Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the task and the data are encoded as axioms of an algebra, and a model is obtained where only these axioms and their logical consequences hold. Although this is not a generalizing model, we show that selecting specific subsets of its breakdown into algebraic atoms obtained via subdirect decomposition gives a model that generalizes. We validate this new learning principle on standard datasets such as MNIST, FashionMNIST, CIFAR-10, and medical images, achieving performance comparable to optimized multilayer perceptrons. Beyond data-driven tasks, the new learning principle extends to formal problems, such as finding Hamiltonian cycles from their specifications and without relying on search. This algebraic foundation offers a fresh perspective on machine intelligence, featuring direct learning from training data without the need for validation dataset, scaling through model additivity, and asymptotic convergence to the underlying rule in the data.

Representations of domains mean in a general way representing a domain as a suitable family endowed with set-inclusion order of some mathematical structures. In this paper, representations of domains via CF-approximation spaces are considered. Concepts of CF-approximation spaces and CF-closed sets are introduced. It is proved that the family of CF-closed sets in a CF-approximation space endowed with set-inclusion order is a continuous domain and that every continuous domain is isomorphic to the family of CF-closed sets of some CF-approximation space endowed with set-inclusion order. The concept of CF-approximable relations is introduced using a categorical approach, which later facilitates the proof that the category of CF-approximation spaces and CF-approximable relations is equivalent to that of continuous domains and Scott continuous maps.

The report suggests the concept of risk, outlining two mathematical structures necessary for risk genesis: the set of outcomes and, in a general case, partial order of preference on it. It is shown that this minimum partial order should constitute the structure of a semilattice. In some cases, there should be a system of semilattices nested in a certain way. On this basis, the classification of risk theory tasks is given in the context of specialization of mathematical knowledge. In other words, we are talking about the development of a new rela-tional risk theory. The problem of political decision making in game-theoretic formulation in terms of having partial order of preference on the set of outcomes for each par-ticipant of the game forming a certain system of nested semilattices is consid-ered as an example of a relational risk concept implementation. Solutions to the problem obtained through the use of various optimality principles are investi-gated.

The functional and the algebraic routing problem are generalizations of the shortest path problem. This paper shows that both problems are equivalent with respect to the concept of profile searches known from time-dependent routing. Because of this, it is possible to apply various shortest path algorithms to these routing problems. This is demonstrated using contraction hierarchies as an example. Furthermore, we show how to use Cousots' concept of abstract interpretation on these routing problems generalizing the idea of routing approximations, which can be used to find approximative solutions and even to improve the performance of exact queries. The focus of this paper lies on vehicle routing while both the functional and algebraic routing models were introduced in the context of internet routing. Due to our formal combination of both fields, new algorithms abound for various specialized vehicle routing problems. We consider two major examples, namely the time-dependent routing problem for public transportation and the energy-efficient routing problem for electric vehicles.

We introduce abstract preference frameworks to study general properties common across a variety of preference formalisms. In particular, we study strong equivalence in preference formalisms and their separability. We identify abstract postulates on preference frameworks, satisfied by most of the currently studied preference formalisms, that lead to characterizations of both properties of interest.