We de ne a new logic programming semantics in terms of abducible atoms. We use it to characterize the stage extensions of an argumentation framework AF by means of an associated normal program PAF . We also de ne the stage semantics for a special type of normal programs and present a similar characterization.
The rest of the paper is divided as follows: In x2, we present some basic concepts w.r.t. logic programming and argumentation theory. In x3, we characterize the stage argumentation semantics of an argumentation framework based on a stable abducible logic programming semantics. In the last section, we present our conclusions. 2
In this section, we review some theory about logic programming semantics and argumentation semantics. 2.1
A signature L is a nite set of elements that we call atoms. A literal is either an atom a, called positive literal ; or the negation of an atom :a, called negative literal. A normal clause is a clause of the form a b1 ^ : : : ^ bn ^ :bn+1 ^ : : : ^ :bn+m where a and each of the bi are atoms for 1 i n + m; a is called the head and b1 ^ : : : ^ bn ^ :bn+1 ^ : : : ^ :bn+m is called the body of the normal clause. If the body of a normal clause is empty, then the clause is known as a fact and can be denoted just by: a or a ⊤. We de ne a normal logic program P , as a nite set of normal clauses. By normal program we will mean a normal logic program when ambiguity does not arise. We de ne an n-program P , as a normal logic program where every clause is of the form a :b. We write LP , to denote the set of atoms that appear in the clauses of P . We want to point out that our negation symbol, :, corresponds to \not" in the standard use of Logic Programming.
From now on, we assume that the reader is familiar with the notion of an
interpretation and validity [
In this paper, a logic programming semantics S is a mapping de ned on the
family of all programs which associates to a given program a subset of its 2-valued
models. We say that M is a minimal model of P if and only if there does not exist
a model M ′ of P such that M ′ M [
The stable model semantics was de ned in terms of the so called Gelfond-Lifschitz
reduction [
De nition 1. Let P be a normal program. For a set M LP we de ne the program P M obtained from P by deleting each clause that has a literal :l in its body with l 2 M , and then all literals of the form :l in the bodies of the remaining clauses. Clearly P M does not contain the symbol :. M is a stable model of P if and only if M is a minimal model of P M . We will denote by stable(P) the set of all stable models of P . 2.3
We review some basic concepts of the stage argumentation semantics de ned by
Verheij [
De nition 2. [
Example 1. Let AF := ⟨fa; b; c; dg; f(a; a); (a; c); (b; c); (c; d)g⟩ be an argumentation framework.
We say that a attacks c (or c is attacked by a) if (a; c) 2 attacks holds. Similarly, we say that a set S of arguments attacks c (or c is attacked by S) if c is attacked by an argument in S.
Intuitively speaking, for a set of arguments to be an extension of an
argumentation framework, it is necessary that all of its arguments are consistent in the sense
that there are no attacks between them. The next de nition states this idea.
De nition 3. [
Now we present the de nition of stage argumentation semantics. Given a set S of arguments, S+ = fb 2 AR j 9a 2 S such that (a; b) 2 attacksg.
De nition 4. [
Example 2. From de nition 4, it follows that the set fb; dg is the only stage extension of the argumentation framework of Example 1. 2.4
In [
De nition 5. [
Example 3. Let AF be the argumentation framework of Example 1. We can see that PAF = f d(a) :d(a); d(c) :d(a); d(c) :d(b); d(d) :d(c)g.
Now, we describe the characterization of stage extensions of an argumentation framework AF in terms of the stage models of the normal program PAF . Here, F acts is de ned as a function on normal programs which returns the facts which are present in a given normal program.
De nition 6. [
We de ne the following mapping from 2LPAF to 2AR as E(M ) = fxjd(x) 2
Theorem 1. [
Example 4. We can see that according to example 3, the normal program PAF of the argumentation framework AF of Example 1 has ve 2-valued models: fd(a); d(b); d(c); d(d)g, fd(a); d(c); d(d)g, fd(a); d(b); d(d)g, fd(a); d(b); d(c)g, and fd(a); d(c)g. Moreover, we can verify that M = fd(a); d(c)g is the only stage model of PAF since M n F acts((PAF )M ) is minimal with respect to set inclusion. Hence, according to Theorem 1, fb; dg is the only stage extension of AF , which coincides with the result showed in example 2.
We remark that de nition 6 of a stage model can be obtained straightforwardly
from the de nition of stage model in [
We use the idea of abducible atom to de ne a semantics on a very special class of normal programs. The obtained semantics allows us to characterize the stage extensions of an argumentation framework by means of the models of the related normal program de ned by this new semantics.
In what follows, the symbol ( means proper subset. 3.1
The classical stable ab-m semantics of a normal program is the particular version
of the stable abducible logic programming semantics [
In this subsection, we de ne our main semantics for normal programs, called classical m-ab-m semantics, which is used to characterize the stage argumentation semantics of an argumentation framework. Moreover, we describe two properties of the classical m-ab-m semantics which are suitable for that characterization.
In order to de ne the classical m-ab-m semantics of normal programs, we de ne a partial order on the set of classical ab-m models and then we nd the set of minimal classical ab-m models; these minimal elements de ne the classical m-ab-m semantics.
We state the following useful fact as an observation.
Observation 1 : For any normal program P whose stable semantics is empty, the program P [ LP has LP as a stable model. Here, each element of LP as a rule in a normal program is interpreted as a fact.
Now, we de ne the classical stable ab-m semantics of a normal program, where the sets of abducible atoms are taken from 2-valued models of the program. We de ne also the stable m-ab-m semantics of a normal program.
We say that the pair ⟨M; Ab⟩ denoted by MAb is a classical stable ab-m model of P if M is a stable model of P [ Ab.
Example 5. Let P = fa :b b :c; c :ag. We can verify that P does not have stable models. Table 1 shows the classical stable ab-m models of P respect to each of the three 2-valued models of P . From Table 1, we can see that the classical stable m-ab-m semantics of P is: f⟨fa; bg; fag⟩; ⟨fb; cg; fbg⟩; ⟨fa; cg; fcg⟩g.
Now, we state two consequences of observation 1 related to the classical m-ab-m semantics of a normal program.
semantics of P coincides with the stable semantics of P .
De nitions 7 and 8 could be extended to more general programs, for example disjunctive programs. 3.2
n-programs Here, we analyze the classical stable m-ab-m semantics for n-programs. We de ne several semantics in terms of abducible atoms and look at some of their properties. Then we characterize the stage extensions of an argumentation framework AF based on one of these new semantics when they are applied to the associated nprogram PAF .
We recall from section 2 that a n-program is a normal logic program where every clause is of the form a :b.
Now, we analyze some consequences of de nitions 7 and 8 for the particular case of n-programs.
classical stable ab-m model of P iff P M [ Ab = M .
Note that (P [ Ab)M = P M [ Ab.
Let us remark that in de nition 8, M and N can be different, so the sets of abducible atoms can be obtained from different sets. A particular case of de nition 8 is given when the partial order among the classical stable ab-m models of a n-program is de ned respect to the set of abducible atoms but these subsets of abducibles are taken from the same set of atoms. This corresponds to the de nition of classical normal stable ab-m semantics of a n-program.
Ab M . We say that the pair ⟨M; Ab⟩ is a classical normal stable ab-m model of P if there does not exist another classical stable ab-m model MAb1 such that Ab1 ( Ab.
The next result tell us that, for a xed model M of P there is always exactly one classical normal stable ab-m model of P .
then M(Ab1\Ab2) is a classical stable ab-m model of P .
classical normal stable ab-m model MAb of P is unique. 3. If MAb is a classical normal stable ab-m model of P then P M [ Ab = M and
P M \ Ab = ∅.
P with Ab M . We say that the pair MAb is a classical normal stable m-ab-m model of P if there does not exist another classical normal stable ab-m model of
Example 6. Let us consider again the program PAF of the argumentation framework AF of Example 1. Table 2 shows the classical stable ab-m models of PAF respect to each of the ve classical 2-valued models of PAF . For convenience, we drop the pre x d of the atoms, for instance, for d(a) we only write a. From Table 2, we can see the following: 1. The only classical stable m-ab-m model of PAF is ⟨fa; cg; fag⟩. 2. The classical normal stable ab-m models of PAF are: ⟨fa; cg; fag⟩, ⟨fa; b; cg; fa; b; cg⟩, ⟨fa; b; dg; fa; bg⟩, ⟨fa; c; dg; fa; dg⟩, and ⟨fa; b; c; dg; fa; b; c; dg⟩. 3. The only classical normal stable m-ab-m model of PAF is ⟨fa; cg; fag⟩.
Finally, we prove that the classical normal stable m-ab-m semantics coincides with the classical stable m-ab-m semantics of a given n-program.
In next subsection, we characterize the stage argumentation semantics of an Argumentation Framework AF based on the classical normal stable ab-m logic programming semantics and classical stable m-ab-m logic programming semantics when they are applied to the associated n-program PAF . 3.3
In this section, we de ne the stage semantics for n-programs. We will see that this semantics corresponds to the stage semantics of an argumentation framework under the mapping presented in de nition 5. Next we will show in one of our main results (Theorem 5), that for n-programs this semantics can be characterized in terms of the classical stable m-ab-m semantics. As a consequence of this result we prove also that the stage semantics of an argumentation framework AF corresponds to the classical m-ab-m semantics of the associated program PAF (Corollary 1). With these results we see that the semantics we have de ned in terms of abducible atoms offer an alternative to de ne the semantics of stage extensions of an argumentation framework as well as the stage semantics for n-programs. The de nition of stage logic programming semantics is based on an auxiliar logic programming semantics, called c-Stage.
pair ⟨M; (M n P M )⟩ denoted by MX is a c-Stage-model of P . Here X = M n P M .
atoms. Then MX is classical normal stable ab-m model of P iff MX is a c-Stagemodel of P .
that the pair MX is a stage-model of P if there does not exist a c-Stage model NX1 of P such that X1 ( X.
As we mentioned before, some of the semantics for logic programs as well as for argumentation frameworks can be de ned in terms of semantics with abducible atoms. The next result, which is one of our main contributions, shows that for nprograms the stage semantics and the classical stable m-ab-m sematics are equivalent.
a classical stable m-ab-m model of P iff MX is a stage-model of P . LP . MX is Example 7. Let us consider again the program PAF of the argumentation framework AF of Example 1. Table 3 shows the c-Stage models of PAF respect to each of the ve classical 2-valued models of PAF .
Model c-Stage model fa; cg ⟨fa; cg; fag⟩ fa; b; cg ⟨fa; b; cg; fa; b; cg⟩ fa; b; dg ⟨fa; b; dg; fa; bg⟩ fa; c; dg ⟨fa; c; dg; fa; dg⟩ fa; b; c; dg ⟨fa; b; c; dg; fa; b; c; dg⟩ From Table 3, we can see the following: 1. The set of c-Stage models of PAF coincides with the set of classical normal stable ab-m models of PAF , as Theorem 4 indicates. 2. The only stage-model of PAF is ⟨fa; cg; fag⟩. 3. The stage-model ⟨fa; cg; fag⟩ is also the only classical stable m-ab-m model of PAF , as Theorem 5 indicates.
We are now ready to present our second result, which is the characterization of the stage extensions of an argumentation framework in terms of the classical stable m-ab-m semantics for n-programs.
Corollary 1. Let AF := ⟨AR; attacks⟩ be an argumentation framework and PAF be its associated n-program, then the stage extension semantics of AF is characterized by the classical stable m-ab-m semantics of PAF .
Example 8. Let us take again the argumentation framework AF de ned in Example 1. As we have seen in example 7 the only classical stable m-ab-m model of PAF is ⟨fa; cg; fag⟩ which is also the only stage model of PAF . According to Theorem 1, the only stage extension of AF is the set fb; dg which is the image of the set fa; cg under the mapping E (see de nition of mapping E in section 2.4). This coincides with the result of example 4.
We recall the de nition of the stage logic programming semantics for a very simple type of normal programs, the n-programs, and we studied the relations between this semantics and the semantics of stage extension for argumentation frameworks. The n-programs we have de ned here, besides having a simple structure, have the property that any of them is the associated normal program of an argumentation framework. We characterized both semantics in terms of an abductive logic programming semantics, the classical stable m-ab-m semantics.