In this paper we present a generalization of the ASPIC argumentation system, where a rule can be associated with a probability. This probability expresses uncertainty about whether or not the particular rule is active. We call this system p-ASPIC. The uncertainty about rules carries over to uncertainty about whether or not a particular argument is active. We generalize the usual Dung-style abstract argumentation semantics, in order to deal with this uncertainty. Our work should be understood as an initial step in the direction of probabilistic, instantiated, Dung-style argumentation, and we discuss a number of directions for future work.
Dung's famous '95 paper [
In this paper we look at a problem that has received little attention so far,
namely the problem of reasoning under uncertainty using the Dung-style
argumentation model. In some respect, non-monotonic reasoning is already reasoning
under uncertainty. That is, we want to derive conclusions under incomplete or
uncertain knowledge. This knowledge comes in the form of defeasible rules, which
we do not want to accept as strict rules, or in the form of assumptions, which
we do not want to accept as facts. This defeasibility, sometimes made more
expressive, as in the ASPIC system, by using of preferences over rules (see also [
On the abstract level, we extend the notion of a framework to that of a probabilistic framework. This is an argumentation framework supplemented with a probability distribution over sets of arguments. Furthermore, we generalize the usual notion of an acceptance function, in order to evaluate the arguments of a probabilistic framework, and we show how it can be used to determine, given a probabilistic framework, the probability of skeptical and credulous acceptance of an argument.
On the instantiated level, we take as a basis a simpli ed version of the ASPIC
argumentation system. This simpli ed version has also been considered in [
The work presented in this paper should be understood as an initial step in the direction of probabilistic, instantiated, Dung-style argumentation. There are many open issues and possible directions for future work.
The outline of this paper is as follows. In section 2 we focus on the abstract level, and present rst the familiar concepts of Dung-style argumentation: argumentation frameworks, labelings and acceptance functions. Then we present our probabilistic generalizations of these concepts, namely that of probabilistic frameworks, along with a method of determining the probability that arguments are skeptically or credulously accepted. In section 3 we turn to the instantiated setting, and we present a generalization of the ASPIC-Lite argumentation system, which we call p-ASPIC. This section de nes how we can generate an instance of a probabilistic framework. In section 4.1, we turn again to the abstract level, where we demonstrate the evaluation of the arguments of a probabilistic framework, generated from a p-ASPIC theory. Finally, before we present our conclusions in section 7, we discuss in section 5 a number of open problems and in section 6 we give an overview of related work. 2 2.1
Dung-style argumentation theory centers on the concept of an argumentation framework (in short framework ). A framework consists of a set of arguments and a binary attack relation, representing con icts among arguments. De nition 1. An argumentation framework F is a pair (Args; Att), where Args is a nite set of arguments, and where Att Args Args is called the attack relation.
Given a framework, we are interested in which arguments we can simultaneously accept. The answer is given in the form of a labeling, that represents an evaluation of the arguments of the framework.
De nition 2. Given a framework F = (Args; Att), a labeling is a function L : Args ! V , where V = fI; U; Og. We denote the set of all labelings of F by LFall. If L is a labeling of F then L 1 : V ! 2Args, where 2Args denotes the power set of Args, is de ned by L 1(v) = fa 2 Args j L(a) = vg.
An argument labeled I, O or U (for in, out or undecided ) is, respectively, accepted, rejected or neither accepted nor rejected. A function that takes as input a framework and returns a set of labelings, is called an acceptance function: De nition 3. An acceptance function is a function A that returns, for any framework F , a set A(F ) LFall.
Di erent types of acceptance functions can be de ned, each corresponding to an argumentation semantics that embodies a particular set of rationality criteria. The properties of con ict-freeness and admissibility are usually considered to be the minimum requirements: De nition 4. Given a framework F = (Args; Att), we say that a labeling L 2 LFall is con ict-free1 i : @(a; b) 2 Att s.t. L(a) = L(b) = I; and admissible i : 8a 2 Args, L(a) = I implies 8(b; a) 2 Att; L(b) = O and L(a) = O implies 9(b; a) 2 Att s.t. L(b) = I. We denote the set of con ict-free and admissible cf labelings of F by LF and AFad, respectively.
Another property is completeness. It states that an argument is labeled I if and only if all attackers are labeled O, and that an argument is labeled O if and only if there is an attacker labeled I.
De nition 5. Given a framework F = (Args; Att), we say that a labeling L is complete if and only if: 1. L(a) = I if and only if 8(b; a) 2 Att, L(b) = O. 2. L(a) = O if and only if 9(b; a) 2 Att, L(b) = I.
Note that complete labelings are also con ict free and admissible. Among
the complete labelings, we can select those that satisfy certain properties, such
as having a maximal set of arguments labeled I or U , or having no argument
labeled U . The main types of acceptance functions are now de ned as follows:
1 Di erent de nitions of con ict-free labelings exist. In [
De nition 6. The complete acceptance function, denoted by Aco, is de ned to return all complete labelings of F . The preferred, grounded and stable acceptance functions, denoted by Apr; Agr and Ast, respectively, are de ned as follows. { Apr(F ) = fL 2 AcFo j @K 2 Aco(F ); K 1(I) { Agr(F ) = fL 2 AcFo j @K 2 Aco(F ); K 1(U ) { Ast(F ) = fL 2 AcFo j L 1(U ) = ;g
Note that we consider acceptance of sets rather than single arguments because we may have that, in the instantiated setting presented in section 3, multiple arguments have the same conclusion. We are then interested in skeptical and credulous acceptance of a conclusion, rather than an argument, and we need to look at the set of all arguments with this conclusion. In the previous section, we introduced the basic concepts of the theory of abstract argumentation theory. We now consider a probabilistic generalization of these concepts. Given a framework (Args; Att), there may be uncertainty about whether or not an argument a 2 Args is active. This uncertainty may arise, for example, from: { Uncertainty of evidence. Individual pieces of evidence, on which an argument is based, may be uncertain. This uncertainty carries over to the argument. So the probability that the argument is active is the probability that the evidence is true. { Opponent modeling. If we use a framework to model the knowledge of an opponent (e.g. in the setting of an argumentation game), we may be uncertain about which arguments the opponent is aware of. So the probability that the argument is active is the probability that the opponent is aware of the argument.
To represent this kind of uncertainty, we introduce the concept of a probabilistic framework : De nition 7. A probabilistic framework is a pair P F = (F; P ), where F = (Args; Att) is a framework and P : 2Args ! [0; 1] is a probability distribution over sets of arguments, called framework states, such that PC22Args P (C) = 1.
A framework state C corresponds to the event that all (and only) the arguments in C are active. We do not know which framework state C 2Args is active; we only know its probability P (C). We say that an argument a is active (resp. inactive) in C if a 2 C (resp. a 62 C).
Example 1. Consider the probabilistic framework (F; P ) with F = (Args; Att), where Args = fa; bg, R = f(a; b); (b; a)g. Thus, we have four framework states: C1 = ;; C2 = fag; C3 = fbg and C4 = fa; bg. Suppose the probability that a is active is 0.4 and that b is active is 0.5, and that these events are independent. The probability distribution over framework states is then de ned by P (C1) = P (C3) = 0:3 and P (C2) = P (C4) = 0:2.
The way we evaluate the arguments of the framework F , given a state C, is straightforward. We apply the criterium of completeness with the proviso that all inactive arguments are labeled O. More precisely, given a state C, an argument is labeled I if and only if it is active and all attackers are labeled O or is labeled O if and only if it is inactive or there is an attacker labeled I. We thus end up with a notion of completeness given a framework state C: De nition 8. Given a framework (Args; Att) and a framework state C Args, we say that a labeling L 2 Lall is complete given the framework state C i : 1. L(a) = I if and only if a 2 C and 8(b; a) 2 Att, L(b) = O. 2. L(a) = O if and only if either a 62 C or 9(b; a) 2 Att, L(b) = I.
We can now de ne the conditionally complete, preferred, grounded and stable acceptance functions, that take as input a framework and a framework state: De nition 9. Given a framework F , the conditionally complete, preferred, grounded pr gr and stable acceptance functions2 CAcFo; CAF ; CAF and CAsFt are de ned by: {{ CCAAcpor((FF;; CC)) == ffLL 22 LCFaAlcFlo(C) j @K 2 CAcFo(C); K 1(I)
j L is complete given Cg. { CAgr(F; C) = fL 2 CAcFo(C) j @K 2 CAcFo(C); K 1(U ) { CAst(F; C) = fL 2 CAcFo(C) j L 1(U ) = ;g
Now we generalize the two basic types of queries. Given a probabilistic framework ((Args; Att); P ), a semantics sem and a set B Args, we can ask: { What is the probability that B is skeptically accepted? { What is the probability that B is credulously accepted?
Loosely speaking, the probabilities of skeptical and credulous acceptance can
be understood as lower and upper bound probabilities, where the di erences
between the two arise from con icts among arguments. The probabilities are
calculated as follows.
2 The concept of a conditional acceptance function presented here is similar to the one
presented in [
Psskeemp(B) = Pcsreemd (B) =
X fC22Argsj8L2CAsem(F;C);9a2B;L(a)=Ig
X fC22Argsj9L2CAsem(F;C);9a2B;L(a)=Ig
P (C) P (C) The probability of credulous acceptance, denoted Pcsreemd (B), is de ned by: In words, the de nitions above amount to: the probability of skeptical (resp. credulous) acceptance of B is the sum probability of all framework states in which B is skeptically (resp. credulously) accepted.
Example 2. (Continued from 1) First, if we look at the framework F , we can see that both fag and fbg are credulously accepted but not skeptically accepted, under the complete semantics. Let us now look at the probability of credulous and skeptical acceptance under the complete semantics, given the probabilistic framework (F; P ). We have that the set fag is skeptically accepted only given framework state C2 and the set fbg only given framework state C3. Hence Psckoep(fag) = P (C2) = 0:2 and Psckoep(fag) = P (C3) = 0:3. The set fag is credulously accepted in the framework states C2 and C4 and fbg in C3 and C4. Hence Pccroed(fag) = P (C2) + P (C4) = 0:4 and Pccroed(fbg) = P (C3) + P (C4) = 0:5.
We are now equipped to deal with uncertainty of arguments on the abstract level, using the concept of a probabilistic argumentation framework. In the following section, we present an instantiation that generates a probabilistic framework, called p-ASPIC. That is, we de ne a logical language to specify a p-ASPIC theory, which includes rules associated with probabilities, and we specify how we can construct a probabilistic framework from such a theory. After that we return to the issue of evaluating the arguments of a probabilistic framework, where we apply the concepts introduced here, to a probabilistic framework generated by the p-ASPIC system. 3
p-ASPIC: ASPIC with probabilities We present here a generalization of the ASPIC-Lite system that supports rules associated with a probability. We call it p-ASPIC. Let L be a logical language which is closed under negation and let : L ! L be de ned by = , if = : , and = : , otherwise. In the ASPIC-Lite system, knowledge is represented using two types of rules: strict rules, of which the conclusion always holds, if the premises hold, and defeasible rules, of which the conclusion holds defeasibly, if the premises hold. Here, we associate the rules with a probability, and we call them p-rules.
De nition 11. A p-rule is a rule of the form !px , where L is the antecedent and the consequent, x 2 fs; dg the type, and p 2 (0; 1] is the probability of the rule. Given a rule !px we say that it is a strict p-rule if x = s and a defeasible p-rule if x = d.
Throughout this text, we either write !px , where x ranges over any of
the two types of rules, or !sp and !pd , to denote a strict and defeasible
p-rule in particular. Our notation is di erent from that in [
The probability p, associated with a p-rule !px , is interpreted as the probability that the rule is active. As we hinted at before, such probabilities may arise from: { Uncertainty of evidence. It may be uncertain whether or not a fact or rule is a valid constituent of an argument. So the probability that a rule is active is the probability that the rule is valid. { Opponent modeling. If we use a framework to model the knowledge of an opponent (e.g. in the setting of an argumentation game), we may be uncertain about which rules the opponent is aware of. So the probability that the argument is active is the probability that the opponent is aware of the rule.
Note that we assume that there are no rules with zero probability. For a rule that is certain, i.e. has a probability of 1, we omit the probability and simply write !x . A p-theory is a set of p-rules: De nition 12. A p-theory is a set R of p-rules. We denote by D(R) the set of defeasible rules in R and by S(R) the set of strict rules in R.
In the ASPIC-Lite system, a theory is assumed to be closed under
transposition of strict rules and to be consistent. These requirements are necessary to
enforce certain coherence requirements on the output of the ASPIC-Lite system
(we refer to [
s { If f 1; : : : ; ng !p 2 R and 1; : : : ; n 2 ClR(P ) then 2 ClR(P ).
De nition 14. A strict rule 0 !sp 0 is a transposition of a strict rule !sp if and only if 9 2 s.t. 0 = ( n f g) [ f g and 0 = . Let R be a ptheory. The closure of R under transposition of strict rules, denoted T r(R), is the smallest set satisfying: { R { If
T r(R) !sp 2 T r(R) and s 0 2 T r(R). 0 !p De nition 16. Let R be a p-theory. A strict rule r 2 S(R) is redundant in R if and only if T r(R) = T r(R n frg). We say that R is non-redundant if and only if 8r 2 S(R), r is not redundant.
Throughout this paper, we assume that R is consistent and non-redundant and, furthermore, that there are no two rules !px ; !px0 2 R such that p 6= p, i.e. there are no two rules with equivalent antecedents/premises, but a di erent p-value. In the rest of this paper, we use the following running example. Example 3. Consider the p-theory R = fr1; : : : ; r10g with: r1 = ; !s aw r5 = fawg !d aj r9 = faj; bj; cjg !s :dj r2 = ; !s0:5 bw r6 = fbwg !d bj r10 = fcj; djg !s :dr5e r3 = ; !s0:6 cw r7 = fcwg !d cj r4 = ; !s0:7 dw r8 = fdwg !d dj
The example can be interpreted as follows: Four friends, let us call them Anne, Bob, Chris and David have expressed their desire to join you on a road trip. The strict rules r1; : : : ; r4 stand for `Anne wants to join', : : :, `David wants to join'. You are not certain whether Bob, Chris and David want to join. This is expressed by the probabilities associated with r2; r3 and r4. The defeasible rules r5; : : : ; r8 express the principle that if your friends want want to join you then, if possible, they can join you. However, your car seats at most three passengers, which is expressed by the rule r9. (Also consider the transpositions of r9, e.g. faj; bj; djg !s :cj.) Furthermore, both Chris and David are in love with Anne, which could lead to unpleasant situations in the cramped car. You decide, therefore, that if Chris and David both join, then Anne cannot join, which is expressed by r10. (Here, d: : :e stands for the objecti cation operator, which maps defeasible rules to elements of the language. We need this to de ne undercut, which we introduce below.)
An ASPIC-Lite theory can be used to instantiate a Dung-style argumentation framework. This framework is then an abstract representation of the reasoning problem posed by the theory. To do this, the ASPIC system speci es how arguments are constructed and how the attack relation is de ned. Moreover, the probabilities associated with the rules of a p-theory, allow us to de ne a probability distribution over sets of arguments. We can thus specify how to instantiate a probabilistic framework. Before turning to this aspect, we start out with arguments and attacks. An argument is a chaining of rules, where the premises of every rule follow from the conclusions of preceding rules. Apart from notation, we de ne argument generation as usual in the ASPIC-Lite system: De nition 17. An argument a is a pair (B; ), where B is a set of p-rules and 2 L is the conclusion. The set of arguments generated by a p-theory R, denoted ArgsR, is inductively de ned as follows: { If f 1; : : : ; ng !px 2 T r(R) and ( 1; 1); : : : ; ( n; n) 2 ArgsR then ( 1 [ : : : [ n [ ff 1; : : : ; ng !px g; ) 2 ArgsR.
Given a set of arguments A, we denote by R(A) the set of rules appearing in the arguments in A and by S(A) and D(A) the set of strict and defeasible rules, respectively, appearing in A.
Notice that the base case of the inductive de nition of ArgsR is the case where a rule has an empty set of premises.
Example 4. (Continued from 3) The set of arguments generated by the p-theory of example 3 is the set ArgsR = fa1; : : : ; a13g with: a1 = (fr1g; aw) a8 = (fr4; r8g; dj) a2 = (fr2g; bw) a9 = (fr2; r3; r4; r6; r7; r8; r9g; :aj) a3 = (fr3g; cw) a10 = (fr1; r3; r4; r5; r7; r8; r9g; :bj) a4 = (fr4g; dw) a11 = (fr1; r2; r4; r5; r6; r8; r9g; :cj) a5 = (fr1; r5g; aj) a12 = (fr1; r2; r3; r5; r6; r7; r9g; :dj) a6 = (fr2; r6g; bj) a13 = (fr3; r7; r4; r8; r10g; :dr5e) a7 = (fr3; r7g; cj)
Two kinds of attack are de ned by the ASPIC-Lite system. The rst is de ned by a conclusion of the attacking argument contradicting a defeasible conclusion of the attacked argument. This is called rebut. For the second type, called undercut, it is assumed, in the ASPIC-Lite system, that there is an objecti cation operator d: : :e that maps defeasible rules to elements of L. Formally:
say:
Given a p-theory R and the generated arguments ArgsR, we { An argument (B; ) 2 ArgsR rebuts (B0; 0) 2 ArgsR if and only if there is a !pd 2 D(B0) and = . { An argument (B; ) 2 ArgsR undercuts (B0; 0) 2 ArgsR if and only if there is a !pd 2 D(B0) and = :d !pd e.
The attack relation AttR ArgsR ArgsR is de ned by ((B; ); (B0; 0)) 2 AttR if and only if (B; ) rebuts or undercuts (B0; 0).
Example 5. (Continued from 4) Some attacks in our running example are: a9 rebuts a5; a10; a11 and a12; a13 undercuts a10; a11; a12 and a13; and both a11 and a12 rebut a13.
Given a p-theory we can now generate a framework: De nition 19. Given a p-theory R, the framework generated by R is the framework FR = (ArgsR; AttR).
Example 6. (Continued from 4) Figure 1 shows the argumentation framework (ArgsR; AttR) generated by the p-theory of our running example. The nodes represent arguments in ArgsR and are labeled with the argument name and the conclusion. The edges represent the attack relation AttR.
We have now described the process of the generation of a framework, given a p-theory R, as in the ASPIC-Lite system. However, we have not done anything with the probabilities associated with the rules. We turn to this in the following section, where we describe the process of generating a probabilistic framework, given a p-theory R. 4
As we said, we interpret a p-rule !px as saying that the probability that the rule !x is active, is p. That is, the rule is active and can be applied, with a probability of p and it is inactive and cannot be applied, with a probability of 1 p. The uncertainty about the rules used in an argument carry over to uncertainty about whether or not the argument is active. That is, the probability that an argument (B; ) is active, is the probability that all p-rules in B are active. To model this, we de ne the notion of a theory state. A theory state is corresponds to the event that a set of rules is active. If a theory state holds, then all and only the rules in the theory state are active. Next, we de ne a probability distribution over the set of theory states. We can, in principle, use any type of distribution but, for simplicity, we assume here that the events of di erent prules being active, are independent. Thus, the probability of the event that a theory state is active, is easily calculated: De nition 20. Let R be a p-theory. A theory state is a set S R. If ( ! ) 2 T r(S), we say that ( ! ) is active in S, otherwise inactive. The probability of S, denoted pS , is de ned by pS =
Y p
Y ( !px )2S ( !px )2RnS (1 p)
Note that, from the way the probability of a theory state is calculated, it is easy to see that the sum probability of all theory states is 1. Also note that a strict rule !s is active in S if and only if all transpositions of !s are active in S. Apart from transposition, we may consider in de nitions 16 and 20 closure under additional properties. For example, we may require that rules are closed under transitivity so that, in a state, two rules fag !s b; fbg !s c are active if and only if fag !s c is active. However, for the current purpose, we consider only transposition.
Example 7. (Continued from 6) There are three uncertain rules in the p-theory of our running example, namely r2; r3 and r4. Let Rcertain = R n fr2; r3; r4g. Every theory state not containing all rules in Rcertain receives zero probability. Table 1 lists the states that receive non-zero probability: Given the probability distribution over theory states, the probability of a framework state (see de nition 7) is de ned by the sum probability of all theory states that give rise to the framework state.
De nition 21. Let R be a p-theory and (ArgsR; AttR) the framework generated by R. We say that a theory state S gives rise to a framework state C 2Args if and only if C = G(S). The probability distribution over framework states generated by R, denoted PR, is de ned by:
PR(C) =
X fS22RjC=G(S)g pS Proposition 1. Given a p-theory R and the framework (ArgsR; AttR) generated by R, the sum probability of all framework states C ArgsR is 1, i.e. PC ArgsR PR(C) = 1.
Note that, while the events of di erent p-rules being active are be independent, the events of di erent arguments being active, are in general not independent. This is not surprising, because the same p-rule may be used in several arguments.
Furthermore, some framework states may receive zero probability. This happens when no theory state gives rise to the particular framework state. These framework states are exactly those that are not well-formed, in the sense that they are not closed under argument construction : De nition 22. We say that a set of arguments C is closed under argument construction if and only if C = AR(C) Proposition 2. Let R be a p-theory, (ArgsR; AttR) the framework generated by R and C ArgsR. If PR(C) > 0 then C is closed under argument construction.
Note that the right-to-left direction of proposition 2 fails to hold only because of rules r 2 R with probability 1. Then there may be a framework state C that is closed under argument construction but receives zero probability because r 62 R(C). tEhxaatmaprlee g8e.nCeroantesiddebry athpis-tthheeoorryy Rare=: af1;=!(s0f:;5 !a;s0a:5 !ags0;:5a)bagn.dTaw2o=ar(gfu;m!ens0t:5s s a; a !0:5 bg; b). We have that the framework states fa1g and fa1; a2g are closed under argument construction, but fa2g is not. There are four theory states: S1 = ;; S2 = fr1g; s3 = fr2g and S4 = fr1; r2g. S1 gives rise to the framework state ;, S2 to fa1g, S3 to ; and S4 to fa1; a2g. No theory state gives rise to fa2g. Hence it receives zero probability.
We can now generate a probabilistic framework: De nition 23. Given a p-theory R, the probabilistic framework generated by R is the probabilistic framework P FR = (FR; PR).
Example 9. (Continued from 7) Given the p-theory of our running example, the probabilistic framework is (FR; PR), where FR is the framework shown in example 6 and where PR is displayed in table 2. (The column C shows the framework states that receive non-zero probability, the column PR(C) shows the corresponding probabilities and in every row, a checkmark means that the argument is active in the corresponding framework state.) In the previous sections we demonstrated how, given a p-theory, we can generate a probabilistic framework. In this section we conclude our running example and show the result of calculating the probability of skeptical and credulous acceptance of a conclusion from our p-theory.
Example 10. If we take the complete semantics, then the di erent labelings correspond to di erent ways in which the constraints, expressed by the rules of our theory, can be satis ed. Focusing on just Anne, we can answer the following questions: What is the lower bound probability that Anne joins? and What is the upper bound probability that Anne joins?. These questions correspond, respectively, to the probabilities of skeptical and credulous acceptance. Regarding skeptical acceptance, we have that a5 is labeled I in all complete labelings, given all framework states except the framework states C7 and C8. Therefore, this probability is the sum probability of the states C1; : : : ; C6, which is 0:7. (Indeed, Anne does not necessarily join because whenever both Chris and David want to join (p = 0:3), it is not sure that Anne will join.) Regarding credulous acceptance, we have that a5 is labeled I in at least one complete labeling, given all possible framework states. Therefore, this probability is 1. (Indeed Anne can always possibly join; even if Bob, Chris and David want to join, it is possible to refuse either Chris or David, so that Anne can join.) 5
In this paper we presented an initial attempt at combining probability with instantiated and abstract argumentation. Many issues are still open and are left for future work. Here we give an overview.
One limitation of our formalization is that we apply probabilities on a metalevel rather than the object-level. That is, we associate probabilities with rules and not with elements of L. We cannot, for example, express in our formalism that the conclusion of a rule holds with some probability, given that the premises hold. We can say that our formalism is strictly about reasoning under uncertainty, rather than reasoning about uncertainty. Furthermore, it is not very clear, from a conceptual point of view, what it means to associate a probability with a rule. However, as our example shows, it is quite natural to associate probabilities with facts (i.e. strict rules with empty sets of premises). This suggests two directions for future work: one the one hand, we can investigate argumentation about uncertainty and on the other hand, we can focus on the `uncertainty about facts' part, and from there, add expressivity.
It is worth mentioning one important issue, when modeling argumentation about uncertainty. Namely, how do we interpret rules? An obvious candidate for such an interpretation is conditional probability, but then we lose context independence, transitivity (i.e. chaining) and transposition.
Furthermore, we have assumed in this paper that the events of individual rules being active are probabilistically independent. While this assumption reduces the complexity, it leads to serious inaccuracy when modeling many realworld problems. In many domains, one has to deal with phenomena such as common causes and e ects, where di erent events are not probabilistically independent. We have already mentioned that, in principle, we can work with arbitrary probability distributions over theory states. An obvious choice to represent such a distribution is by using Bayesian belief networks. Since both Bayesian belief networks and abstract argumentation frameworks are intuitive and easy to understand graphical representations of knowledge, an interesting question is whether the two can be fused in a meaningful way. 6
Our concept of a probabilistic framework is similar to that of a
probabilistic framework, as introduced in [
Other related work includes [
An interesting approach to probabilistic argumentation, which is not related
to Dung-style argumentation theory, is presented in [
Also mentioned should be work that approaches the problem from another
side, namely by associating numerical weights [
In this paper we extended the model of Dung-style argumentation with uncertainty, both on the abstract level and the instantiated level. On the instantiated level, we presented an extension of the ASPIC argumentation system, which we call p-ASPIC. This system extends ASPIC in a simple manner, by allowing probabilities associated with rules. This probability is interpreted as being associated with the event that the rule is active. To capture this uncertainty on the abstract level, we extended the notion of an argumentation framework to a probabilistic framework. We have showed how to instantiate a probabilistic framework using the p-ASPIC system, and we demonstrated the system with a number of examples. This work should be understood as an initial step in the direction of probabilistic, instantiated, Dung-style argumentation, and we have discussed a number of directions for future work.