Defeasible argumentation has advanced as a solid theoretical research discipline for inference under uncertainty. Scholars have predominantly focused on the construction of argument-based models for demonstrating non-monotonic reasoning adopting the notions of arguments and con icts. However, they have marginally attempted to examine the degree of explainability that this approach can o er to explain inferences to humans in real-world applications. Model explanations are extremely important in areas such as medical diagnosis because they can increase human trustworthiness towards automatic inferences. In this research, the inferential processes of defeasible argumentation and non-monotonic fuzzy reasoning are meticulously described, exploited and qualitatively compared. A number of properties have been selected for such a comparison including understandability, simulatability, algorithmic transparency, post-hoc interpretability, computational complexity and extensibility. Findings show how defeasible argumentation can lead to the construction of inferential non-monotonic models with a higher degree of explainability compared to those built with fuzzy reasoning.
Knowledge-driven approaches have been extensively used in the eld of Arti
cial Intelligence (AI) for producing inferential models of reasoning. Among them,
fuzzy reasoning [
The remainder of this paper is organised as it follows: Section 2 rstly outlines defeasible argumentation and non-monotonic fuzzy reasoning. Secondly, it introduces related work on Explainable Arti cial Intelligence (XAI) presenting a number of properties useful for assessing model explainability. The design of a comparative research study and the inferential processes of defeasible argumentation and non-monotonic fuzzy reasoning are detailed in Section 3. Section 4 provides a qualitative comparison of the selected properties followed by a discussion, while Section 5 concludes the research study. 2
Defeasible (non-monotonic) reasoning has emerged as a solid theoretical
approach within AI for modeling non-monotonic activities under fragmented,
ambiguous and con icting knowledge. In a non-monotonic reasoning process,
conclusions do not necessarily increase monotonically, but instead they can be
withdrawn as new information arises [
Another type of non-monotonic reasoning can be achieved by employing fuzzy
logic and reasoning. This allows the creation of computational models with a
robust representation of linguistic information provided by domain experts by
employing the notion of degree of truth. Fuzzy reasoning consists of a fuzzi
cation module, responsible for assigning to each proposition or linguistic fuzzy
term, provided by an expert, a degree of truth; an inference engine accountable
for ring rules and aggregating fuzzy terms; and a defuzzi cation module, which
translates this aggregation using the original natural language employed in the
underlying reasoning [
Previous studies have attempted to analyse the inferential capacity of
defeasible argumentation in the context of other approaches of quantitative reasoning
under uncertainty [17{19]. However, so far, such analysis has been brought
forward only by means of predictive accuracy. It has been demonstrated that the
evaluation of predictive accuracy alone might not be su cient for a model to
be employed and trusted by domain experts. For instance, in [
In order to investigate the explanatory capacity provided by defeasible
argumentation and non-monotonic fuzzy reasoning, a knowledge-base was selected and
operationalized employing two mechanisms for non-monotonic reasoning:
defeasible argumentation and non-monotonic fuzzy reasoning. This knowledge-base
was produced by a clinician. The reasoning models built upon it aimed at
predicting the risk of mortality in elderly individuals by using information related
to their biomarkers. The rst inferential approach, defeasible argumentation, is
structured over 5 layers as in [
Non-monotonic fuzzy reasoning 1. Fuzzi cation 2. Inference engine 3. Defuzzi cation
Inference
Data of one individual Set of knowledge-base rules
Selection of activated rules Defeasible argumentation 1. Structure of arguments 2. Con icts of arguments 3. Evaluation of con icts 4. Dialectical status 5. Accrual of arguments
Inference Comparison of prop
erties (Table 1) Fifty-one biomarkers were described by a clinician and their association with mortality risk levels was provided through the use of `IF premises THEN risklevel '. Some biomarkers were described by natural language terms such as low or high. This applies also to risk levels (no, low, medium, high and extremely high). Numerical ranges had to be de ned for these terms and were used in di erent ways within the defeasible argumentation and fuzzy reasoning approaches. Contradictions among biomarkers were also made explicit as rules of the form `IF premises THEN conclusion'. Eventually, some preferences among biomarkers were provided. A contradiction refers to a situation in which some biomarker should not be logically employed, while a preference occurs when a biomarker should be used instead of another biomarker. Since the full knowledge-base contains many rules, contradictions and preferences, it cannot be presented in this paper but it can be accessed online1. A dataset2 was obtained in a primary health care European hospital and the survival status of the 93 patients was recorded 5 years after data collection. One random individual was picked for a detailed analysis and the associated data can be seen in Table 2. From this information, a set of rules, contradictions and preferences was activated as shown in Table 3. Note that rules, contradictions and preferences activation depend on the patient's data. A rule designed for female will not be activated for males. A contradiction is not evaluated if its premises or conclusion are not activated. Similarly, a preference is evaluated if both its terms are activated. Fuzzi cation module Rules in the form \IF ... THEN ..." and contradictions rules were constructed from data in Table 3 and depicted in Fig. 2-A on page 7. Afterwards, fuzzy membership functions (FMF) were de ned for linguistic variables such as BMI low (low body mass index) and FE high (high serum iron). Each category of risk had an associated FMF (Fig. 2-B) with input in the range [0, 100] 2 R. Because of that the input variables (biomarkers) had to be normalised for the same range according to their possible minimum and maximum values. 1 http://dx.doi.org/10.6084/m9. gshare.7028480 2 https://doi.org/10.6084/m9. gshare.7028516.v1
N ec(A) = min(N ec(A); :N ec(Q1); : : : ; :N ec(Qn)) (1) where :N ec(Q) = 1 N ec(Q). In addition, an order of precedence has to be de ned when applying equation 1. In this study, contrarily to usual fuzzy control systems, the reasoning is done in a single step with all the activated rules red at once. Nonetheless, it is possible to organise exceptions in a tree structure in which the consequent of an exception is the antecedent of the next exception. Fig. 2-E illustrates this structure which allows equation 1 to be applied from the roots to the leaves. The updated truth values of those rules subject to refutation by other rules are listed in Fig. 2-F. The last step of the inference engine is to aggregate all the truth values of the membership functions associated to each risk category (grouped by the same category), by using the fuzzy-OR operator (as per gure 2-G). The output of this can be graphically represented (Fig. 2-H).
Defuzzi cation module A single defuzzi ed scalar which represents the nal mortality risk inferred has to be computed. Two common methods are selected: mean of max and centroid. The former returns the average of all x coordinates (mortality risks) whose respective y coordinates (membership grades) are maximum in the graphical representation (Fig. 2-H). The latter returns the coordinates of the centre of gravity of the same graphical representation (the x coordinate is the nal scalar). Fig. 2-I lists all the nal inferences produced for the patient under analysis. 3 Given propositions a; b, then fuzzy-and and fuzzy-or are \min(a,b)", \max(a,b)". 4 Product's fuzzy-and and fuzzy-or are respectively \a b" and \a + b - a b". 5 Lukasiewicz's fuzzy-and and fuzzy-or are \max(a + b - 1, 0)" and \min(a + b, 1)". (I) Defuzzification of graphical representations (H) and final inference Defuzzi cation
Centroid Mean of max
Zadeh Lukasiewicz Product (54.12, 0.31) (51.10, 0.32) (51.77, 0.31)
56.25 56.25 56.25 3.3
Layer 1 - De nition of the internal structure of arguments The rst step of a defeasible argumentation process is to de ne a set of arguments. Internally these are generally composed by a set of premises and a conclusion derivable by applying an inference rule !. A typical version of this is known as forecast argument in which, from a set of premises, a conclusion can be reasonably forecasted. Examples can be found in Table 3 (left) where premises reasonably forecast a degree of risk of mortality (as also listed in Fig. 3-A). Note that, in contrast to fuzzy rules, the natural language linguistic terms associated to the premises are not quantitatively exploited. Instead, the premises are evaluated true or not if input values are within certain ranges.
Layer 2 - De nition of the con icts of arguments Given a set of forecast
arguments, the next step for modelling an underlying knowledge-base, is to de ne
the con icts between arguments. The goal is to evaluate potential inconsistencies
and identify invalid arguments through the notion of attack (con ict). In this
research, the notion of undercutting attack [
Layer 3 - Evaluation of the con icts of arguments After con icts
formalisation, these can be evaluated using di erent approaches such as considering
the strength of attacks or the notion of preferentiality of arguments [
Layer 4 - De nition of the dialectical status of arguments Given an
argumentation framework and a notion of con ict, it is necessary to de ne the set
of defeated arguments. An argument A is defeated by B if there is a valid attack
from A to B. A well-known approach has been proposed by [
Layer 5 - Accrual of acceptable arguments Having a set of acceptable forecast arguments, it is necessary to accrue them in case a nal inference is required. If no quantity can be associated to an argument, then the conclusion supported by the highest number of arguments could be chosen as nal inference. In case arguments can be quantitatively evaluated (they carry a value as in this study), then several approaches can be used, including the selection of measures of central tendency such as average (used in this study). Fig. 3-F illustrates the value associated to each argument and the nal inference which is their average. (A) Forecast arguments from activated
IF-THEN rules Activated rules
A comparative qualitative analysis of the explanatory capacity of the defeasible argumentation and non-monotonic fuzzy reasoning processes is performed by using the properties listed in Table 1 (Section 2).
{ Non-monotonic fuzzy reasoning
The inferential process is aligned to the
expert's knowledge and natural language for most of its parts, which makes it
generally intuitively understandable by humans. However, this does not apply
for some parts, such as the normalisation of the input values, the selection of
fuzzy logic and the defuzzi cation mechanism. Some mathematical reasoning
is required to select suitable parameters of these parts.
{ Defeasible argumentation - The initial reasoning steps (layers 1-3) are built
upon the same natural language terms provided by the domain expert in the
knowledge-base. In layer 4 the grounded semantics was selected. This
particular semantics is not a complex algorithm to understand: intuitively, an
argument is only rejected if it is attacked by an accepted argument. In layer
5, the accrual of accepted arguments can be done by an intuitive measure
of central tendency (average here). In case more complex (less intuitive)
semantics, such as preferred [
Computational complexity
{ Non-monotonic fuzzy reasoning - The full inferential process, in the worst
case, is linear in the number of rules.
{ Defeasible argumentation - Layers 3 and 5 are linear in the number of
arguments and attacks relations. However, for layer 4 (application of
acceptability semantics for the computation of the dialectical status of arguments),
complexity can range from linear (example the grounded semantics) to
exponential (example the preferred semantics) [
Algorithmic transparency { Non-monotonic fuzzy reasoning - The inferential process can be applied across di erent domains. A knowledge-base is a formalisation of a reasoning activity for a speci c underlying domain, thus it can be re-used or extended provided the new domains are similar. However, it is important to highlight that traditional fuzzy reasoning has not been designed for application in those domains requiring non-monotonic reasoning. In fact, in this study, the traditional fuzzy reasoning process has been extended through the incorporation of Possibility Theory in order to deal with non-monotonicity. { Defeasible argumentation - The inferential process can be applied across di erent domains. By nature, defeasible argumentation is suitable for application in domains requiring non-monotonic reasoning activities. However, in the absence of con icts, the inferential process can still be applied as it is. The analysis of the two reasoning approaches suggests that defeasible argumentation might lead to explanations that are more suitable to understand for humans, both for a domain expert and a lay person. In fact, through the comparison performed above, on one hand, without some comprehension of fuzzy logic and its membership functions, the understandability/post-hoc interpretability, simulatability of non-monotonic fuzzy reasoning and the extendibility of its models is compromised. On the other hand, defeasible argumentation tends to use the same natural language terms, provided by the domain expert, throughout the whole inferential process, except in the con ict resolution layer (semantics). Semantics vary in computational complexity (linear or exponential in the number of arguments), allowing fuzzy reasoning to o er an equal or lower complexity, since its fuzzi cation-engine-defuzzi cation layers are always linear in the number of rules. However, Possibility Theory always requires the speci cation of a precedence order of exceptions in the inference engine of fuzzy reasoning. Contrarily to acceptability semantics that do not require any precedence order of attacks for solving con icts, thus it has a higher algorithmic transparency. 5
Despite theoretical advances in defeasible argumentation, to the best of our knowledge, there is lack of research devoted to the examination of the degree of explainability that this reasoning approach can o er to illustrate inferences to humans in real-world applications. Therefore, this research focused on a qualitative comparison of the degree of explainability of defeasible argumentation and non-monotonic fuzzy reasoning in a real-world setting: prediction of mortality of elderly people by using biomarkers. The inferential processes behind the two selected reasoning techniques were meticulously illustrated and exploited. The comparison was performed using six properties for explainability extracted from the literature. A qualitative discussion of these properties show how defeasible argumentation has a greater potential for tackling the problem of explainability of reasoning activities under uncertainty, partial and con ictual information. The contribution of this study is to situate defeasible argumentation among similar approaches for reasoning under uncertainty in terms of degree of explainability.
Lucas Middeldorf Rizzo would like to thank CNPq (Conselho Nacional de Desenvolvimento Cient co e Tecnologico) for his Science Without Borders scholarship, proc n. 232822/2014-0.