In the field of explainable artificial intelligence (XAI), causal models and abstract argumentation frameworks constitute two formal approaches that provide definitions of the notion of explanation. These symbolic approaches rely on logical formalisms to reason by abduction or to search for causalities, from the formal modeling of a problem or a situation. They are designed to satisfy properties that have been established as necessary based on the study of human-human explanations. As a consequence they appear to be particularly interesting for human-machine interactions as well. In this paper, we show the equivalence between a particular type of causal models, that we call argumentative causal graphs (ACG), and abstract argumentation frameworks. We also propose a transformation between these two systems and look at how one definition of an explanation in the argumentation theory is transposed when moving to ACG. To illustrate our proposition, we use a very simplified version of a screening agent for COVID-19.
the performance of the human-agent pair [
consists in drawing inspiration from human cognitive and social mechanisms, in particular the
ones related to the explanation process, and deriving desirable properties and behaviors from
them. In [
A first formal framework is based on the work of Joseph Halpern and Judea Pearl [
tion. Introduced by Phan Minh Dung in 1995 [
After briefly presenting these two frameworks in Sections 2 and 3, we establish an equivalence between them through a transformation allowing us to go from argumentative graphs to argumentative causal graphs and vice versa (Sections 4 and 5, respectively). To the best of our knowledge, there is no work in this direction, which is why we propose transformations to link the two fields, which is the main contribution of the paper. The objective is not to present a new method nor a new framework, but instead to propose a method to move from one to the other and thus to allow for the exploitation of the interesting properties of each framework in the other one.
A causal model as introduced by J. Halpern [
which we propose to call Argumentative Causal Graphs (ACGs). These are triplets = ( , , ℱ ) for which (i) the variables are Boolean variables, and the structural equations are therefore written as logical formulas; (ii) these formulas do not contain disjunctions; (iii) the related graph is acyclic.
Let = ( , , ℱ ) be a causal model. A context, written u, is an assignment of the variables of . The pair (, u) is called a world. Throughout the paper denotes a set of contexts.
Let X be a set of variables of , X = x denotes an assignment of the variables of X with the values of x.
Let u ∈ ,(, u) |= X = x holds if X = x is the unique solution to the structural equations of ℱ in u.
Let X ∈ and x be values of X. We denote by X=x the set of contexts u′ of such that (, u′) |= X = x.
Let u ∈ , (, u) |= [X = x](Y = y) holds if, in the world ( ′, u) defined as (, u) in which the structural equations of ℱ determining the variables of X are replaced by the assignment X = x, it holds that ( ′, u) |= Y = y.
Example 1. (from [
a formula , the assignment X = x is an actual cause of in the world (, u) if the three following conditions are verified: AC1 (, u) |= (X = x) ∧ , i.e. both the cause and the consequence are true in the considered world.
AC2 There exists a set W of endogenous variables with values w and a setting x′ for the variable X such that if (, u) |= (W = w) then
(, u) |= [X = x′, W = w]¬ AC3 X is minimal: there is no strict subset of X that verifies AC1 and AC2. This condition aims to avoid having useless variables in the cause.
Condition AC2 formalizes a counterfactual reasoning and checks whether, if the presumed cause X = x had not occurred (i.e. if X had values x′ ̸= x) and possibly other events had occurred (i.e. W = w), the consequence would still occur.
Example 1. (continued) – Intuitively, one cause of the bottle shattering is the fact that Suzy threw the stone. Indeed, it was her stone that hit the bottle and thus broke it. However, if we ask the question: if Suzy had not thrown her rock, would the bottle have broken? The answer is ‘yes’ because Billy would have hit the bottle ( = ∧ ¬). Therefore the following counterfactual must be considered: if Suzy had not thrown her rock knowing that Billy did not hit the bottle, would the bottle have shattered? In this case, the answer is ‘no’, i.e. the fact that Suzy threw her stone is indeed an actual cause of the bottle shattering.
Let be a set of contexts and u ∈ . The assignment X = x is a suficient cause
world (, u) if the following four conditions are satisfied [
SC2 There exist a part of X, = , and another conjunction (Y = y) (possibly empty) such that ( = ) ∧ (Y = y) is an efective cause of in (, u), i.e. some part of X is part of an actual cause in the considered world.
SC3 (, u′) |= [X = x] for all contexts u′ ∈ , i.e. if X = x then holds regardless of the context.
SC4 X is a minimal set that satisfies SC1, SC2 and SC3.
Remark 1. There exists another version of the definition of suficient cause proposed by T.Miller
in [
is dedicated. This person is called the explainee. For this reason, the search for actual cause and suficient cause is constrained to a set of contexts determined by what the explainee considers as possible.
The assignment X = x is an explanation of relative to the set of contexts if the
following three conditions are verified [
EX2 X is minimal.
EX3 (X=x)∧ ̸= ∅, i.e. at least one of the contexts considered as possible by the explainee is compatible with the explanation.
EX4 (, u′) |= ¬(X = x) for some context u′ ∈ .
The set of contexts is determined by the explainee. Thus, it is possible that there are no suficient causes in any context of (i.e. (X=x)∧ = ∅) and then, there is no possible explanation.
There exists a more general definition of explanation proposed by J. Halpern [
An AAF is a pair = (, ) such that is a finite set of arguments and is a binary relation on × , that is called the attack relation: an argument ∈ attacks ∈ if (, ) ∈ , also denoted (, ). Since is a binary relation with a finite support, an AAF can be represented as a graph.
This formalism does not impose any constraint on the internal structure of an argument, nor
on the nature of an attack: an argument can simply be a statement in natural language. It can
also be a formula defined in a certain language according to rules, as in the case of the ASPIC+
system [
Example 2. This example considers a simple scenario of an agent helping to screen for COVID-19. Let us imagine that a user, Billy, wakes up with some aches and pains. He decides to consult the agent. The agent asks a number of questions about his health. Indeed, having aches and pains is not enough to justify going for a PCR test, a self-test could be enough for example. The agent asks Billy to taste a condiment with a strong taste (salt, sugar, vinegar, etc.) to test whether he lost his sense of taste or not. Finally, the agent also checks whether he has been in close contact with someone who has COVID-19.
Their conversation can be represented by the following abstract argumentation framework, illustrated in Figure 2: = {: “A PCR test is necessary”, : “No symptoms”, : “Vaccinated”, : “Aches and pains”, : “Loss of taste”, : “Close contact”} and = {(, ), (, ), (, ), (, ), (, ), (, )}.
In the case where Billy does not feel that he has any particular symptom or is vaccinated, a PCR test is not necessary. This is represented by the first two attack relations (, ) and (, ). However, if he has aches and pains or a loss of taste, it is no longer possible to say that he does not have symptoms: (, ), (, ). In the same way, if he is a close contact or no longer has any taste, the fact that he is vaccinated no longer justifies not going for a PCR test: (, ), (, ). In particular, being vaccinated does not prevent one from getting COVID-19.
The graph shown in Figure 2 represents the case where Billy has aches and pains, lost taste and is close contact (, , ). According to this graph, is only attacked by non-accepted arguments (because they are attacked by non-attacked arguments) and can therefore be accepted. Thus, a PCR test must be performed.
Let denote the set of direct attackers of for the attack relation :
= { ∈ | (, )} When only one attack relation is defined, the notation is simplified to . A set of arguments is conflict-free if there is no pair (, ) ∈ 2 such that (, ) ∈ : ∀(, ) ∈ 2, (, ) ∈/ .
An argument ∈ is acceptable by a set if attacks all the attackers of : ∀ ∈ , ∃ ∈ ∩ A set of argument is said to be admissible if is conflict-free and any element of is acceptable by :
∀(, ) ∈ 2, (, ) ∈/ and ∀ ∈ , ∀ ∈ , ∃ ∈ ∩ A set of arguments is said to be related admissible if it is admissible and at least one of its arguments is attacked:
is admissible and ∃ ∈ such that ̸= ∅.
Such an argument is referred to as a topic of .
Example 2. (continued) – Let us look for a related admissible set with as a topic. Since is attacked by , must contain an attacker of . Let us take for example: is not attacked so it is acceptable by . Besides, is also attacked by . So we have to add an attacker of to . Let us add for example : is not attacked, so it is also acceptable by . Finally, all attackers of are attacked by an element of , so is acceptable by . We have thus constructed = {, , }. Considering all possibilities, the set of (related) admissible sets is: = {{}, {}, { }, {, }, {, }, {, }, {, , },
{a,d,e,f}, {a,d,f}, {a,e,f}, {a,d,e}, {a,e}}.
tion: let ∈ be an argument of , an explanation of is a related admissible set with as a topic. The explanation is compact if it is minimal for the inclusion relation; it is verbose if it is maximal for the inclusion relation.
Example 2. (continued) – Argument has two compact explanations: “a PCR test is needed” because Billy has “a loss of taste”, i.e. {, }, or because he has “aches and pains” and “is a close contact”, i.e. {, , }. There also exists a verbose explanation: “loss of taste”, “aches and pains” and “contact cases”, i.e. {, , , }.
Remark 2. In the context of abstract argumentation, the objective is not to model the explainee. Instead, an AAF is rather a transcription of an exchange of arguments between several entities. The explanation thus serves to justify why an argument can be accepted by referring to the diferent arguments that are used to defend it: there is no notion of context. In particular, it is assumed that all the arguments and their interactions are known.
Let = (, ) be an AAF and its associated graph which is assumed to be acyclic.
For each argument ∈ , a Boolean variable is created such that = 1 is read as “Argument is accepted”. These variables constitute the set of endogenous variables. Moreover, for any unattacked argument ∈ , another Boolean variable ˜ is created. These variables constitute the set of exogenous variables. Formally, let us define: • = { | ∈ },
˜ • = { | ( ∈ ) ∧ ( = ∅)}, • ℱ = { | ∈ } with: ◇ ∀ ∈ such that ̸= ∅, = ⋀︀
∈ ◇ ∀ ∈ such that = ∅, = ˜.
¬,
The triplet = ( , , ℱ ) is a causal model, acyclic and for which the structural equations in ℱ do not use disjunction. This model is therefore an ACG.
Remark 3. For each unattacked argument, we propose to build two variables, an endogenous one
and an exogenous one. This duplication allows us to choose whether an unattacked argument is
˜
accepted or not through its exogenous representative by initializing it to 0 or 1. Moreover, in
the framework defined by J. Halpern and J. Pearl [
Example 2. (continued) – The application of the proposed transformation leads to the construction of six endogenous variables: = {, , , , , }, and three exogenous variables, corresponding to the three unattacked arguments (, , ): = {˜, ˜, ˜ }. In addition, the attack relations are transformed into structural equations. For example, is attacked by and , so = ¬ ∧ ¬. With these transformations, we obtain the argumentative causal graph displayed in Figure 3.
We call default context of the argumentation the unique context u* such that all exogenous variables are set to 1. It represents the situation described by the argumentative graph in which all unattacked arguments are accepted.
˜
˜ ˜
Proposition 1. Let = (, ) be an abstract argumentation framework whose associated graph is acyclic. Let * ∈ be an argument such that there is an admissible set of which * is the topic. Let be a compact explanation of * . Let = ( , , ℱ ) be the argumentative causal graph built from the transformation described in Section 4.1.
Let us consider: • = (* = 1), • = ∖ {* } and X = { | ∈ }, • a set of contexts that contains the default context u* (u* ∈ ).
Then X = 1 is a non-minimal causal explanation of relative to , i.e. X = 1 satisfies EX1 and EX3 in .
This proposition reintroduces the notion of explainee. Indeed, we create the set of contexts considered by the explainee. We only impose that u* belongs to . This assumption seems reasonable because it is the only context considered when working from a purely argumentative point of view.
Proof 1. Let us prove that X = 1 satisfies EX1 and EX3 in .
(EX3) Let us first show that u* belongs to (X=1)∧ .
By assumption, u* belongs to . (i) Let us prove by contradiction that u* belongs to X=1.
Let us suppose that (, u* ) |= ¬(X = 1). Then, ∃ ∈ X such that = 0. Now ̸= ∅, and as = ( ⋀︀ ¬), ∃ ∈ such that = 1.
∈ However is admissible thus ∃ ∈ ∩ .
If = ∅ then by definition of u* , = 1. It is not possible because = 1. This leads to a contradiction.
Otherwise, as = 1 then = 0. We can then recursively apply the same reasoning to . As the graph is finite and acyclic, then it necessarily leads to a case where the defender ( here) is unattacked which leads to the same contradiction as before.
This shows that indeed u* is included X=1. (ii) Let us prove that u* belongs to .
As is admissible with * as a topic and as the graph is acyclic, ∀ ∈ * , ∃ ∈ ∩ . Now, (, u* ) |= X = 1, so = 1 hence = 0. Thus, ∀ ∈ * , = 0 hence * = ⋀︀ ¬0 = 1.
∈* Therefore u* belongs to .
Thus, u* belongs to (X=1)∧ , so this set is not empty, which shows that EX3 is satisfied. (EX1) Let us prove that X = 1 is a suficient cause in , i.e. it verifies SC1, SC2 and SC3, for all u ∈ (X=1)∧ . Let u ∈ (X=1)∧ .
SC4 is a minimality condition which relates to the suficient cause but which in the case of
explanations is equivalent to EX2 [
2) Let us prove by contradiction that SC3 is satisfied. Let u′ be a context such that (, u′) |= [X = 1]¬ .
As ¬ holds (i.e. * = 0), then according to ℱ for attacked argument (* is a topic of so * ̸= ∅) ∃ ∈ , such that ∈ * and = 1.
Now, is admissible hence ∃ ∈ ∩ . Moreover, the graph is acyclic therefore ̸= * hence ∈ X. As X = 1, it holds especially that = 1 and hence = 0 following . This leads to a contradiction.
3) Finally, let us show that SC2 is verified. To do so, we first build an actual cause of in u and then show that this set does contain at least one element of X.
(i) Let ∈ * and Zb = ⋃︀ { | (, u) |= ( = 1)}.
∈ As u ∈ (X=1)∧ , then (, u) |= * = 1, hence ∈ * leads to = 0.
Now, is admissible so in particular, as ∈ * and * ∈ , ∃ ∈ ∩ . As = 0 it holds that = 1. Thus, ∈ Zb. It follows that Zb is not empty.
Let us set Z = ⋃︀ Zb.
∈
Z = 1 is not the set we are looking for to be an actual cause. Nevertheless, let us show that it satisfies AC1 and AC2 for = ( = 1) in the world (, u):
AC1 is verified by construction of Zb.
AC2: By construction of Z, if we force Z = 0, then it holds that ∀ ∈ * , ∀ ∈ , = 0. Thus, = ⋀︀ ¬ = ⋀︀ ¬0 = 1. Therefore it holds that (, u) |= [Z = 0]¬ so it ∈ ∈ follows that AC2 is satisfied with W = ∅.
Let us note Zm a minimal subset of Z such that (Zm = 1) verifies AC1 and AC2. It is well defined and not empty because (Z = 1) satisfies AC1 and AC2. Moreover, Zm satisfies AC3 by definition of Zm. Therefore, (Zm = 1) is an actual cause of .
(ii) Let us now show that we can build an actual cause (Z′ = z′) of such that Z′ ∩ X ̸= ∅. If Zm ∩ X ̸= ∅ then Z′ = Zm works.
Otherwise, i.e. if Zm ∩ X = ∅, then let ∈ * be an attacker of * : ∃ ∈ Zm such that ∈ , (, u) |= ( = 1) and ∈/ X.
As is admissible and the graph is acyclic, ∃′ ∈ X such that ′ ∈ . Moreover, Zm ∩ X = ∅, hence ′ ∈/ Zm. Finally, as u ∈ we have (, u) |= (′ = 1).
Let Zm′ = (Zm∖{})∪{′ }. Zm′ also verifies AC1 and AC2. As Zm is minimal by construction, then if Zm′ is not minimal, ∃Z′ ⊆ Zm′ such that Z′ ⊈ Zm. However, Zm′ ∖ Zm = {′ } so ′ ∈ Z′ and therefore we have Z′ ∩ X ̸= ∅. Thus, we have built a set Z′ verifying AC1 and AC2, minimal for the inclusion relation (AC3) and such that Z′ ∩ X ̸= ∅. Therefore, Z′ satisfies SC2.
We have proved that whatever u ∈ , X = 1 satisfies SC1, SC2, SC3. Therefore, EX1 is satisfied by X = 1.
We proved that X = 1 satisfies EX1 and EX3, hence X = 1 is an non-minimal causal explanation of . □ Example 2. (continued) – Let us illustrate this proposition with Example 2. (i) We set * = , = {, , }, X = {, } and = ( = 1).
By definition, u* = (˜ = 1, ˜ = 1, ˜ = 1). It follows immediately that (, u* ) |= ( = 1 ∧ = 1 ∧ = 1). In particular, it holds that (, u* ) |= (X = 1). As a result, = 0 and = 0, hence = 1. Therefore, (, u* ) |= . Thus it holds that u* ∈ (X=1)∧ so EX3 is indeed verified. (ii) Let be a set of contexts, and u ∈ (X=1)∧ ((X=1)∧ is not empty as it contains u* ). First, it holds that (, u) |= (X = 1) ∧ since u ∈ (X=1)∧ . Secondly, let u′ ∈ . In (, u′) we have (, u′) |= [X = 1]( = 0 ∧ = 0), hence (, u′) |= [X = 1]( = 1). Finally, we set Y = {} and = {}. As = 0 ∧ Y = 0, it holds that = 1, hence = 0. Thus, (, u) |= [ = 0 ∧ Y = 0]¬ . This shows that ∀u ∈ , X = 1 is a suficient cause of , i.e. EX1 is satisfied.
Thus, we illustrate in this example the fact that a compact explanation in the AAF of Example 2 is indeed an non minimal causal explanation in its associated ACG.
Given an ACG = ( , , ℱ ) and a set of contexts , the proposed transformation builds the AAF (′,′) defined as: ′ = { | ∈ },
For any couple (, ) ∈ 2 of endogenous variables, let Y = ∖ {, }. If for any context u ∈ , (, u) |= [ = 1, Y = 0]( = 0) then (, ) ∈ ′.
Example 2. (continued) – For the case of the argumentative causal graph presented in Figure 3, we have = {, , , , , }, thus we set ′ = {, , , , , }.
Let be a set of contexts. Let u ∈ be a context of . We have = ¬ ∧ ¬. Thus, (, u) |= [ = 1]( = 0) with ∈ {, }.
Therefore (, u) |= [ = 1, Y = 0]( = 0) with ∈ {, } and Y = ∖ {, }. Thus (, ) ∈ ′ and (, ) ∈ ′.
Applying the same reasoning to all structural equations of ℱ leads to {(, ), (, ), (, ), (, )} ∈ ′.
Now let us consider Y = ∖ {, } with ∈ {, , }.
(, u) |= [ = 1, Y = 0]( = 1) holds. Indeed, all structural equations have been replaced by = 0 except for and .
As = 1 and = ¬ ∧ ¬, then = ¬0 ∧ ¬0 = 1.
Therefore (, ) ∈/ ′.
As a result, ′ = {(, ), (, ), (, ), (, ), (, ), (, )}.
Proposition 2. Let = (, ) be an abstract argumentation framework, = ( , , ℱ ) the argumentative causal graph generated from by the transformation presented in Section 4 and ′ = (′, ′) the abstract argumentative framework resulting from the inverse transformation of . Then:
= ′.
Proof 2. Let = (, ) be an AAF, = ( , , ℱ ) be the ACG built from and ′ = (′, ′) the AAF resulting from . = ′ by construction; let us prove that = ′ by double inclusion.
1. Let (, ) ∈ 2 be two arguments of such that (, ). By definition, = ¬ ∧ ( ⋀︀ ¬) and therefore = 0 if = 1. ∈∖{} Thus, for all contexts u, (, u) |= [ = 1, Y = 0]( = 0) with Y = ∖ {, }, therefore (, ) ∈ ′ and ⊆ ′. 2. Let (′, ′) ∈ ′2 be two arguments of ′ such that (′, ′) ∈ ′.
Let Y = ∖ {′ , ′ }. By definition:
(, u) |= [′ = 1, Y = 0](′ = 0) Now = ′, so ∀ ∈ , = ′ . In particular it thus holds that, (, u) |= [′ = 1, Y = 0]( = 0), hence ̸= ∅.
Moreover, = ⋀︀ ¬. If ′ ∈/ then with Y = 0, ′ = ⋀︀ ¬0 = 1, that ∈ ∈ contradicts the hypothesis.
It follows that ′ ∈ and as a result, ′ ⊆ .
We proved that ⊆ ′ and ′ ⊆ , i.e. = ′. □
argumentation frameworks. We also proposed explicit transformations to go from one to the other. This allows us to use all the work already done on both sides and select what we are looking for on each one opening new direction for enriching the representation of interactions: dynamic modeling ofered by AAF and dynamic representation ofered by ACG with the notion of context.
values of the variables as one wishes, and thus ofers a dynamic framework. Moreover, it allows us to take into account the knowledge of the agents. Furthermore, the work of J. Pearl and
as a general definition which in addition provides knowledge of the model to the explainee.
point of view [
interaction situations, which can facilitate their implementation for systems interacting with humans. Thus, one approach could be to dynamically model an interaction with an AAF, to compute a result or an action and then to perform the transformation into ACG in order to generate explanations with the desired properties.
other relations between arguments, beyond the attack one, to model better complex interactions.
order to increase their confidence in AI systems but also to facilitate human-machine interactions.