Nowadays is critical to complement Artificial Intelligence (AI) systems, such as those based on Deep Learning and Generative AI, by robust and trustworthy methodologies like different useful mathematical tools, such as (fuzzy) logic, formal concept analysis, rough set theory, etc. Multi-adjoint algebras are flexible structures considered in many of these mathematical tools, which are fundamental for obtaining traceable and reliable information from data sets. Along this line, this paper aims at introducing an algebraizable logic having the quasi-variety of these algebras as its associated equivalent algebraic semantics. To do so, we first introduce a logic associated with bounded lattices with an implication and show it is algebraizable in the sense of Blok-Pigozzi. We then expand this base logic to another algebraizable logic able to properly capture the multi-adjoint framework. Furthermore, we also consider different extensions of the logics and their properties analysed.

Datasets have been interpreted in (fuzzy) rough set theory as decision tables to obtain useful information to be used, for example, in decision making. These tables have been modeled through a collection of decision rules, which was called decision algorithm by Pawlak. These algorithms are analyzed by the notion of efficiency, which evaluates their quality of classification. This paper presents two different approaches for defining the notion of efficiency in the fuzzy framework. The first approach is a direct generalization to the classical case, while the second one is focused on obtaining a bounded efficiency preserving the philosophy of the classical framework. Both approaches are illustrated by means of different properties and examples.

The computation of (if-then) rules relating the most representative variables of a given dataset is a relevant goal in different areas of data analysis such as stock market prediction, disease diagnosis and census data analysis, among others. In Formal Concept Analysis (FCA), attribute implications follow the philosophy of the if-then rules establishing relationships between sets of attributes. This paper will focus on the theoretical development of attribute implications through the tools provided by the multi-adjoint concept lattice framework. We will show different notions to compute the validity of attribute implications, analyzing their properties and comparing the proposed alternatives. The use of linguistic hedges, and truth-stressing hedges in particular, is an interesting way of incorporating truth functions into attribute implications. Moreover, they have been considered in the computation of the concepts of a given context. This paper also introduces and studies the main properties of the multi-adjoint concept lattices framework with truth-stressing hedges.

Extended multi-adjoint logic programming is a non-monotonic logic programming framework whose semantics is defined in terms of stable models. This paper will focus on the theoretical development of a syntactical sufficient condition for the existence and uniqueness of stable models in extended multi-adjoint logic programming, through the notion of stratification. Namely, we will detail a constructive method for the computation of the unique stable model of a given stratified extended multi-adjoint logic program. As a consequence, this study will be an interesting alternative to the semantical sufficient conditions for the existence and the uniqueness of stable models, given in the literature for multi-adjoint normal logic and extended multi-adjoint logic programs.

A method to solve a latticed optimization problem constrained by a bipolar fuzzy relation equation is presented in this paper, under the hypothesis of a partially monotonic objective function. The solving strategy consists of transforming the problem into optimizing an order-preserving function in all arguments subject to another bipolar fuzzy relation equation. As a result, all the solutions of the original optimization problem can be deduced from the extremal elements of the feasible domain of the transformed problem. The presented approach embraces the particular case of linear optimization constrained by bipolar fuzzy relation equations.

Bipolar max-⁎ equations, with ⁎ a triangular norm, have recently become a popular research topic embedded in the broad field of fuzzy relational equations. In this paper, we lift the work from the restrictive setting of the real unit interval — obfuscating the underlying lattice-theoretical essence — to the general setting of complete distributive symmetric residuated lattices, allowing to build upon the vast body of knowledge on unipolar sup-⁎ equations on complete distributive residuated lattices. We determine the full solution set, with particular emphasis on the extremal solutions, of a bipolar sup-⁎ equation in case the right-hand side is a join-irreducible element. The results are illustrated by means of ample examples.

The problem of reducing information in databases is an important topic in formal concept analysis, which has been studied in several articles. In this article, we consider the fuzzy environment of the multi-adjoint concept lattices, since it is a general fuzzy framework that allows us to easily establish degrees of preference on the elements of the considered database. We introduce algorithms to discover the information contained in the relational system. By means of these algorithms, we classify the attributes of a multi-adjoint context, and build a minimal subset of attributes preserving the information of the original knowledge system.

Bousi Prolog is a first-order fuzzy logic programming language whose operational semantics is an adaptation of the SLD resolution principle and whose fuzzy unification algorithm is based on proximity relations. This programming language is suitable for dealing with query answering processes, advanced pattern matching, flexible deductive databases, knowledge-based systems and approximate reasoning. Specifically, Bousi Prolog has already been used in interesting real applications such as text cataloguing, knowledge discovery and linguistic feedback in computer games.This paper presents an initial study on the incorporation of the weak unification with fuzzy functor/arity mismatch, recently introduced by Aït-Kaci and Pasi, in the Bousi Prolog system. This fuzzy unification algorithm will allow to Bousi Prolog to obtain answers for query processes in which first-order terms with different arity and terms ordering are considered. Any system using unification mechanisms on imperfect domains, whose data can be inaccurate or uncertain, can be benefited from the approach presented in this paper.

Bipolar fuzzy relation equations are given from the fuzzy relation equations introduced by Sanchez in the 1980s considering a negation operator in the equations. Numerous applications require variables that show a bipolar character such as decision making and revenue management, hence the importance of studying bipolar fuzzy relation equations. According to the literature, bipolar max-min equations have already been studied and a characterization of their solutions, by means of a finite set of maximal and minimal solution pairs, has been provided. This paper will present a first study on bipolar max-product fuzzy relation equations with one equation containing different variables, which includes different interesting properties in order to guarantee both their solvability and the existence of the greatest (least) solution or maximal (minimal) solutions. Moreover, a characterization of the solvability of a particular system of two bipolar max-product fuzzy relation equations is given.

Fuzzy relation equations (FRE) are an important decision support system (DSS), for example, in fuzzy logic. FRE have recently been extended to a more general framework, called multiadjoint relation equations (MARE). This paper shows MARE as a fundamental DSS in multi-adjoint logic programming. For that purpose, multi-adjoint logic programs will be interpreted as a MARE, and the solvability of them will be given in terms of concept lattice theory. Furthermore, two approximations (optimistic and pessimistic approximations) of unsolvable equations will be obtained from a multiadjoint object-oriented concept lattice. Finally, a real-life example will be studied.

Recently α-cut irreducible and δ1δ2-multi-adjoint concept lattices have been introduced as two different methodologies focus on reducing the size of a given fuzzy concept lattice. The philosophy of both methodologies is completely different and so, the obtained lattices too. This paper analyzes the differences and proposes that the best is to combine both methodologies in order to obtain new procedures to reduce the information retrieval, only considering the important information for the user and with the advantages of both philosophies.