack b c . Since b was already assumed to be accepted, thissuffices. At move m , PRO defends itself from the attack a b bynegating it. This restricts the focus on the abducible AFs G and G .The dialogue proves that these two abducible AFs explain creduloussupport for the observation { b } . Finally, the skeptical explanationdialogues from example 2 are also credulous explanation dialogues. In this section we show that AAFs can be instantiated with abductivelogic programs, in the same way that regular AFs can be instanti-ated with regular logic programs. In sections 5.1 and 5.2 we recallthe necessary basics of logic programming and the relevant resultsregarding logic programming as instantiated argumentation. In sec-tion 5.3 we present a model of abductive logic programming basedon Sakama and Inoue’s model of extended abduction [13, 14], and insection 5.2 we show how this model can be instantiated using AAFs.
A logic program P is a finite set of rules, each rule be-ing of the form C ← A , . . . , A n , ∼ B , . . . , ∼ B m where C, A , . . . , A n , B , . . . , B m are atoms . If m = 0 then the rule iscalled definite . If both n = 0 and m = 0 then the rule is called a fact and we identify it with the atom C . We assume that logic programsare ground. Alternatively, P can be regarded as the set of ground in-stances of a set of non-ground rules. We denote by At P the set of all(ground) atoms occurring in P . The logic programming semanticswe focus on can be defined using [19]: Definition 11.
A 3-valued interpretation I of a logic program P isa pair I = ( T, F ) where T, F ⊆ At P and T ∩ F = ∅ . An atom A ∈ At P is true (resp. false , undecided ) in I iff A ∈ T (resp. A ∈ F , A ∈ At P \ ( T ∪ F ) ). The following definition of a partial stable model is due to Przy-musinski [19]. Given a logic program P and 3-valued interpretation I of P , the GL-transformation PI is a logic program obtained by re-placing in every rule in P every premise ∼ B such that B is true (resp.undecided, false) in I by the atoms (resp. , ), where (resp. , ) are defined to be false (resp. undecided, true) in every interpreta-tion. It holds that for all 3-valued interpretations I of P , PI is definite(i.e., consists only of definite rules). This means that PI has a unique least ( T, F ) with minimal T and maximal F that satisfies all rules. That is, for all rules C ← A , . . . , A n , in PI , C is true (resp. not false) in ( T, F ) if for all i ∈ { , . . . , n } , A i istrue (resp. not false) in ( T, F ) . Given a 3-valued interpretation I , theleast 3-valued interpretation of PI is denoted by Γ( I ) . This leads tothe following definition of a partial stable model of a logic program,along with the associated notions of consequence. Definition 12. [19] Let P be a logic program. A 3-valued interpre-tation I is a partial stable model of P iff I = Γ( I ) . We say that anatom C is a skeptical (resp. credulous) consequence of P iff C is truein all (resp. some) partial stable models of P . It has been shown that the above defined notion of skeptical con-sequence coincides with the well-founded semantics [19].
Wu et al. [22] have shown that a logic program P can be transformedinto an AF F in such a way that the consequences of P under thepartial stable semantics can be computed by looking at the completeextensions of F . The idea is that an argument consists of a conclusion C ∈ At P , a set of rules R ⊆ P used to derive C and a set N ⊆ At P of atoms that must be underivable in order for the argument tobe acceptable. The argument is attacked by another argument witha conclusion C ′ iff C ′ ∈ N . The following definition, apart fromnotation, is due to Wu et al. [22]. Definition 13.
Let P be a logic program. An instantiated argumentis a triple ( C, R, N ) , where C ∈ At P , R ⊆ P and N ⊆ At P . Wesay that P generates ( C, R, N ) iff either: • r = C ← ∼ B , . . . , ∼ B m is a rule in P , R = { r } and N = { B , . . . , B m } . • (1) r = C ← A , . . . , A n , ∼ B , . . . , ∼ B m is a rule in P , (2) P generates, for each i ∈ { , . . . , n ] an argument ( A i , R i , N i ) such that r R i , and (3) R = { r } ∪ R ∪ . . . ∪ R n and N = { B , . . . , B m } ∪ N ∪ . . . ∪ N n .We denote the set of arguments generated by P by A P . Furthermore,the attack relation generated by P is denoted by P and is definedby ( C, R, N ) P ( C ′ , R ′ , N ′ ) iff C ∈ N ′ . The following theorem states that skeptical (resp. credulous) ac-ceptance in ( A P , P ) corresponds with skeptical (resp. credulous)consequences in P as defined in definition 12. It follows from theo-rems 15 and 16 due to Wu et al. [22]. Theorem 3.
Let P be a logic program. An atom C ∈ At P is askeptical (resp. credulous) consequence of P iff some ( C, R, N ) ∈ A P is skeptically (resp. credulously) accepted in ( A P , P ) . The model of abduction in logic programming that we use is based onthe model of extended abduction studied by Inoue and Sakama [13,14]. They define an abductive logic program (ALP) to consist of alogic program and a set of atoms called abducibles . Definition 14.
An abductive logic program is a pair ( P, U ) where P is a logic program and U ⊆ At P a set of facts called abducibles. Note that, as before, the set U consists of ground facts of the form C ← (identified with the atom C ) and can alternatively be regardedas the set of ground instances of a set of non-ground facts. A hy-pothesis, according to Inoue and Sakama’s model, consists of both apositive element (i.e., abducibles added to P ) and a negative element(i.e., abducibles removed from P ). Definition 15.
Let ALP = (
P, U ) be an abductive logic program. Ahypothesis is a pair (∆ + , ∆ − ) such that ∆ + , ∆ − ⊆ U and ∆ + ∩ ∆ − = ∅ . A hypothesis (∆ + , ∆ − ) skeptically (resp. credulously)explains a query Q ∈ At P if and only if Q is a skeptical (resp.credulous) consequence of ( P ∪ ∆ + ) \ ∆ − . Note that Sakama and Inoue focus on computation of explanationsunder the stable model semantics of P , and require P to be acyclicto ensure that a stable model of P exists and is unique [14]. We,however, define explanation in terms of the consequences accordingto the partial stable models of P , which always exist even if P is notacyclic [19], so that we do not need this requirement.The following example demonstrates the previous two definitions. Example 4.
Let ALP = (
P, U ) where P = { ( p ← ∼ s, r ) , ( p ←∼ s, ∼ q ) , ( q ← ∼ p ) , r } and U = { r, s } . The hypothesis ( { s } , ∅ ) skeptically explains q , witnessed by the unique model I =( { r, s, q } , { p } ) satisfying I = Γ( I ) . Similarly, ( { s } , { r } )) skepti-cally explains q and ( ∅ , { r } )) credulously explains q . In this section we show that an AAF ( F, I ) can be instantiated on thebasis of an abductive logic program ( P, U ) . The idea is that everypossible hypothesis (∆ + , ∆ − ) maps to an abducible AF generatedby the logic program ( P ∪ ∆ + ) \ ∆ − . The hypotheses for a query Q then correspond to the abducible AFs that explain the observation X consisting of all arguments with conclusion Q . The constructionof ( F, I ) on the basis of ( P, U ) is defined as follows. Definition 16.
Let ALP = (
P, U ) be an abductive logic program.Given a hypothesis (∆ + , ∆ − ) , we denote by F (∆ + , ∆ − ) the AF ( A ( P ∪ ∆ + ) \ ∆ − , ( P ∪ ∆ + ) \ ∆ − ) . The AAF generated by ALP is de-noted by M
ALP and defined by M
ALP = (( A P , P ) , I ALP ) , where I ALP = { F (∆ + , ∆ − ) | ∆ + , ∆ − ⊆ U, ∆ + ∩ ∆ − = ∅} . The following theorem states the correspondence between the ex-planations of a query Q in an abductive logic program ALP and theexplanations of an observation X in the AAF M ALP . Theorem 4.
Let ALP = (
P, U ) be an abductive logic program, Q ∈ At P a query and (∆ + , ∆ − ) a hypothesis. Let M ALP = (
F, I ) .We denote by X Q the set { ( C, R, N ) ∈ A P | C = Q } . It holds that (∆ + , ∆ − ) skeptically (resp. credulously) explains Q iff F (∆ + , ∆ − ) skeptically (resp. credulously) explains X Q .Proof of theorem 4. Via theorem 3 and definitions 15 and 16.This theorem shows that our model of abduction in argumentationcan indeed be seen as an abstraction of abductive logic programming.
Example 5.
Let ALP = (
P, U ) be the ALP as de-fined in example 4. All arguments generated by ALP are: a = ( p, { ( p ← ∼ s, r ) , r } , { s } ) d = ( r, { r } , ∅ ) b = ( q, { ( q ← ∼ p ) } , { p } ) e = ( s, { s } , ∅ ) c = ( p, { ( p ← ∼ s, ∼ q ) } , { s, q } ) iven these definitions, the AAF in example 1 is equivalent to M ALP .In example 4 we saw that q is skeptically explained by ( { s } , ∅ ) and ( { s } , { r } ) , while ( ∅ , { r } ) only credulously explains it. Indeed, look-ing again at example 1, we see that G = F ( { s } , ∅ ) and G = F ( { s } , { r } ) explain skeptical support for the observation { b } = X q ,while G = F ( ∅ , { r } ) only explains credulous support. Remark 2.
This method of instantiation shows that, on the abstractlevel, hypotheses cannot be represented by independently selectableabducible arguments. The running example shows e.g. that a and d cannot be added or removed independently. (Cf. remark 1.) We already discussed Sakama’s [20] model of abduction in argumen-tation and mentioned some differences. Our approach is more gen-eral because we consider a hypothesis to be a change to the AF thatis applied as a whole, instead of a set of independently selectableabducible arguments. On the other hand, Sakama’s method supportsa larger range semantics, including (semi-)stable and skeptical pre-ferred semantics. Furthermore, Sakama also considers observationsleading to rejection of arguments, which we do not.Some of the ideas we applied also appear in work by Wakaki etal. [21]. In their model, ALPs generate instantiated AFs and hypothe-ses yield a division into active/inactive arguments.Kontarinis et al. [15] use term rewriting logic to compute changesto an abstract AF with the goal of changing the status of an argument.Two similarities to our work are: (1) our production rules to generatedialogues can be seen as a kind of term rewriting rules. (2) their ap-proach amounts to rewriting goals into statements to the effect thatcertain attacks in the AF are enabled or disabled. These statementsresemble the moves
PRO: x + y and PRO: x − y in our sys-tem. However, they treat attacks as entities that can be enabled ordisabled independently. As discussed, different arguments (or in thiscase attacks associated with arguments) cannot be regarded as inde-pendent entities, if the abstract model is instantiated.Goal oriented change of AFs is also studied by Baumann [2], Bau-mann and Brewka [3], Bisquert et al. [4] and Perotti et al. [5]. Fur-thermore, Booth et al. [8] and Coste-Marquis et al. [11] frame it asa problem of belief revision . Other studies in which changes to AFsare considered include [6, 10, 16, 18]. We developed a model of abduction in abstract argumentation, inwhich changes to an AF act as explanations for skeptical/creduloussupport for observations. We presented sound and complete dialog-ical proof procedures for the main decision problems, i.e., findingexplanations for skeptical/credulous support. In addition, we showedthat our model of abduction in abstract argumentation can be seen asan abstract form of abduction in logic programming.As a possible direction for future work, we consider the incorpo-ration of additional criteria for the selection of good explanations,such as minimality with respect to the added and removed argu-ments/attacks, as well as the use of arbitrary preferences over dif-ferent abducible AFs. An interesting question is whether the prooftheory can be adapted so as to yield only the preferred explanations.
Richard Booth is supported by the Fonds National de la Recherche,Luxembourg (DYNGBaT project).
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