Standard view on using probability of arguments. In this view, we provide properties for the probability function that ensure that the epistemic extensions coincide with Dung’s definitions for extensions. Key properties include coherence (if A attacks B, then P (A) 1 P (B)) and foundation (if A has no attackers, then P (A) = 1). The advantage of using a probability function instead of Dung’s definitions is that we can also specify the degree to which each argument is believed.
Non-standard view on using probability of arguments. In this view, we consider alternative properties for the probability function. This means that the resulting epistemic extensions may not coincide with Dung’s definitions for extensions. The framework that we present in this paper is appealing theoretically as it provides further insights into semantics for abstract argumentation, and it offers a finer-grained representation of uncertainty in arguments. Perhaps more importantly, our framework for probability functions is appealing practically because we can now model how audiences judge argumentation. Consider for example how a member of the audience of a debate hears arguments and counterarguments, but is unable (or does not want) to express arguments. Here it is natural to consider how that member of the audience considers which arguments she believes, thereby constructing an epistemic extension. More generally, if we want to make computational models of argument where we can capture persuasion, we need to take account of the belief that participants or audiences have in the individual arguments that are posited. We see the non-standard view as being particularly important in addressing this need.
This paper builds on previous work [
An abstract argumentation framework AF is a tuple AF = (Arg; !) where Arg is a set of arguments and ! is a relation ! Arg Arg. For two arguments A; B 2 Arg the relation A ! B means that argument A attacks argument B. For A 2 Arg define AttAF(A) = fB j B ! Ag.
Semantics are given to abstract argumentation frameworks by means of extensions
[
Arguably, the most important property of a semantics is its admissibility. A labelling L is called admissible if and only if for all arguments A 2 Arg 1. if L(A) = out then there is B 2 Arg with L(B) = in and B ! A, and 2. if L(A) = in then L(B) = out for all B 2 Arg with B ! A, and it is called complete if, additionally, it satisfies 3. if L(A) = undec then there is no B 2 Arg with B ! A and L(B) = in and there is a B0 2 Arg with B0 ! A and L(B0) 6= out.
The intuition behind admissibility is that an argument can only be accepted if there are no attackers that are accepted and if an argument is not accepted then there has to be some reasonable grounds. The idea behind the completeness property is that the status of an argument is only undec if it cannot be classified as in or out. Different types of classical semantics can be phrased by imposing further constraints. Let AF = (Arg; !) be an abstract argumentation framework and L : Arg ! fin; out; undecg a complete labelling. Then – L is grounded if and only if in(L) is minimal, – L is preferred if and only if in(L) is maximal, – L is stable if and only if undec(L) = ;, and – L is semi-stable if and only if undec(L) is minimal.
All statements on minimality/maximality are meant to be with respect to set inclusion.
Note that a grounded labelling is uniquely determined and always exists [
We now go beyond classical three-valued semantics of abstract argumentation and turn
to probabilistic interpretations of the status of arguments. Let 2X denote the power set
of a set X . A probability function P on some finite set X is a function P : 2X ! [0; 1]
with PX X P (X) = 1. Let P : 2Arg ! [0; 1] be a probability function on Arg. We
abbreviate P (A) = PA2E Arg P (E). This means that the probability of an argument
is the sum of the probabilities of all sets of arguments that contain that argument. The
following definition is a generalization of the notion of epistemic extensions given in
[
An epistemic labelling can be used to give either a standard semantics (as we will investigate in Section 4) or a non-standard semantics (see Section 5).
To further illustrate the epistemic extensions, consider the graph given in Figure 1. Here, we may believe that, say, A is valid and that B and C are not valid. In which case, with this extra epistemic information about the arguments, we can resolve the conflict and so take the set fAg as the “epistemic” extension. In contrast, there is only one admissible set which is the empty set. So by having this extra epistemic information, we get a more informed extension (in the sense that it has harnessed the extra information in a sensible way).
B = Bob will go to the party and this means that Chris will not go to the party
A = Ann will go to the party and this means that Bob will not go to the party
C = Chris will go to the party and this means that
Ann will not go to the party
In general, we want epistemic extensions to allow us to better model the audience of argumentation. Consider, for example, when a member of the audience of a TV debate listens to the debate at home, she can produce the abstract argument graph based on the arguments and counterarguments exchanged. Then she can identify a probability function to represent the belief she has in each of the arguments. So she may disbelieve some of the arguments based on what she knows about the topic. Furthermore, she may disbelieve some of the arguments that are unattacked. As an extreme, she is at liberty of completely disbelieving all of the arguments (so to assign probability 0 to all of them). If we want to model audiences, where the audience either does not want to or is unable to add counterarguments to an argument graph being constructed in some form of argumentation, we need to take the beliefs of the audience into account, and we need to consider which arguments they believe or disbelieve. 4
We now consider some constraints on the probability function which may take different aspects of the structure of the argument graph into account. We will show how these constraints are consistent with Dung’s notions of admissibility.
For the remainder of this paper let AF = (Arg; !) be an abstract argumentation
framework and P : 2Arg ! [0; 1]. Consider the following properties (note that COH is
from [
FOU P is founded wrt. AF if P (A) = 1 for every A 2 Arg with AttAF(A) = ;. SOPT P is semi-optimistic wrt. AF if P (A) 1 PB2AttAF(A) P (B) for every A 2
Arg with AttAF(A) 6= ;.
OPT P is optimistic wrt. AF if P (A) 1 PB2AttAF(A) P (B) for every A 2 Arg. JUS P is justifiable wrt. AF if P is coherent and optimistic.
TER P is ternary wrt. AF if P (A) 2 f0; 0:5; 1g for every A 2 Arg.
COH ensures that if argument A attacks argument B, then the degree to which A is believed caps the degree to which B can be believed; SFOU ensures that if an argument is not attacked, then the argument is not disbelieved (i. e. P (A) 0:5); FOU ensures that if an argument is not attacked, then the argument is believed without doubt (i. e. P (A) = 1); SOPT ensures that the belief in A is bounded from below if the belief in its attackers is not high; OPT ensures that if an argument is not attacked, then the argument is believed without doubt (i. e. P (A) = 1) and that the belief in A is bounded from below if the belief in its attackers is not high; JUS combines COH and OPT to provide bounds on the belief in an argument based on the belief in its attackers and attackees; and TER ensures that the probability assignment is a three-valued assignment. Example 1. Consider AF = (Arg; !) depicted in Fig. 2 and the probability functions depicted in Table 1 (note that the probability functions there are only partially defined by giving the probabilities of arguments). The following observations can be made: 1.) P1 is semi-founded, founded, but neither coherent, optimistic, semi-optimistic, ternary, nor justifiable, 2.) P2 is coherent and semi-optimistic, but neither semi-founded, founded, optimistic, ternary, nor justifiable, 3.) P3 is coherent, semi-optimistic, semi-founded, founded, optimistic, and justifiable, but not ternary, 4.) P4 is semi-founded, founded, optimistic, and semi-optimistic, but neither coherent, justifiable, nor ternary, and 5.) P5 is coherent, semi-founded, semi-optimistic, and ternary but neither optimistic, justifiable, nor founded. Let P be the set of all probability functions, P(AF) be the set of all probability functions on Arg, and Pt(AF) be the set of all t-probability functions with t 2 fCOH,SFOU,FOU, OPT,SOPT,JUS,TERg. We obtain the following relationships between the different classes of probability functions.
A1
A2
A3 A5
A6 Proposition 1. Let AF = (Arg; !) be an abstract argumentation framework.
For the proof of item 1.) of the above proposition see [
For all probability functions P such that LP is admissible in the classical sense, we have that P assigns some degree of belief to each argument that is unattacked, thereby P satisfies the SFOU constraint.
Proposition 2. For all probability functions P , if LP is admissible then P 2 PSFOU(AF). We can further constrain a probability assignment, so that the epistemic labelling straightforwardly captures the standard semantics (i. e. Dung’s semantics). By setting the probability function appropriately, its epistemic labelling corresponds to grounded, complete, stable, preferred, or semi-stable labellings. All we require is a three-valued probability function that simulates each complete labelling function. For this, we provide the following definition that provides the counter-part in our framework for a complete labelling.
Definition 1. Let AF = (Arg; !) be an argumentation framework. Then a complete probability function P 2 P(AF) for AF is a probability function P such that for every A 2 Arg the following conditions hold: 1. P 2 PTER(AF); 2. if P (A) = 1 then P (B) = 0 for all B 2 Arg with B ! A; 3. if P (B) = 0 for all B with B ! A then P (A) = 1; 4. if P (A) = 0 then there is B 2 Arg with P (B) = 1 and B ! A; 5. if P (B) = 1 for some B with B ! A then P (A) = 0.
Note that the above definition straightforwardly follows the definition of completeness for classical semantics. Therefore, we have that P is a complete probability function if and only if there is a complete labeling L and P L.
Completeness of probability functions can be characterized by the aforementioned properties as follows.
Proposition 3. For an argument graph AF, and for P 2 P(AF), P is a complete probability function iff P 2 PCOH(AF) \ PFOU(AF) \ PTER(AF).
In the same way that Caminada and Gabbay [
Proposition 4. Let AF = (Arg; !) be an abstract argumentation framework and P 2 P(AF). If P is a complete probability function for AF and the restriction specified in Table 2 holds for P , then the corresponding type of epistemic labelling is obtained.
Restriction on probability function P
No restriction No arguments A such that P (A) = 0:5 Maximal no. of A such that P (A) = 1 Maximal no. of A such that P (A) = 0 Maximal no. of A such that P (A) = 0:5 Minimal no. of A such that P (A) = 1 Minimal no. of A such that P (A) = 0 Minimal no. of A such that P (A) = 0:5
Classical semantics complete extensions
stable preferred preferred grounded grounded grounded semi-stable For an argumentation framework AF we can identify specific probability functions in P 2 PJUS(AF) that are congruent with admissible labellings, grounded labellings, or stable labellings, for AF as follows.
Proposition 5. (From [
P 2 arg minQ2PJUS(Arg) H(Q) with L 1. For every admissible L there is P 2 PJUS(AF) with L P . 2. Let L be the grounded labelling and let3 P = arg maxQ2PJUS(Arg) H(Q). Then
L P . 3. Let stable labellings exist for AF and let L be a stable labelling. Then there is P .
So Proposition 4 and Proposition 5 provide two ways to identify probability functions that capture specific types of labellings. Each of these results show that standard notions of classical semantics (e. g. admissibility) can be captured using probability functions.
The next result shows that using probability functions to capture labellings gives a finer-grained formalization of classical semantics.
Proposition 6. For each complete labelling L, if there is A such that L(A) 6= undec, then there are infinitely many probability functions P such that LP = L. Obviously, for every probability function P , there is by definition exactly one epistemic labelling LP . This means that using a probability function to identify which arguments are in, undec, or out, subsumes using labels. Furthermore, the probability function captures more information about the arguments. The granularity can differentiate between 3 Define the entropy H(P ) of P as H(P ) =
PE Arg P (E) log P (E) for example a situation where A is believed (i. e. it is in) with certainty by P (A) = 1 from a situation where A is only just believed (i. e. it is only just in) for example by P (A) = 0:51. Similarly, we can differentiate a situation where an attack by B on A is undoubted when P (B) = 1 and P (A) = 0 from a situation where an attack by B on A is very much doubted when for example P (B) = 0:55 and P (A) = 0:45.
In conclusion, we have shown how axioms can be used to constrain the probability function, and thereby constrain the epistemic labelings and the epistemic extensions. This allows us to subsume Dung’s notions of extensions as epistemic extensions. Furthermore, we get a finer-grained representation of the labelling of arguments by representing the belief in each of the arguments. 5
Before exploring the notion of non-standard epistemic extensions, we will augment the set of properties we introduced in the previous section with the following properties. Let AF = (Arg; !) be an abstract argumentation framework and P : 2Arg ! [0; 1]. RAT P is rational wrt. AF if for every A; B 2 Arg, if A ! B then P (A) > 0:5 implies P (B) 0:5.
NEU P is neutral wrt. AF if P (A) = 0:5 for every A 2 Arg.
INV P is involutary wrt. AF if for every A; B 2 Arg, if A ! B, then P (A) = 1 P (B).
MAX P is maximal wrt. AF if P (A) = 1 for every A 2 Arg.
MIN P is minimal wrt. AF if P (A) = 0 for every A 2 Arg.
We explain these constraints as follows: RAT ensures that if argument A attacks argument B, and A is believed (i. e. P (A) > 0:5), then B is not believed (i. e. P (B) 0:5); NEU ensures that all arguments are neither believed nor disbelieved (i. e. P (A) = 0:5 for all arguments); INV ensures that if argument A attacks argument B, then the belief in A is the inverse of the belief in B; MAX ensures that all arguments are completely believed; and MIN ensures that all arguments are completely disbelieved. Example 2. We continue Example 1, the framework AF = (Arg; !) depicted in Fig. 2, and the probability functions depicted in Table 1. The following observations can be made: 1.) P2 and P3 are rational but neither neutral, involutary, maximal, nor minimal, 2.) P1 and P4 are neither rational, neutral, involutary, maximal, nor minimal, and 3.) P5 is rational, neutral, and involutary but neither maximal nor minimal.
As before let Pt(AF) be the set of all t-probability functions with t 2 fCOH,SFOU, FOU, SOPT, OPT, JUS, TER, RAT, NEU, INV, MAX, MINg. We extend the classification from Proposition 1 as follows.
Proposition 7. Let AF = (Arg; !) be an abstract argumentation framework.
PSOPT(AF) PTER(AF) PCOH(AF) PINV(AF) PNEU(AF) ;
PSFOU(AF) PFOU(AF)
POPT(AF) PMIN(AF) PJUS(AF) PMAX(AF) Together with Examples 1 and 2 we obtain the strict classification of classes of probability functions as depicted in Figure 3.
The RAT constraint is a weaker version of the COH constraint, and it can be used to capture each admissible labelling as a probability function.
Proposition 8. If L is an admissible labelling, then there is a P 2 PRAT(AF) such that L P .
Furthermore, the epistemic labelling corresponding to each probability function that satisfies the RAT property is conflict-free.
Proposition 9. Let AF = (Arg; !) be an abstract argumentation framework. For each P 2 PRAT(AF ), in(LP ) is a conflict-free set of arguments in AF.
When the argument graph has odd cycles, there is no probability function that is involutary, apart from a neutral probability function.
Proposition 10. Let AF = (Arg; !) be an abstract argumentation framework. If AF contains an odd cycle (i. e. there is a sequence of attacks A1 ! A2 ! :::::: ! Ak where A1 = Ak and k is an even number), and P 2 PINV(AF) then P 2 PNEU(AF). Even when the graph is acyclic, it may be the case that there is no involutary probability function (apart from the neutral probability function). Consider for example an argument graph containing three arguments A, B and C, with A attacking both B and C, and B attacking C. For this, there is no involutary probability function (apart from the neutral probability function). If we restrict consideration to trees, then we are guaranteed to have a probability function that is involutary and not neutral. But even here there are constraints as captured in the next result.
Proposition 11. If P 2 PINV(AF), then for all Bi; Bj 2 AttAF(A) we have P (Bi) = P (Bj ).
When P 2 PMAX(AF), the probability function does not take the structure of the graph into account. Hence, there is an incompatibility between a probability function being maximal and a probability function being either rational or coherent. However, there is compatibility between a probability function being maximal and a probability function being founded since each P 2 PMAX(AF) is in PFOU(AF).
Proposition 12. Let AF = (Arg; !) be an abstract argumentation framework. If there are A; B 2 Arg such that A ! B, then PRAT(AF ) \ PMAX(AF ) = ;. In conclusion, we have identified epistemic extensions that are obtained from rational probability functions as being an appealing alternative to extensions obtained by Dung’s definitions. Rational probability functions are more general than coherent probability functions, and allow the audience more flexibility in expressing their beliefs whilst taking the structure of the argument graph into account. We have also considered alternatives such as the involutary probability functions but these are over-constrained. 6
Assigning a probability value to an argument can be useful for a variety of purposes such as representing the belief that the premises of the argument are valid, or the belief that the claim is valid given that the premises are valid, or the belief that both the premises and claim are valid, or the belief that the argument should appear in the argument graph, etc. However, given an argument graph, it may be difficult for a user to assign a value to every argument. The user might have knowledge in order to identify a value for some arguments, but the user may be unable or unwilling to make assignments to the remaining arguments. This means that the user can only provide a partial assignment. If this is the case, then it would be desirable to have techniques to handle this incomplete information. Ideally, we would like to identify an appropriate assignment for all the arguments based on the assignment to the subset of arguments.
More specifically, a partial function : Arg ! [0; 1] on Arg is called a partial
probability assignment. A probability function P 2 P(Arg) is -compliant if for every
A 2 dom we have (A) = P (A). Let P (AF) P(AF) be the set of all -compliant
probability functions. The question that arises is that given an abstract argumentation
framework AF = (Arg; !) and a partial probability assignment , how do we
determine P 2 P(Arg) that is most compatible with both AF and , i. e., which P 2 P(Arg)
do we select as a meaningful representative? This question has also been addressed in
similar ways for partial probabilistic information without argumentation, cf. e. g. [
Proposition 13. Let AF = (Arg; !) be an abstract argumentation framework. 1. The sets P(AF), PCOH(AF), PTER(AF), PNEU(AF), PINV(AF), PSFOU(AF), PFOU(AF),
POPT(AF), PSOPT(AF), PJUS(AF), PMIN(AF), PMAX(AF) are convex and closed. 2. The set PRAT(AF) is not convex but closed.
The above proposition suggests that most of our properties are suitable for convex optimization problems such as maximum entropy estimation. Moreover, the same is true for the notion of partial probability assignments.
Proposition 14. For every partial probability assignment the set P (AF) is convex and closed.
Let t be any one of our properties which lead to a convex and closed set of
probability functions (or any combination of those). If it is the case that there is at least one
-compliant P in Pt(AF) then (thanks to the convexity properties) we have that the
intersection of P (AF) and Pt(AF) is convex and closed as well, cf. [
B (a)
A
A B
C (b) (c)
B C some examples to illustrate the definitions and to investigate some of our concerns in dealing with partial assignments.
Example 3. For the argumentation framework depicted in Figure 4(a) consider 1 with 1(A) = 1. Obviously, the most reasonable choice for a 1-compliant P 2 P(AF) would be P (A) = 1 and P (B) = 0 (by obeying the property of involution). Furthermore, for 2(B) = 0:3 we would have P (A) = 0:7 and P (B) = 0:3 following the same rationale.
For the argumentation framework depicted in Figure 4(b) consider 3 with 3(C) = 0:4. A possible choice for P would be P (A) = 0:6, P (B) = 0:4, and P (C) = 0:4 (having thus a maximally committed function that is coherent). But note that the set P (AF) \ PCOH(AF) does contain more than this single probability function. Furthermore, for 4 with 4(B) = 0:7 and 4(C) = 0:6 one would only guess P (A) 0:3 but due to the “inconsistency” of 4 (violating the coherence condition), what is the best choice?
For the argumentation framework depicted in Figure 4(c) consider 5 with 5(A) = 0:4 and the following four selections P1; P2; P3; P4 2 P(AF):
P1(A) = 0:4 P1(B) = 0:6 P1(C) = 0:4
P2(A) = 0:4 P2(B) = 0:4 P2(C) = 0:6
P3(A) = 0:4 P3(B) = 0:5 P3(C) = 0:5
P4(A) = 0:4 P4(B) = 0:2 P4(C) = 0:3 All of the above probability functions are 5-compliant and coherent. Function P4 is not maximally committed and as such is perhaps not a good choice. Both P1 and P2 are “extreme points of view” and model some kind of probabilistic stable semantics. The function P3 is as unbiased as possible but still “reasonable” as it models probabilistic grounded semantics. Note that P3 is also the probability function with maximal entropy in P (AF) \ PCOH(AF).
Given P (AF) and Pt(AF), we can either select P 2 P (AF) that is “as close as possible to” Pt(AF) or P 2 Pt(AF) that is “as close as possible to” P (AF). In future work, we will investigate definitions for “as close as possible to”, and we will explore the pros and cons of each of these alternatives for selecting P . 7
In this paper, we have investigated the use of a probability function to represent belief in an argument. We use this to identify an epistemic labelling, and thereby an epistemic extension. We have considered various constraints on the probability function leading to two views on using the probability functions, namely the standard view, and the nonstandard view. We applied our classification of properties of probabilistic argumentation to the problem of completing partial probability assignments. A first investigation leads us to believe that maximizing entropy within probability functions of a specific type gives appropriate results for this problem. In future work, we will investigate this issue in more depth.
The work in this paper contrasts with other research on introducing a probability
assignment to arguments such as [