A fuzzified BRAIN algorithm for learning DNF from incomplete data

Aim of this paper is to address the problem of learning Boolean functions from training data with missing values. We present an extension of the BRAIN algorithm, called U-BRAIN (Uncertainty-managing Batch Relevance-based Artificial INtelligence), conceived for learning DNF Boolean formulas from partial truth tables, possibly with uncertain values or missing bits. Such an algorithm is obtained from BRAIN by introducing fuzzy sets in order to manage uncertainty. In the case where no missing bits are present, the algorithm reduces to the original BRAIN

‎Lattices of (Generalized) Fuzzy Ideals in Double Boolean Algebras

This paper develops the notion of fuzzy ideal and generalized fuzzy ideal on double Boolean algebra (dBa)‎. ‎According to Rudolf Wille‎, ‎a double Boolean algebra $\underline{D}:=(D‎, ‎\sqcap‎, ‎\sqcup‎, ‎\neg‎, ‎\lrcorner‎, ‎\bot‎, ‎\top)$ is an algebra of type $(2‎, ‎2‎, ‎1‎, ‎1‎, ‎0‎, ‎0),$ which satisfies a set of properties‎. ‎This algebraic structure aimed to capture the equational theory of the algebra of protoconcepts‎. ‎We show that collections of fuzzy ideals and generalized fuzzy ideals are endowed with lattice structures‎. ‎We further prove that (by isomorphism) lattice structures obtained from fuzzy ideals and generalized fuzzy ideals of a double Boolean algebra D can entirely be determined by sets of fuzzy ideals and generalized fuzzy ideals of the Boolean algebra $D_{\sqcup}.

Formalization of Human Categorization Process Using Interpolative Boolean Algebra

Since the ancient times, it has been assumed that categorization has the basic form of classical sets. This implies that the categorization process rests on the Boolean laws. In the second half of the twentieth century, the classical theory has been challenged in cognitive science. According to the prototype theory, objects belong to categories with intensities, while humans categorize objects by comparing them to prototypes of relevant categories. Such categorization process is governed by the principles of perceived world structure and cognitive economy. Approaching the prototype theory by using truth-functional fuzzy logic has been harshly criticized due to not satisfying the complementation laws. In this paper, the prototype theory is approached by using structure-functional fuzzy logic, the interpolative Boolean algebra. The proposed formalism is within the Boolean frame. Categories are represented as fuzzy sets of objects, while comparisons between objects and prototypes are formalized by using Boolean consistent fuzzy relations. Such relations are directly constructed from a Boolean consistent fuzzy partial order relation, which is treated by Boolean implication. The introduced formalism secures the principles of categorization showing that Boolean laws are fundamental in the categorization process. For illustration purposes, the artificial cognitive system which mimics human categorization activity is proposed