# Symbolic AI for XAI: Evaluating LFIT Inductive Programming for Explaining Biases in Machine Learning

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## Abstract

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## 1. Introduction

**LFIT**) [21] is one of the most promising approaches of ILP.

**LFIT**induces a logical representation of dynamical complex systems by observing their behaviour as a black box under some circumstances. This logic version can be considered as a white-box digital twin of the system under consideration. The most general of

**LFIT**algorithms is

**GULA**(general usage LFIT algorithm).

**PRIDE**is an approximation to

**GULA**with polynomial performance. These approaches will be introduced in depth in the following sections.

**LFIT**belongs to this kind of approach.

**LFIT**additionally guarantees that the set of conditions of each propositional clause (rule) is minimal. These two guarantees informally mean that the complexity of the theory learned by

**LFIT**depends exclusively on the complexity of the observed data.

**LFIT**is an inductive propositional logic programming model. It is a well-known fact (and was previously mentioned) that propositional logic theories end up being less readable than others, such as those of first-order logic.

**LFIT**in different scenarios. However, it is out of our scope to compare

**LFIT**with other numerical/statistical rules-based learners, and also to try to increase the readability of

**LFIT**results in comparison with, for example, first-order logic. We plan to try to face these pending questions in future experiments.

**PRIDE**of a given black-box classifier.

- We have proposed a method to provide declarative explanations and descriptions using
**PRIDE**of typical machine learning scenarios. Our approach guarantees a logical equivalent version to explain how the outputs are related with the inputs in a general machine learning setup. - We have applied our proposal to two different domains to check the generality of our approach.
- –
- A multimodal machine learning test-bed around automatic recruitment including different biases (by gender and ethnicity) on synthetic datasets.
- –
- A real dataset about adult incomes taken from US census whose possible biases to get higher earnings are found and shown.

- We have updated the state of the art methods applicable to XAI.
- We have enriched the introduction to
**LFIT**with examples for a more general audience. - We have checked the expressiveness of our approach (based on
**LFIT**) extending it to a dataset about adult income level from the 1994 US census. In this domain, we have not used any deep learning algorithms to compare, showing that the proposed approach is also applicable under this circumstance.

**LFIT**,

**GULA**, and

**PRIDE**. Section 4 presents the experimental framework, including the datasets and experiments conducted. Section 5 presents our results. Finally Section 6 and Section 7 respectively discuss our work and describe our conclusions and further research lines.

## 2. Related Works

#### 2.1. Explainable AI (XAI): Declarative Approaches

**LFIT**and machine learning algorithms focused on rule sets (including fuzzy logic ones) and decision trees because their outputs syntactically look similar.

**LFIT**and other declarative approaches. As we have previously explained, the current research is not interested in these kinds of numerical/statistical approaches.

- Statistical approaches need huge amounts of data to extract valid knowledge, while declarative ones are usually able to minimise the set of examples and counterexamples to get the same.
- Statistical approaches are usually compatible with noisy and poorly labelled data, while for declarative ones, this is a circumstance difficult to overcome.
- Statistical approaches do not offer, in the general case, clear explanations about the decisions they make (usually considered as weak machine learning algorithms), while declarative approaches (due to the declarative nature of the formal models that support them) are designed to be at least strong.
- Declarative approaches are supported by formal models like functional programming or formal logic. The theoretical properties of these models make it possible that the learned knowledge exhibits some characteristics (such as logical equivalence, minimisation, etc.)
- Hybrid approaches try to take advantage of both possibilities. Hybridisation can mix a declarative learning engine with numerical components or the opposite. The characteristics of the learned model depend on the type of hybridisation: equivalent noise-tolerant versions of the observed data can be learned by logical engines with numerical input components, and quasi-equivalent logical theories can be approximately induced by numerical/statistical machine learning algorithms that implement differentiable versions of logical operators and inference rules.

#### 2.2. Inductive Programming for XAI

**LFIT**as especially relevant for explainable AI (XAI). Although

**LFIT**learns propositional logic theories instead of first-order logic, the aforementioned ideas about

**ILP**are still valid. In the next section, we will describe the fundamentals of

**LFIT**and its

**PRIDE**implementation, which will be tested experimentally for XAI in the experiments that will follow.

#### 2.3. Learning from Interpretation Transition (LFIT)

**LFIT**) [49] has been proposed to automatically construct a model of the dynamics of a system from the observation of its state transitions. Given some raw data, like time-series data of gene expression, a discretisation of those data in the form of state transitions is assumed. From those state transitions, according to the semantics of the system dynamics, several inference algorithms modelling the system as a logic program have been proposed. The semantics of a system’s dynamics can indeed differ with regard to the synchronism of its variables, the determinism of its evolution and the influence of its history.

**LFIT**framework proposes several modelling and learning algorithms to tackle those different semantics. To date, the following systems have been tackled: memory-less deterministic systems [49], systems with memory [50], probabilistic systems [51], and their multi-valued extensions [52,53]. The work [54], proposes a method that deals with continuous time series data, the abstraction itself being learned by the algorithm.

**LFIT**was extended to learn system dynamics independently of its update semantics. That extension relies on a modeling of discrete memory-less multi-valued systems as logic programs in which each rule represents a variable that takes some value at the next state, extending the formalism introduced in [49,57]. The representation in [55,56] is based on annotated logics [58,59]. Here, each variable corresponds to a domain of discrete values. In a rule, a literal is an atom annotated with one of these values. It represents annotated atoms simply as classical atoms, and thus, remains at a propositional level. This modelling characterises optimal programs independently of the update semantics. It allows modelling the dynamics of a wide range of discrete systems, including our domain of interest in this paper.

**LFIT**can be used to learn an equivalent propositional logic program that provides explanations for each given observation.

## 3. Methods

#### 3.1. General Methodology

**LFIT**of a given black-box classifier. We can see that our purpose is to get declarative explanations in parallel (in a kind of white-blox digital twin) to a given neural network classifier. In the present work, for our first set of experiments, we used the same neural network and datasets described in [8], excluding the face images as explained in the following sections. In our second set of experiments (income prediction) we did not consider any machine learning algorithms to compare with. Therefore, the black box of Figure 2 is not considered in that case, although the rest of the figure is still applicable. In that set of experiments, we explore declarative explanations of the input/output relation of the training/testing datasets.

#### 3.2. PRIDE Implementation of LFIT

**GULA**[55,56] and

**PRIDE**[60] are particular implementations of the

**LFIT**framework [49]. In the present section we introduce and describe, first informally and then formally, the notation and the fundamentals of both methods.

**LFIT**concepts.

#### 3.2.1. Multi-Valued Logic

**LFIT**translates them into propositional ones (binary) creating as many propositional (binary) variables as possible values for each attribute. Although we keep the typical functional notation $var\left(value\right)$, each combination is in fact the propositional variable $va{r}^{value}$.

#### 3.2.2. Rules

**LFIT**expresses the theory it learns as a set of propositional Horn clauses with exactly one positive literal, that is, as logical implications between a conjunction of propositional atoms in the following form:

$h\left(va{l}_{h}^{i}\right)$ :- ${p}_{1}\left(va{l}_{1}^{{i}_{1}}\right)$ ,…, ${p}_{n}\left(va{l}_{n}^{{i}_{n}}\right)$. |

class(0) :- age(3), education(6), marital-status(0), occupation(0). |

class(0) :- age(4), workclass(0), education(1), occupation(8), relationship(0), native-country(0). |

class(1) :- education(7), marital-status(5). |

class(1) :- age(2), education(8), occupation(10). |

class(1) :- age(1), education(3), marital-status(2), occupation(9). |

#### 3.2.3. Rule Domination

**LFIT**learning process, rule domination is an important concept. Roughly speaking, when they have the same head, a rule dominates another if its body is contained in the others.

${R}_{1}$: class(0) :- education(6), marital-status(0). |

${R}_{2}$: class(0) :- age(3), education(6), marital-status(0), occupation(0). |

**LFIT**.

#### 3.2.4. States and Rule-State Matching

**LFIT**starts from the design of a body that fits as many examples as possible. This is done by rule-state matching. Informally, a state is a conjunction of atoms (positive literals, that is, associations between attributes and specific values) that could describe one or more examples.

${s}_{1}$: age(3), education(6), marital-status(0), occupation(0) |

${R}_{1}$: class(0) :- education(6), marital-status(0). |

**Example**

**1.**

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

**LFIT**corresponds to a data sample in the problems we tackle in this paper: a couple of input features and a target label. When a rule matches a state, it can be considered as a possible explanation to the corresponding observation. The final program we want to learn should both:

- Match the observations in a complete (all observations are explained) and correct (no spurious explanation) way;
- Represent only minimally necessary interactions (according to Occam’s razor: no overly-complex bodies of rules).

**GULA**[55,56] and

**PRIDE**[60] will learn a set of rules P such that all observations are explained: $\forall (s,{s}^{\prime})\in T,\forall {\mathrm{v}}^{val}\in {s}^{\prime},\exists R\in P,R\sqcap s,h\left(R\right)={\mathrm{v}}^{val}$. All rules of P are correct w.r.t. T: $\forall R\in P,\forall (s1,s2)\in T,R\sqcap s1\Rightarrow \exists (s1,s3)\in T,h\left(R\right)\in s3$ (if T is deterministic, $s2=s3$). All rules are minimal w.r.t. $\mathcal{F}$: $\forall R\in P,\forall {R}^{\prime}\in \mathcal{M}\mathrm{V}\mathrm{L}\mathrm{P},{R}^{\prime}$ correct w.r.t. T it holds that $R\le {R}^{\prime}\Rightarrow {R}^{\prime}=R$.

**GULA**and

**PRIDE**can also be used in order to predict and explain from unseen feature states by learning additional rules that encode when a target variable value is not possible as shown in the experiments of [56].

**LFIT**) in some scenarios with numerical aspects: in the FairCV db case we are generating white-box explanations to a deep-learner black-box; in the US census case we are explaining a dataset that could be typically tackled by numeric (statistical) approaches.

**LFIT**, the focus is on the qualitative guarantee of learning a logical version equivalent to the observed system. Regarding equivalence, the version is equivalent or it is not. If the model fails in 1% of the examples, equivalence is lost in the same way than if it had failed in 20% or 60% of the examples.

**LFIT**does not matter; but this is not exactly true.

**LFIT**can easily collect qualitative information, such as how many states (input examples) match each rule. This numerical information can be used as weights, both to better explain and understand the process, but also to incorporate predicting capabilities to the declarative version. This option has been explained and explored in [61,62].

## 4. Experimental Framework

**PRIDE**to explain machine learning domains we have designed several experiments using the FairCVdb dataset [8] and the data about adult incomes from the 1994 US census [63].

#### 4.1. FairCVdb Dataset

#### Experimental Protocol: Towards Declarative Explanations

**PRIDE**on the FairCVdb dataset described in the previous section.

#### 4.2. Adult Income Level Dataset

**PRIDE**with the complete dataset.

**PRIDE**expressiveness when trying to find a data-driven explanation for this common belief.

#### Experiments Design

- To prepare the dataset for
**PRIDE**by preprocessing:- Removing those entries with some unknown attribute. Only 45,222 entries remain after this step.
- Discretising continuous attributes (those marked as continuous in Table 1).

- To get a logical version equivalent to the data to analyse the effect of the attribute sex considering the income level as a function of the other attributes.
- To get a logical version equivalent to the data to analyse the effect of the attribute ethnicity considering the income level as a function of the other attributes.

## 5. Results

**LFIT**guarantees: the learned propositional logic theory is equivalent to the observed data, and the conditions of each clause (rule) are minimal. These properties allow for estimating the complexity of the observed data from the complexity of the learned theory—the simpler the dataset the simpler the theories. In the future, we would like to explore the possibility of defining some kind of complexity measure of the datasets from the complexity of the theories learned by

**LFIT**. It could be something similar to Kolmogorov’s compression complexity [64].

**LFIT**models. Propositional logic excludes the use of variables. Although functional notation has been used (for example in $sex\left(0\right)$) each pair of functions and one specific value of its argument, represents a proposition ($se{x}^{0}$ in our example). The use of variables by other declarative models, such as first-order logic, allows a more compact notation by grouping different values of the same attribute by means of a well defined variable. However, there is no trivial translation from one model to another. It is important to realise that this circumstance is an inherent characteristic of propositional logic that can not be overcome inside the propositional realm. It is true that more compact notations could be more readable and, hence, they can offer more easily understandable explanations. However, the increase of the readability of

**LFIT**results by translating them into another model is out of the scope of the current contribution.

#### 5.1. FairCVdb Dataset

#### 5.1.1. Example of Declarative Explanation

scores(3) :- gender(1), education(5), experience(3). |

scores(3) :- education(4), experience(3). |

#### 5.1.2. Quantitative Summary of the Results

#### 5.1.3. Quantitative Identification of Biased Attributes in Rules

#### 5.1.4. Quantitative Identification of the Distribution of Biased Attributes

**PRIDE**to identify random indirect perturbations of other attributes in the bias is a relevant achievement of our proposal.

#### 5.2. Adult Income Level Dataset

**LFIT**catches the structure of the dataset because there are no significative differences when excluding gender or ethnicity to study their biases. It is also interesting to mention that these attributes do not contribute the most to income level.

**PRIDE**supports and explains the common belief about the relationship among sex (idem. ethnicity) and higher income level:

## 6. Discussion

**PRIDE can****explain****algorithms learnt by neural networks.**The theorems that support the characteristics of PRIDE allow a set of propositional clauses logically equivalent to the systems observed when facing the input data provided. In addition, each proposition has a set of conditions that is minimum. Thus, regarding the FairCVdb case, once the scorer is learnt, PRIDE translates it into a logical equivalent program. This program is a list of clauses like the one shown in Listing 5. Logical programs are declarative theories that explain the knowledge on a domain.**PRIDE can****explain****what happens in a specific domain.**Our experimental results discover these characteristics of the domain:- –
**Insights into the structure of the FairCVd dataset**. We have seen (and further confirmed with the authors of the datasets) characteristics of the datasets, e.g., (1) All attributes are needed for the score. We have learnt the logical version of the system starting from only two input attributes and including one additional attribute at a time and only reached an accuracy of 100% when taking into account all of them. This is because removing some attributes generates indistinguishable CVs (all the remainder attributes have the same value) with different scores (that correspond to different values in some of the removed attributes). (2) Gender and ethnicity are not the most relevant attributes for scoring: The number of occurrences of these attributes is much smaller than others in the conditions of the clauses of the learnt logical program. (3) While trying to catch the biases we have discovered that some attributes seem to increase their relevance when the score is biased. For example, the competence in some specific languages (attribute i7) seems to be more relevant when the score has gender bias. After discussing with the authors of the datasets, they confirmed a random perturbation of these languages into the biases, that explained our observations.- –
**Biases in the training FairCVdb datasets were detected.**We have analysed the relationship between the scores and the specific values of the attributes used to generate the biased data. We have proposed a simple mathematical model based on the effective weights of the attributes that concludes that higher values of the scores correspond to the same specific values of gender (for gender bias) and ethnic group (for ethnicity bias). On the other hand, we have performed an exhaustive series of experiments to analyse the increase of the presence of the gender and ethnicity in the conditions of the clauses of the learnt logical program (comparing the unbiased and biased versions).- –
**Insights into the structure of dataset about the adult income from the US census**. In this case, there is no unbiased version to compare with, as in the FairCVdb dataset. In addition, we do not have any machine learning approach to be considered for the black-box explanation. Nevertheless, there exists a common belief about the presence of biases (gender and ethnicity) in the income level.**PRIDE**has been used considering the dataset itself as a black-box, understanding the income level as a function of the other attributes. We have obtained a logic theory that supports this common belief.

**LFIT**, and in particular

**PRIDE**, are able to offer explanations to the algorithm learnt in the domain under consideration. The resulting explanation is, as well, expressive enough to catch training biases in the models learnt with neural networks.

**PRIDE**is still able to explain the structure of the datasets considering themselves as the black-box that has to be explained.

## 7. Further Research Lines

**Increasing understandability.**Two possibilities could be considered in the future: (1) to ad hoc post-process the learned program for translating it into a more abstract form, or (2) to increase the expressive power of the formal model that supports the learning engine using, for example, ILP based on first-order logic.**Adding predictive capability. PRIDE**is actually not aimed to predict but to explain (declaratively) by means of a digital twin of the observed systems. Nevertheless, it is not really complicated to extend**PRIDE**functionality to predict. It should be necessary to change the way in which the result is interpreted as a logical program: mainly by adding mechanisms to chose the most promising rule when more than one is applicable.Our plan is to test an extended-to-predict**PRIDE**version to this same domain and compare the result with the classifier generated by deep learning algorithms.**Handling numerical inputs.**[8] included as input the images of the faces of the owners of the CVs. Although some variants to**PRIDE**are able to cope with numerical signals, the huge amount of information associated with images implies performance problems. Images are a typical input format in real deep learning domains. We would like to add some automatic pre-processing steps for extracting discrete information (such as semantic labels) from input images. We are motivated by the success of systems with similar approaches but different structure like [65].**Generating and combining multiple explanations.**The present work has explored a way to provide a single human-readable explanation of the behavior of an AI model. An extension we have in mind is generating multiple explanations by different complementary methods and parameters of those methods and then generating a combined explanation [66,67].**Measuring the accuracy and performance of the explanations.**As far as the authors know, there is no standard procedure to evaluate and compare different explainability approaches. We will incorporate in future versions some formal metric.**Analysing other significant problems where non-explainable AI is now the common practice for good explanations.**The scenario studied here (automatic tools for screening in recruitment and estimating the income level based on demographic information) are only two of the many application areas where explanations of the action of AI systems are really needed. Other areas that will significantly benefit from this kind of approaches are e-learning [70], e-health [71,72], and other human-computer interaction applications [73,74].**Proposing metrics for the complexity of the datasets.**Due to the formal properties that the general**LFIT**model gives to the learned theories, the complexity of the original data could be estimated from the complexity of the propositional logic equivalent theory. This approach is inspired by some implementations of Kolmogorov’s complexity by means of file compressors [64].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Architecture of the proposed approach for generating an explanation of a given black-box classifier (1) using

**PRIDE**(2) with a toy example (3). Note that the resulting explanations generated by

**PRIDE**are in propositional logic.

**Figure 2.**Experimental framework:

**PRIDE**is fed with all the data available (train + test) for increasing the accuracy of the equivalence. In our experiments we consider the classifier (see [8] for details) as a black box to perform regression from input resume attributes (atts.) to output labels (recruitment scores labelled by human resources experts). LFIT gets a digital twin to the neural network providing explainability (as human-readable white-box rules) to the neural network classifier.

**Figure 3.**Structure of the experimental tests. There are 4 datasets for analysing gender (named g) and ethnicity (e) bias separately. Apart from gender and ethnicity, there are 12 other input attributes (named from $i1$ to $i12$). There is a couple of (biased and unbiased) datasets for each one: gender and ethnicity. We have studied the input attributes by increasing complexity starting with $i1$ and $i2$ and adding one at each time. Thus, for each couple we considered 11 different scenarios (named from $s1$ to $s11$). This figure shows their structure (${s}_{i}$ is included in all ${s}_{j}$ for which $i<j$).

**Figure 6.**Percentage of the absolute increment (comparing scores with and without bias for ethnicity) of each attribute for scenarios s1, s2, s3, s4, s5 and s6 ($AI{P}_{us1-6,ebs1-6}$). The graphs link the points corresponding to all the input attributes considered in each scenario.

**Figure 10.**Normalised percentage of frequency in scenario s11 of each attribute: g, i1 to i11 ($N{P}_{s11}$). No bias (blue), gender-biased scores (red).

**Figure 11.**Normalised percentage of frequency in scenario s11 of each attribute: g, i1 to i11 ($N{P}_{s11}$). No bias (blue), gender-biased scores (red).

**Table 1.**Names, values and codification of the dataset about incomes. Attributes of type C take integer or real continuous values and they are uniformly discretised. Attributes of type D are originally discrete and are numerically coded from 0 to the maximum needed value.

Attribute | Meaning | Type | Codification |
---|---|---|---|

Age | Age of the individual (years) | C | $\{0,1,\dots ,7\}$ |

Workclass | Work type (self employment, private, …) | D | $\{0,1,\dots ,6\}$ |

Fnlwgt | Demographic weight (row) from census | D | $\{0,1,\dots ,14\}$ |

Education | Highest academic degree | D | $\{0,1,\dots ,15\}$ |

Marital status | Civil status | D | $\{0,1,\dots ,3\}$ |

Occupation | Individual’s job sector | D | $\{0,1,\dots ,13\}$ |

Relationship | Present individual’s relationship | D | $\{0,1,\dots ,5\}$ |

Ethnicity | Ethnic group | D | $\{0,1,\dots ,4\}$ |

Sex | D | $\{0,1\}$ | |

Capital gain | Increase in individual’s capital asset | C | $\{0,1,\dots ,9\}$ |

Capital loss | Decrease in individual’s capital asset | C | $\{0,1,\dots ,4\}$ |

Hours per week | Spent on work (average) | D | $\{0,1,\dots ,9\}$ |

Native country | Country of origin | D | $\{0,1,\dots ,40\}$ |

Income level | Individual’s class of income (≤50, >50) | D | $\{0,1\}$ |

**Table 2.**Names, values, and codification of the FairCVdb dataset. Attributes of type C take continuous real values and are uniformly discretised. Attributes of type D are discrete and are numerically coded from 0 to the maximum needed value. For all values the higher is considered the better.

Attribute | Meaning | Type | Codification |
---|---|---|---|

Ethnicity | Ethnic group | D | $\{0,1,2\}$ |

Gender | D | $\{0,1\}$ | |

Education | Education level | D | $\{0,1,\dots ,5\}$ |

Experience | Work experience | C | $\{0,1,\dots ,4\}$ |

Availability | Time for being ready to start | D | $\{0,1,\dots ,5\}$ |

Foreign languages | Level of 8 possible languages | D | $\{0,1,\dots ,3\}$ |

Score | Unbiased value assigned | C | $\{0,1,\dots ,3\}$ |

Gender biased score | (Gender) biased value assigned | C | $\{0,1,\dots ,3\}$ |

Ethnicity biased score | (Ethnicity) biased value assigned | C | $\{0,1,\dots ,3\}$ |

e | i1 | i2 | i3 | i4 | i5 | i6 | |
---|---|---|---|---|---|---|---|

Ethnic bias | 3221 | 3648 | 2802 | 1789 | 2951 | 3300 | 1520 |

No ethnic bias | 1682 | 2398 | 1822 | 1065 | 1846 | 2032 | 1023 |

i7 | i8 | i9 | i10 | i11 | i12 | #Rules | |
---|---|---|---|---|---|---|---|

Ethnic bias | 1449 | 1404 | 1214 | 1044 | 870 | 652 | 7886 |

No ethnic bias | 892 | 875 | 805 | 683 | 544 | 397 | 2732 |

g | i1 | i2 | i3 | i4 | i5 | i6 | |
---|---|---|---|---|---|---|---|

Gender bias | 1150 | 2164 | 1671 | 1006 | 1642 | 1830 | 884 |

No gender bias | 992 | 1943 | 1524 | 861 | 1445 | 1663 | 832 |

i7 | i8 | i9 | i10 | i11 | i12 | #Rules | |
---|---|---|---|---|---|---|---|

Gender bias | 874 | 807 | 681 | 630 | 537 | 347 | 2449 |

No gender bias | 714 | 714 | 633 | 557 | 470 | 320 | 2200 |

#Rules | Age | Workclass | Education | ed.# | Civil-Status | Occu. | |
---|---|---|---|---|---|---|---|

ethnc. | 7948 | 5478 | 4612 | 7007 | 6902 | 3737 | 5860 |

Relationship | Ethnc/Sex | Cap-Gain | Cap-Loss | h/Week | Country | |
---|---|---|---|---|---|---|

ethnc. | 1656 | 1263 | 374 | 813 | 1605 | 554 |

#Rules | Age | Workclass | Education | ed.# | Civil-Status | Occu. | |
---|---|---|---|---|---|---|---|

gender | 7735 | 5353 | 4522 | 6821 | 6633 | 3696 | 5634 |

Relationship | Ethnc/Sex | Cap-Gain | Cap-Loss | h/Week | Country | |
---|---|---|---|---|---|---|

gender | 1620 | 478 | 374 | 832 | 1685 | 810 |

ethnc(0) | ethnc(1) | ethnc(2) | ethnc(3) | ethnc(4) | |
---|---|---|---|---|---|

class(0) | 452 | 71 | 14 | 16 | 132 |

class(1) | 334 | 84 | 23 | 10 | 127 |

Sex(0) | Sex(1) | |
---|---|---|

class(0) | 101 | 169 |

class(1) | 126 | 82 |

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## Share and Cite

**MDPI and ACS Style**

Ortega, A.; Fierrez, J.; Morales, A.; Wang, Z.; de la Cruz, M.; Alonso, C.L.; Ribeiro, T.
Symbolic AI for XAI: Evaluating LFIT Inductive Programming for Explaining Biases in Machine Learning. *Computers* **2021**, *10*, 154.
https://doi.org/10.3390/computers10110154

**AMA Style**

Ortega A, Fierrez J, Morales A, Wang Z, de la Cruz M, Alonso CL, Ribeiro T.
Symbolic AI for XAI: Evaluating LFIT Inductive Programming for Explaining Biases in Machine Learning. *Computers*. 2021; 10(11):154.
https://doi.org/10.3390/computers10110154

**Chicago/Turabian Style**

Ortega, Alfonso, Julian Fierrez, Aythami Morales, Zilong Wang, Marina de la Cruz, César Luis Alonso, and Tony Ribeiro.
2021. "Symbolic AI for XAI: Evaluating LFIT Inductive Programming for Explaining Biases in Machine Learning" *Computers* 10, no. 11: 154.
https://doi.org/10.3390/computers10110154