On stationary classes of three-value logic functions

This work is devoted to the properties of transformations of the vector of values of three-value logic functions to the vector of coefficients of their polynomials. A similar transformation of Boolean functions is used in cryptology, and its properties have been thoroughly studied. Stationary classes of three-value logic functions are introduced, and their hierarchy and the exact number of functions in them are obtained.     

Balanced biorthogonal scaling vectors using fractal function macroelements on

Geronimo, Hardin et al. have previously constructed orthogonal and biorthogonal scaling vectors by extending a spline scaling vector with functions supported on [0,1]. Many of these constructions occurred before the concept of balanced scaling vectors was introduced. This paper will show that adding functions on [0,1] is insufficient for extending spline scaling vectors to scaling vectors that are both orthogonal and balanced. We are able, however, to use this technique to extend spline scaling vectors to balanced, biorthogonal scaling vectors, and we provide two large classes of this type of scaling vector, with approximation order two and three, respectively, with two specific constructions with desirable properties in each case. The constructions will use macroelements supported on [0,1], some of which will be fractal functions.     

On the distribution of three-valued functions over pre-complete classes

A system of 36 ??axioms?? binding pre-complete classes of a three-valued logic is proposed. On the basis on these axioms (and their duals), a set of 406 rows is built in which all possible cases of distributions of three-valued functions over pre-complete classes are presented.     

On function classes in P 3 precomplete with respect to a strengthened closure operator

A classical theorem of Post [1] describes five precomplete classes in the set of Boolean functions. In [2], it was shown that there exist 18 precomplete classes of functions of three-valued logic. In [1, 2], the closure of sets of functions with respect to the substitution operator was studied. We consider two closure operators on functions of three-valued logic, which are obtained by supplementing the substitution operator by closures with respect to two identifications of function values, and prove the existence of three precomplete classes for one of these operators and five precomplete classes for the other.     

Orders of discriminator classes in multivalued logic

Mathematical Notes - For k ≥ 2, discriminator classes, that is, closed classes of functions of k-valued logic containing the ternary discriminator p, are considered. It is proved that any...     

计量逻辑学中的雪崩逻辑公式

将密码学中满足严格雪崩准则的布尔函数的概念引入到计量逻辑学之中,提出了雪崩逻辑公式的概念,并研究了雪崩逻辑公式的真度及其性质。证明了至少含有三个原子公式的雪崩逻辑公式的真度之集为H1={k/2n-12n-3≤k≤3×2n-3;n=3,4,…},在此基础上,通过引入函数ξ建立了n(n≥3)元雪崩布尔函数个数的表达式,给出了雪崩逻辑公式的构造方法。最后,研究了反射变换下k阶雪崩逻辑公式的性质。     

FE classification of functions of many-valued logic

On the set of functions of many-valued logic, we consider the closure operator defined on the basis of systems of functional equations (the operator of FE closure). This operator generates an FE classification of the functions of many-valued logic, whose kernel consists of classes of S _G -type defined by groups of permutations of G. A number of results are obtained to guarantee FE precompleteness of classes of only this type in the classes of S _G type. The results obtained are illustrated by examples of functions of 2-, 3-, and 4-valued logics.     

Classes generated by monotone symmetric functions in the three-valued logic

Closed classes of functions of the three-valued logic whose generating systems consist of monotone symmetric functions taking values in the set {0, 1} are considered. Criteria for existence of a basis and for existence of a finite basis are obtained for closed classes from the family under consideration     

Closed classes generated by symmetric functions in the three-valued logic

Closed classes of functions in the three-valued logic whose generating systems consist of symmetric functions taking values in the set {0, 1} are studied in the paper. Criteria for the existence of a basis and for the existence of a finite basis are obtained for closed classes from the considered family.     

Finding the optimal metric in classification problems with ordinal features

A method for finding the optimal distance function for the classification problem with two classes in which the objects are specified by vectors of their ordinal features is proposed. An optimal distance function is sought by the minimization of the weighted difference of the average intraclass and interclass distances. It is assumed that a specific distance function is given for each feature, which is defined on the Cartesian product of the set of integer numbers in the range from 0 to N − 1 and takes values from 0 to M. Distance functions satisfy modified metric properties. The number of admissible distance functions is calculated, which enables one to significantly reduce the complexity of the problem. To verify the appropriateness of metric optimization and to perform experiments, the nearest neighbor algorithm is used.     

Some classes of analytic and multivalent functions involving a linear operator

By using a linear operator, which is defined here by means of the Hadamard product (or convolution), the authors introduce some new classes of analytic and multivalent functions in the open unit disk and investigate their inclusion relationships and convolution properties. Integral transforms of functions in these classes are also discussed.     

Generalized arcwise-connected functions and characterizations of local-global minimum properties

In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.Dedicated to R. BellmanThe authors are indebted to I. Kozma, N. Megiddo, and A. Tamir for valuable discussions and to S. Schaible for valuable remarks. This research was partially supported by the Fund for the Encouragement of Research at the Technion.     

Closed classes of the three-valued logic generated by systems containing symmetric functions

Closed classes of functions of the three-valued logic whose generating systems include nonmonotone symmetric functions taking values in the set {0, 1} are studied. It is shown that in some cases the problems of existence of a basis and existence of a finite basis can be reduced to similar problem for reduced generating systems.     

Ultra‐differentiable classes and intersection theorems

The aim of the paper is to study the relation between ultra‐differentiable classes of functions defined in terms of estimates on derivatives on one hand and in terms of growth properties of Fourier transforms of suitably localized functions in the class on the other hand. We establish this relation for the ultra‐differentiable classes introduced in 6 , 16 , and show that the classes of 6 , 16 , can be regarded as inhomogeneous Gevrey classes in the sense of 22 . We also discuss a number of properties of the weight functions used to define the respective classes and of their Young conjugates.     

On Monodromy of Generalized Analytic Functions

We discuss a number of topics concerned with certain boundary-value problems in the context of generalized analytic functions. Solution of the classical Riemann-Hilbert problem and the linear conjugation problem for analytic functions is described in appropriate function classes and the same scheme is applied to generalized analytic functions and vectors. In particular, we describe solution of the Riemann-Hilbert problem for generalized analytic functions and obtain an explicit analytic presentation of monodromy matrices in the case of generalized analytic vectors. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.     

Approximation of smooth vectors of a closed operator by entire vectors of exponential type

Some classes of infinitely differentiable vectors of a normal operator in a Hilbert space are described in terms of the rate of convergence to zero of the best approximation of these vectors by entire vectors of exponential type. As a special case, we obtain many well-known results in the theory of approximations of continuous functions by algebraic or trigonometric polynomials or by entire functions of exponential type. The proofs of these results are essentially based on the theory of spaces with positive and negative norm developed by our teacher, Prof. Yu. M. Berezanskii, to whom we dedicate this paper as a sign of our deep gratitude.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 616–628, May, 1995.     

Modal characterisation theorems over special classes of frames

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Benthem’s theorem, which says that a first-order formula is invariant under bisimulation if, and only if, it is equivalent to a formula of basic modal logic. The present investigation primarily concerns ramifications for specific classes of structures. We study in particular model classes defined through conditions on the underlying frames, with a focus on frame classes that play a major role in modal correspondence theory and often correspond to typical application domains of modal logics. Classical model theoretic arguments do not apply to many of the most interesting classes-for instance, rooted frames, finite rooted frames, finite transitive frames, well-founded transitive frames, finite equivalence frames-as these are not elementary. Instead we develop and extend the game-based analysis (first-order Ehrenfeucht-Fraïssé versus bisimulation games) over such classes and provide bisimulation preserving model constructions within these classes. Over most of the classes considered, we obtain finite model theory analogues of the classically expected characterisations, with new proofs also for the classical setting. The class of transitive frames is a notable exception, with a marked difference between the classical and the finite model theory of bisimulation invariant first-order properties. Over the class of all finite transitive frames in particular, we find that monadic second-order logic is no more expressive than first-order as far as bisimulation invariant properties are concerned — though both are more expressive here than basic modal logic. We obtain ramifications of the de Jongh-Sambin theorem and a new and specific analogue of the Janin-Walukiewicz characterisation of bisimulation invariant monadic second-order for finite transitive frames.     

Generalized skew bisubmodularity: A characterization and a min–max theorem

Huber, Krokhin, and Powell (2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain. In this paper we consider a natural generalization of the concept of skew bisubmodularity and show a connection between the generalized skew bisubmodularity and a convex extension over rectangles. We also analyze the dual polyhedra, called skew bisubmodular polyhedra, associated with generalized skew bisubmodular functions and derive a min–max theorem that characterizes the minimum value of a generalized skew bisubmodular function in terms of a minimum-norm point in the associated skew bisubmodular polyhedron.     

A topology for galois types in abstract elementary classes

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model‐theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Positively closed classes of three-valued logic

We state some theoretical prerequisites and indicate approaches to constructing a positive classification of the set of functions of k-valued logic. Basing on this, we find all 194 positively closed classes of three-valued logic. We describe them using the semigroups of endomorphisms and indicating positive bases.