Computability theorists have introduced multiple hierarchies to measure the complexity of sets natural numbers. The Kleene Hierarchy classifies according first-order their defining formulas. Ershov $\Delta^0_2$ with respect number mistakes that are needed approximate them. Biacino and Gerla extended realm fuzzy sets, whose membership functions range in a complete lattice $L$ (e.g., real interva...

It is well known that free Boolean algebra on n generators isomorphic to the of functions variables. The distributive lattice monotone In this paper we introduce concept De Morgan function and prove Keywords: Antichain, function, algebra.

1. Closed ideals. (1.1) Definition. Let 5 be an arbitrary set, and 2 a Boolean ring of subsets of 5. The algebra B(S, 2) consists of all uniform limits of finite linear combinations of characteristic functions of sets in 2. The norm in B(S, 2) is given by the formula ||/|| =SupseS \f(s) | • This definition is slightly more general than that given in [l ] where 2 is required to be a field of set...

It is well known that Exclusive Sum-Of-Products (ESOP) expressions for Boolean functions require on average the smallest number of cubes. Thus, a simple complexity measure for a Boolean function is the number of cubes in its simplest ESOP. It will be shown that this structure-oriented measure of the complexity can be improved by a unique complexity measure which is based on the function. Thus, ...

We consider the classification problem to construct a classifier c : {0, 1}n → {0, 1} from a given set of examples (training set), which (approximately) realizes the hidden oracle y : {0, 1}n → {0, 1} describing the phenomenon under consideration. For this problem, a number of approaches are already known in computational learning theory; e.g., decision trees, support vector machines (SVM), and...

The notion of extended filter of a filter associated to a subset of Rl-monoids is defined and related properties are investigated. Rachunek and Salounova proved that positive implicative filters and Boolean filters in Rl-monoids coincide with a condition in Rachunek and Salounova (Acta Univ Palacki Olomuc Fac rer nat Math 48(1):93–107, 2009). In this paper, we prove that positive implicative fi...

Abstract In the independent works by Kalgin and Idrisova Beierle, Leander Perrin, it was observed that Gold APN functions over $$\mathbb {F}_{2^5}$$ F 2 5 give rise to a quadratic function in dimension 6 having maximum possible li...

Boolean-valued predicates over a state space are isomorphic to its characteristic functions into {0, 1}. Enlarging that range to {−1, 0, 1} allows the definition of extended predicates whose associated transformers generalise the conventional wp and wlp. The correspondingly extended healthiness conditions include the new ‘sub-additivity’, an arithmetic inequality over predicates.

The subject of this work is the possibility of private distributed computations of nargument functions defined over the integers. A function f is t private if there exists a protocol for computing f , so that no coalition of 5 t participants can infer any additional information from the execution of the protocol. It is known that over finite domains, every function can be computed prieven n pri...

It has been pointed out that an n-variable Boolean function f has optimal resistance against fast algebraic attacks if and only if there does not exist a nonzero n-variable Boolean function g of degree lower than n 2 such that fg = h and deg(g) + deg(h) < n. In this corresponding, we show that there does not exist an n-variable Boolean function with optimal resistance against fast algebraic att...