Set Theory Research Articles

Set Theory Revisited: Related Theorems and Realms and Their Development

Set Theory Revisited: Related Theorems and Realms and Their Development

Application of dominance-based rough set approach in vaginal dosage forms optimization

The objective of this study was to identify cause-effect relationships between various data related to vaginal dosage forms, such as composition, process parameters (condition attributes), and quality (decision attribute), by employing a mathematical data mining approach that utilizes Rough Set Theory. The analyzed data were organized in a tabular format known as an information system. Objects were labeled as rows, and attributes as columns, with attribute-values serving as entries. Each formulation was described by its condition attributes (composition, manufacturing process, quality parameters), while the decision attribute indicated the quality of the formulation (expressed as pH after vaginal tablet or pessary application) and enabled classification as correct/incorrect or good/bad. The decision rules generated by the study provide insights into relationships between the condition attributes and the decision attribute. They highlight the importance of LA content and LA:polymer ratios in maintaining physiological pH. Additionally, the findings confirm the favorable impact of chitosan on pH regulation while indicating that the type of formulation (pessaries or tablets) does not significantly affect the classification of objects. Decision rules with high confirmation coefficients of condition attributes can help developing optimal formulations. These decision rules contain essential information that describes the cause-and-effect relationships and are free of redundant information. They form the basis for understanding and optimizing processes, and establishing a design space.

An oversampling algorithm of multi-label data based on cluster-specific samples and fuzzy rough set theory

Imbalanced class distributions are common in real-world scenarios, including datasets with multiple labels. One widely acknowledged approach to addressing imbalanced distributions is through oversampling, a technique that both balances the class distribution and improves the effectiveness of classification models. However, when generating synthetic data for multi-label datasets, complexities arise due to the presence of multiple-label sets, which require careful placement and labeling. We propose MLCSMOTE-FRST, an algorithm for synthetic data generation based on label-specific clustering and fuzzy rough set theory. Generation ratios and dependency samples are provided by clusters specific to each label, with a focus on the overall label distribution and the distribution within each cluster. The labels are supported by intra-cluster positive samples, determined using fuzzy rough set theory, which helps to capture the consensus label set. Experimental results on multi-label datasets using four classifiers demonstrate the effectiveness of the proposed method in terms of macro-F1 and micro-F1 scores.

THE IMPACT OF DIGITAL DISINFORMATION ON QUALITY OF LIFE: A FUZZY MODEL ASSESSMENT

Quality of Life (QoL) is a multifaceted concept encompassing economic, social, environmental, psychological, and physical dimensions of an individual’s life, including personal living conditions, happiness, well-being, and life satisfaction. As a vital criterion for sustainable development and active social policy in countries, QoL has been significantly influenced by the dynamic technological evolution of social media. However, the comprehensive impact of social media, including its role in disseminating disinformation – a major social and socio-economic concern – on QoL remains underexplored. This research aims to develop a novel fuzzy model to assess the level of disinformation on digital platforms and its correlation with the population’s QoL. Employing a mathematical approach rooted in expert evaluation, this study leverages intellectual knowledge analysis and fuzzy set theory. Grounded in data from real respondents and knowledge-based models, this study pioneers an information model to evaluate inhabitants’ QoL, incorporating factors such as financial concerns, perception of disinformation, and its influence on digital platforms. The fuzzy estimation model, verified with data from 3,036 respondents, quantitatively assesses citizens’ QoL. An illustrative application of the model demonstrates its effectiveness. The findings are particularly valuable for policymakers, experts in economic and innovative development, aiding the creation of regulatory and monitoring mechanisms to foster sustainable economic growth and devise effective development strategies.

Consistency approximation: Incremental feature selection based on fuzzy rough set theory

Fuzzy Rough Set Theory (FRST)-based feature selection has been widely used as a preprocessing step to handle dynamic and large datasets. However, large-scale or high-dimensional datasets remain intractable for FRST-based feature selection approaches due to high space complexity and unsatisfactory classification performance. To overcome these challenges, we propose a Consistency Approximation (CA)-based framework for incremental feature selection. By exploring CA, we introduce a novel significance measure and a tri-accelerator. The CA-based significance measure provides a mechanism for each sample in the universe to keep members with different class labels within its fuzzy neighbourhood as far as possible, while keeping members with the same label as close as possible. Furthermore, our tri-accelerator reduces the search space and decreases the computational space with a theoretical lower bound. The experimental results demonstrate the superiority of our proposed algorithm compared to state-of-the-art methods on efficiency and classification accuracy, especially for large-scale and high-dimensional datasets.

Network intrusion detection system for IoT security using machine learning and statistical based hybrid feature selection

AbstractThe widespread adoption of Internet of Things (IoT) devices has revolutionized daily life, offering convenience and efficiency. However, this growth has also brought new security challenges. With the escalating use of Internet and network technology, the number of cyber‐attacks has increased, intensifying the focus on Intrusion Detection Systems (IDS) among researchers. Network intrusion detection (NID) plays a crucial role in securing IoT networks, becoming essential for modern security infrastructure. Recently, machine learning algorithms have shown promise in providing IDS solutions. Yet, IoT IDS systems face challenges due to their functional and physical diversity, making comprehensive feature utilization impractical. Therefore, effective feature selection becomes imperative. In this research, a novel feature selection methodology for anomaly‐based NIDS is proposed. The methodology commences by employing two filter‐based techniques, namely 1‐way ANOVA and the Pearson correlation coefficient, to meticulously identify and extract pertinent features from the dataset. These methods serve as initial filters to discern the most relevant attributes, ensuring that only the most informative features are retained for subsequent analysis. Subsequently, the most optimal features identified by both methodologies are extracted utilizing the principles of union and intersection in mathematical set theory. Using the NSL‐KDD and UNSW‐NB15 datasets, we exemplify how our model can outperform conventional ML classifiers in terms of detection rate, precision, recall. In our study, intrusion detection is carried out by SVM, kNN, Decision Tree, Logistic Regression and Random Forest using trained attack patterns. The demonstrated results highlight the exceptional performance of the proposed system, with an impressive accuracy rate of 99.6% on the NSL‐KDD dataset and a substantial 97.7% on the UNSW‐NB15 dataset, clearly surpassing the performance of contemporary methods.

Integrating fuzzy metrics and negation operator in FCM algorithm via genetic algorithm for MRI image segmentation

In this study, we redefine FCM algorithm by integrating fuzzy set theory, fuzzy metrics, and Sugeno negation principles. This innovative approach overcomes the limitations inherent in conventional machine learning models, especially in situations characterized by uncertainty, noise, and ambiguity. Our model utilizes the membership degrees from fuzzy set theory, and transforms the concept of proximity defined by fuzzy metrics into a minimization problem. This transformation is achieved using a linguistic negation operator, which is crucial for optimizing FCM algorithm's objective function. A significant innovation in our research is the use of GA for optimizing parameters within the contexts of fuzzy metrics and Sugeno negation. The precise optimization capabilities of GA greatly enhance the sensitivity and adaptability of FCM algorithm, thereby improving overall performance. By leveraging the meticulous parameter adjustments provided by GA, our approach has shown superior results in practical applications, such as brain MRI image segmentation, surpassing traditional methods. Experimental results highlight the considerable enhancements our proposed FCM algorithms bring over existing methods across various performance metrics. In conclusion, this study makes a valuable addition to the field of fuzzy-based machine learning methodologies. It combines the optimization strength of GA with the flexible classification capabilities of fuzzy logic. The integration of Sugeno negation and fuzzy metrics not only improves the accuracy and precision of FCM algorithm but also provides significant benefits in handling complex and ambiguous datasets. This research signifies a major advance in machine learning and fuzzy logic, setting the stage for future applications and studies.

Optimal design of robust control for belt conveyor systems based on fuzzy dynamic model and Nash game

An optimized robust control method is proposed in this paper to deal with the disturbances of nonlinear factors such as uncertainty in belt conveyor and to realize the speed control of belt conveyor. By controlling the subsystems to track the same desired speed, belt breaks and overlaps caused by varying belt speeds from one section to another are limited. First, a fuzzy model is developed by applying fuzzy set theory to describe uncertainties of belt conveyor such as masses, stiffness, damping and frictions. Second, a robust control is designed for the nominal system and compensating for the uncertainty. With the robust control, the belt conveyor system is proved to be uniformly bounded and uniformly ultimately bounded by the Lyapunov approach. Third, the parameters of controller are optimized based on a non-cooperative game approach. The optimal solution is the Nash equilibrium. Optimal solution allows the controller system to perform better and decrease the cost of control. Finally, by comparing with LQR control and different parameters, the feasibility of robust control and the rationality of the optimal parameters are proven.

METHOD OF TESTING LARGE NUMBERS FOR PRIMALITY

The current stage of scientific and technological development entails ensuring information security across all domains of human activity. Confidential data and wireless channels of remote control systems are particularly sensitive to various types of attacks. In these cases, various encryption systems are most commonly used for information protection, among which large prime numbers are widely utilized. The subject of research involves methods for generating prime numbers, which entail selecting candidates for primality and determining the primality of numbers. The subject of research involves methods for generating prime numbers, which choice selecting candidates for primality and determining the primality of numbers. The objective of the work is the development and theoretical justification of a method for determining the primality of numbers and providing the results of its testing. The aim to address the following main tasks: analyze the most commonly used and latest algorithms, methods, approaches, and tools for primality testing among large numbers; propose and theoretically justify a method for determining primality for large numbers; and conduct its testing. To achieve this aim, general scientific methods have been applied, including analysis of the subject area and mathematical apparatus, utilization of set theory, number theory, fields theory, as well as experimental design for organizing and conducting experimental research. The following results have been obtained: modern methods for selecting candidates for primality testing of large numbers have been analyzed, options for generating large prime numbers have been considered, and the main shortcomings of these methods for practical application of constructed prime numbers have been identified. Methods for determining candidates for primality testing of large numbers and a three-stage method for testing numbers for primality have been proposed and theoretically justified. The testing conducted on the proposed primality determination method has demonstrated the correctness of the theoretical conclusions regarding the feasibility of applying the proposed method to solve the stated problem. Conclusions. The use of a candidate primality testing strategy allows for a significant reduction in the number of tested numbers. For numbers of size 200 digits, the tested numbers is reduced to 8.82%. As the size of the tested numbers increases, their quantity will decrease. The proposed method for primality testing is sufficiently simple and effective. The first two stages allow for filtering out all composite numbers except for Carmichael numbers. In the first stage, using the first ten prime numbers filters out over 80 percent of the tested numbers. In the second stage, composite numbers with factors greater than 29 are sieved out. In the third stage, Carmichael numbers are sieved out. The test is polynomial, deterministic, and unconditional.

Enhancing breast cancer treatment selection through 2TLIVq-ROFS-based multi-attribute group decision making.

Breast cancer is an extremely common and potentially fatal illness that impacts millions of women worldwide. Multiple criteria and inclinations must be taken into account when selecting the optimal treatment option for each patient. The selection of breast cancer treatments can be modeled as a multi-attribute group decision-making (MAGDM) problem, in which a group of experts evaluate and rank alternative treatments based on multiple attributes. MAGDM methods can aid in enhancing the quality and efficacy of breast cancer treatment selection decisions. For this purpose, we introduce the concept of a 2-tuple linguistic interval-valued q-rung orthopair fuzzy set (2TLIVq-ROFS), a new development in fuzzy set theory that incorporates the characteristics of interval-valued q-rung orthopair fuzzy set (IVq-ROFS) and 2-tuple linguistic terms. It can express the quantitative and qualitative aspects of uncertain information, as well as the decision-makers' level of satisfaction and dissatisfaction. Then, the 2TLIVq-ROF weighted average (2TLIVq-ROFWA) operator and the 2TLIVq-ROF weighted geometric (2TLIVq-ROFWJ) operator are introduced as two new aggregation operators. In addition, the multi-attribute border approximation area comparison (MABAC) method is extended to solve the MAGDM problem with 2TLIVq-ROF information. To demonstrate the efficacy and applicability of the suggested model, a case study of selecting the optimal breast cancer treatment is presented. The results of the computations show that the suggested MAGDM model is able to handle imprecision and subjectivity in complicated decision-making scenarios and opens new research scenarios for scholars.

INT-FUP: Intuitionistic Fuzzy Pooling

Convolutional Neural Networks (CNNs) are a kind of artificial neural network designed to extract features and find out patterns for tasks such as segmentation, recognizing objects, and drawing up classification. Within a CNNs architecture, pooling operations are used until the number of parameters and the computational complexity are reduced. Numerous papers have focused on investigating the impact of pooling on the performance of Convolutional Neural Networks (CNNs), leading to the development of various pooling models. Recently, a fuzzy pooling operation based on type-1 fuzzy sets was introduced to cope with the local imprecision of the feature maps. However, in fuzzy set theory, it is not always accurate to assume that the degree of non-membership of an element in a fuzzy set is simply the complement of the degree of membership. This is due to the potential existence of a hesitation degree, which implies a certain level of uncertainty. To overcome this limitation, intuitionistic fuzzy sets (IFS) were introduced to incorporate the concept of a degree of hesitation. In this paper, we introduce a novel pooling operation based on intuitionistic fuzzy sets to incorporate the degree of hesitation heretofore neglected by a fuzzy pooling operation based on classical fuzzy sets, and we investigate its performance in the context of image classification. Intuitionistic pooling is performed in four steps: bifuzzification (by the transformation of data through the use of membership and non-membership maps), first aggregation (through the transformation of the IFS into a standard fuzzy set, second aggregation (through the transformation and use of a sum operator), and the defuzzification of feature map neighborhoods by using a max operator. IFS pooling is used for the construction of an intuitionistic pooling layer that can be applied as a drop-in replacement for the current, fuzzy (type-1) and crisp, pooling layers of CNN architectures. Various experiments involving multiple datasets demonstrate that an IFS-based pooling can enhance the classification performance of a CNN. A benchmarking study reveals that this significantly outperforms even the most recent pooling models, especially in stochastic environments.

Research and Evaluation of Coal Mine Ventilation Management

Abstract The main scientific and practical results of the study and assessment of ventilation control in coal mines using the decomposition method based on data from the analysis of aerodynamic parameters of air distribution are presented. This will lead to increased safety and reduced accident rates in mining operations at coal mines. Methods of decomposition, statistical dynamics, set theory, fundamental laws of mining aerodynamics, and discrete mathematics are described. The studies were carried out based on the experimental data of the authors using various mathematical and statistical methods. A universal technique is proposed that can be used in mine workings for analysis, assessment of ventilation processes and associated aerodynamic factors, as well as optimization of the air distribution control system under various operating modes. A methodology and application software developed on its basis have been compiled. Automated geographic information systems such as GIS K-MINE® (Krivoy Rog, Ukraine) and VENTSIM are recommended. This system has been approved by regulatory authorities in Ukraine and is utilized by specialists from industry institutes, State Mining Supervision authorities, mines, and others. They are adapted for coal mines, ensuring increased safety and efficiency of work. This will ensure increased safety, labor protection, and reduced accident rates in coal mining operations.

Reflexive Event-B: Semantics and Correctness the EB4EB Framework

The Event-B method enables correct by construction modeling of systems. It relies on set theory and first-order logic, to describe a series of refined system models expressed as a set of events modifying state variables. Invariants and theorems are introduced to express system properties submitted to the proof system associated with Event-B. While Event-B has proven its efficiency for the proof of this type of property, it does not offer powerful means allowing the explicit description of properties other than safety and specific forms of reachability. Checking other properties such as deadlock-freeness, liveness, or event scheduling requires ad hoc modeling techniques and external tools such as model checkers or other proof systems. This article presents <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">EB4EB</i> , a new modeling framework offering the capability to introduce formally defined Event-B extensions, in particular new proof obligations corresponding to new properties. It is based on metamodeling techniques. It includes a theory (a metatheory) modeling Event-B and offers means for explicit manipulation of Event-B features and an extension mechanism to explicitly formalize and prove other properties. This reflexive framework relies on a trace-based semantics of Event-B and introduces a set of Event-B theories defining data types, operators, well-defined conditions, theorems, and proof rules to define Event-B constructs and their semantics. Deep and shallow instantiation mechanisms are set up to instantiate the obtained metatheory. The EB4EB framework and its instantiation mechanisms are developed in Event-B using the Rodin platform ensuring correctness and internal consistency of the defined theories. Lamport's clock example, instantiating EB4EB in both shallow and deep mechanisms, is used to evaluate the proposed approach.

From real-life to very strong axioms. Classification problems in Descriptive Set Theory and regularity properties in Generalized Descriptive Set Theory

AbstractThis thesis is divided into three parts, the first and second ones focused on combinatorics and classification problems on discrete and geometrical objects in the context of descriptive set theory, and the third one on generalized descriptive set theory at singular cardinals of countable cofinality.Descriptive Set Theory (briefly: DST) is the study of definable subsets of Polish spaces, i.e., separable completely metrizable spaces. One of the major branches of DST is Borel reducibility, successfully used in the last 30 years to solve and compare many classification problems. One of our goals is the classification of knots, very familiar and tangible objects in everyday life, which also play an important role in modern mathematics. The study of knots and their properties is known as knot theory. Our plan is to gain insight into knots using discrete objects, such as linear and circular orders. This approach was already exploited in [6]. The first part of this work is therefore devoted to countable linear orders and the study of the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results. We also expand our research to the case of circular orders.Another objective of this first part is to extend the notion of convex embeddability on countable linear orders. We provide a family of quasi-orders of which embeddability is a particular case as well. We study these quasi-orders from a combinatorial point of view and analyse their complexity with respect to Borel reducibility. Furthermore, we extend the analysis of these quasi-orders to the set of uncountable linear orders.The second part of the project deals with classification problems on knots and $3$ -manifolds. The goal here is to apply the results obtained in the first part to the study of proper arcs and knots, establishing lower bounds for the complexity of some natural relations between these geometrical objects. We also obtain some combinatorial results which are particularly interesting when we restrict to the set of wild proper arcs and wild knots, classes which haven’t received much attention so far. These parts are included in the two preprints [4, 5] in collaboration with my supervisor Alberto Marcone, Luca Motto Ros (University of Torino), and Vadim Weinstein (University of Oulu).The second part of this work also includes the classification of non-compact $3$ -manifolds up to homeomorphism (the case of compact $3$ -manifolds has already been solved: indeed, there are only countably many $3$ -manifolds up to homeomorphism; see [7]), and that of Cantor sets of $\mathbb {R}^3$ up to conjugation (answering to Question 5.5 of [3]). Here we resort to algebraic tools. Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras (see [1]). The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this work we introduce a weaker concept which we call “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration beyond totally disconnected. As an application of this method, we show that both homeomorphism on non-compact $3$ -manifolds and conjugation of Cantor sets in $\mathbb {R}^3$ are completely classifiable by countable structures. These results are part of an upcoming paper in collaboration with Vadim Weinstein.The last part of this thesis concerns the natural generalization of descriptive set theory that occurs when countable is replaced by uncountable, called Generalized Descriptive Set Theory (briefly: GDST). In particular, we focus on the case of GDST for a singular cardinal $\kappa $ of countable cofinality. The goal here is to study the generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $\mathcal {P}(\kappa )$ . We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms, like I2, and classes of sets definable by $\Sigma _1$ -formulas with parameters from various collections of sets. The obtained results are included in [2], a joint work with my co-supervisor Vincenzo Dimonte and Philipp Lücke (University of Hamburg).Abstract prepared by Martina IannellaE-mail: martina.iannella@tuwien.ac.at.URL: https://air.uniud.it/handle/11390/1262824.

Local Theory of Regular Systems and Delone Sets

Local Theory of Regular Systems and Delone Sets

System design of a microstrip antenna by dimension and substrates optimization

The microstrip antennas matured and improved over the last 25 years. Throughout these years, the limitations, and specifications of the said antennas have been overcome and significantly improved. Known to be lowprofile, suitable for mobile, and lightweight, these microstrip antennas are the focus of this research. In this paper, the researchers designed a microstrip antenna with varying lengths and substrates. They tested the changes and the effects in microstrip antennas of different lengths, along with altering substrates. To verify the differences, the researchers compared the performance parameters maximum gain (dBi), minimum gain (dBi), and S11 graph on each tested length and changed substrate. The rough set theory was used to determine the optimal design via MATLAB. From there, the researchers analyzed the results gathered and drew their respective conclusions. Additionally, they saw and compared each data result to know what antenna has the best performance parameters. From the results, a change in the dimension will result in a decrease in the said performance parameters. Furthermore, the change in substrate thickness also diminishes these changes.

Incorporating multi-source remote sensing in the detection of earthquake-damaged buildings based on logistic regression modelling

After an earthquake, efficiently and accurately acquiring information about damaged buildings can help reduce casualties. Earth observation data have been widely used to map affected areas after earthquakes. However, fine post-earthquake assessment results are needed to manage recovery and reconstruction and to estimate economic losses. In this paper, for quantification and precision purposes, a method of earthquake-induced building damage information extraction incorporating multi-source remote sensing data is proposed. The method consists of three steps： (1) Analysis of multisource features that describe texture, colour, and geometry, (2) rough set theory is carried out to further determine the feature parameters, (3) Logistic regression model (LRM) was built to describe the relationship between the occurrence and absence of destroyed buildings within an individual object. Old Beichuan County (centered at approximately 31.833︒N, 104.459° E), China, the area most devastated by the Wenchuan earthquake on May 12, 2008, is used to test the proposed hypothesis. Multi-source remote sensing imagery include optical data, synthetic aperture radar (SAR) data, and digital surface model (DSM) data generated by interpolating light detection and ranging (LiDAR) point cloud data. Through comparison with the ground survey, the experimental results show that the detection accuracy of the proposed method is 94.2 %; the area under the receiver operating characteristic (ROC) curve is 0.827. The efficiency of the proposed method is demonstrated using 6 modes of data combination acquired from the same area in old Beichuan County. The approach is one of the first attempts to extract damaged buildings through the fusion of three types of data with different features. The approach addresses multivariate regression methodologies and compares the potential of features for application in the damage detection field.

Optimized long short-term memory with rough set for sustainable forecasting renewable energy generation

Research and development in the field of renewable energy is receiving more attention as a result of the growing demand for clean, sustainable energy. This paper proposes a model for forecasting renewable energy generation. The proposed model consists of three main phases: data preparation, feature selection-based rough set and nutcracker optimization algorithm (NOA), and data classification and cross-validation. First, the missing values are tackled using the mean method. Then, data normalization and data shuffling are applied in the data preparation phase. In the second phase, a new feature selection algorithm is proposed based on rough set theory and NOA, namely RSNOA. The proposed RSNOA is based on adopting the rough set method as the fitness function during the searching mechanism to find the optimal feature subset. Finally, a custom long-short-term memory architecture with the k-fold cross-validation method is utilized in the last phase. The experimental results revealed that the proposed model is very competitive. It is achieved with 4.2113 root mean square error, 0.96 R², 2.835 mean absolute error, and 4.6349 mean absolute percentage error. The findings also show that the proposed model has great promise as a useful tool for accurately forecasting renewable energy generation across various sources.

Management of space surveillance radar temporal resource on fuzzy set theory

Management of space surveillance radar temporal resource on fuzzy set theory

Many problems, different frameworks: classification of problems in computable analysis and algorithmic learning theory

AbstractIn this thesis, we study the complexity of some mathematical problems: in particular, those arising in computable analysis and algorithmic learning theory for algebraic structures. Our study is not limited to these two areas: indeed, in both cases, the results we obtain are tightly connected to ideas and tools coming from different areas of mathematical logic, including for example descriptive set theory and reverse mathematics.After giving the necessary preliminaries, we first study the uniform computational strength of the Cantor–Bendixson theorem in the Weihrauch lattice. This work falls into the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. We concentrate on problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. As far as we know, this is the first systematic study of problems at the level of $\mathbf {\Pi }^1_1\mathsf {-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.We continue considering problems related to (induced) subgraphs. We provide results on the (effective) Wadge complexity of sets of graphs, that are also used to determine the Weihrauch degree of certain decision problems. The decision problems we consider are defined for a fixed graph G, and they take as input a graph H, answering whether G is an (induced) subgraph of H: we also consider the opposite problem (i.e., answering whether H is an induced subgraph of G). We conclude this part on (induced) subgraphs considering the Weihrauch degree of “search problems.”These problems are defined for a fixed graph G, and they take as input a graph H such that G is an (induced) subgraph H: the output is a copy of G in H. In both cases, we highlight differences and analogies between the subgraph and the induced subgraph relation.We then move our attention to algorithmic learning theory, and we present the framework we use to study the learnability of families of algebraic structures: here, given a countable family of pairwise nonisomorphic structures $\mathfrak {K}$ , a learner receives larger and larger pieces of an arbitrary copy of a structure in $\mathfrak {K}$ and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. We say that $\mathfrak {K}$ is learnable if there exists a learner which eventually stabilizes to a correct guess. The framework was lacking a method for comparing the complexity of nonlearnable families, and so we propose a solution to this problem using tools coming from invariant descriptive set theory. To do so, we first prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on infinite binary sequences and then, replacing $E_0$ with Borel equivalence relations of higher complexity, we obtain a new hierarchy of learning problems. This leads to the notion of E-learnability, where a family of structures $\mathfrak {K}$ is E-learnable, for a Borel equivalence relation E, if there is a continuous reduction from the isomorphism relation associated with $\mathfrak {K}$ to E. It is then natural to ask how the notion of E-learnability interacts with “classical” learning paradigms.Finally, we study the number of mind changes that a learner needs to learn a given family, both from a topological and a combinatorial point of view, and we consider how the complexity of a learner (in terms of Turing reducibility) affects the number of mind changes for learning a given family.Abstract prepared by Vittorio Cipriani.E-mail: vittorio.cipriani17@gmail.com.URL: https://air.uniud.it/handle/11390/1252226.