K-valued Functions Research Articles

Synthesis of Optimal Correction Functions in the Class of Disjunctive Normal Forms

The paper proposes to consider individual heuristics as unreliably operating parts of the information processing system. In a separate case, several different heuristics are adopted to solve the same problem, and the results obtained are adjusted in a certain way. In this case, problems arise that are close in methodology to the problems of synthesizing reliable circuits from unreliable elements or making a collective expert decision. The work solves the problem of constructing an optimal correction function based on control material; classes of functions of k-valued logic under monotonicity restrictions are studied. A theorem on the completeness of the class of monotonic functions of k-valued logic for arbitrary k is proved, and a basis in the given class is proved and constructed. The problem of constructing an optimal corrector in the class of disjunctive normal forms of k-valued functions is solved.

On some classes of formulas in S5 which are pre-complete relative to existential expressibility

Existential expressibility for all k-valued functions was proposed by A. V. Kuznetsov and later was investigated in more details by S. S. Marchenkov. In the present paper, we consider existential expressibility in the case of formulas defined by a logical calculus and find out some conditions for a system of formulas to be closed relative to existential expressibility. As a consequence, it has been established some pre-complete as to existential expressibility classes of formulas in some finite extensions of the paraconsistent modal logic S5.

On the universality of product for classes of linear functions of two variables

Abstract Earlier, the author introduced the concept of a universal function and proved the existence of universal functions for classes of linear k-valued functions of two variables for k ≥ 5. In this paper, we show that the product modulo k is a universal function for the class of linear k-valued functions of two variables if and only if k = 6l ± 1.

О нахождении оценки сложности дискретных k-значных функций

О нахождении оценки сложности дискретных k-значных функций

Hardware Implementation of Code Converters Designed to Reduce the Length of Binary Encoded Words

The problem of synthesis of combinational circuits of code converters designed to reduce the length of words from a given set of encoded binary words is considered. The encoding assumes that different binary words will be encoded by different binary codes of shorter length. Code converters of this type are designed to reduce the length of binary words transmitted in digital systems over data buses when the bit depth of the transmitted words exceeds the bit depth of the data bus. For example, 18-bit or 17-bit words need to be transmitted over a 16-bit data bus. Each such word can be transmitted in two cycles of operation of a digital system, however, this approach reduces the overall performance of the system. One of the approaches to solve such problems is the development of combinational circuits that convert long binary encoded words into shorter ones. The proposed methods for solving the problem of synthesizing circuits of code converters are based on the compilation and logical minimization of such forms of systems of incompletely specified Boolean functions as disjunc­tive normal forms (DNF) and binary decision diagrams (BDD). Using BDD to minimize representations of k-valued functions that depend on Boolean variables is also proposed. Technologically independent logical minimization of functional descriptions of the designed code converters is proposed to be performed by programs for minimizing systems of Boolean functions in the DNF class and programs for joint minimization of BDD representations of systems of completely specified Boolean functions. Minimization of functional descriptions is aimed at reducing the hardware complexity of combinational circuits in the basis of library elements or programmable FPGA elements implementing code converters of the class in question.

Regularity of certain commutative Banach rings

Let (A,‖⋅‖) be a commutative unital Banach ring. Let M(A) be the Berkovich spectrum of A, i.e., the set of all bounded multiplicative non-zero semi-norms defined on A which is endowed with the pointwise convergence topology. As in the case of complex Banach algebras, we introduce the notions of the hull-kernel topology on M(A) and the regularity of a commutative unital Banach ring. We show that the Banach algebra of all bounded continuous k-valued functions defined on a zero-dimensional topological space is regular, where k is a complete valuation field. Hence, we see that the Berkovich spectrum of this algebra is a Hausdorff space in the hull-kernel topology. Next, we use the tool of Gelfand transform to characterize a Banach ring of finite Berkovich spectrum. From this, we see that if A is uniform, i.e., ‖f2‖=‖f‖2 for all f in A, and has finite Berkovich spectrum, then it is regular.

Intermediate Rings of a Class of Ordered Field Valued Continuous Functions

Intermediate rings A(X, K) of K-valued continuous functions lying between B(X, K) and C(X, K), defined over a zero dimensional space X, are investigated and studied in this article, here K stands for a countable subfield of ℝ. It is realized that the structure space of A(X, K) is β 0 X, the Banaschewski compactification of X. The Hewitt realcompactification analogue υK (X) of υX is defined. Several equivalent descriptions of pseudocompactness of X via υK (X), β 0 X and the uniform topology and the m-topology on C(X, K) are given. The article ends after studying an intercorrelation between z-ideals and z°-ideals in the rings C(X, K) and Cc (X), the functionally countable subalgebra of C(X). This study eventually leads to a characterization of P -spaces and almost P-spaces in terms of z-ideals and z°-ideals in C(X, K).

Calculating the Number of Functions with a Given Endomorphism

An iterative procedure is proposed for calculating the number of k-valued functions of n variables such that each one has an endomorphism different from any constant and permutation. Based on this procedure, formulas are found for the number of three-valued functions of n variables such that each one has nontrivial endomorphisms. For any arbitrary semigroup of endomorphisms, the power is found of the set of all three-valued functions of n variables such that each one has endomorphisms from a specified semigroup.

Minor complexity of discrete functions

In this paper we study a class of complexity measures, induced by a new data structure for representing k-valued functions (operations), called minor decision diagram. When assigning values to some variables in a function the resulting functions are called subfunctions, and when identifying some variables the resulting functions are called minors. The sets of essential variables in subfunctions of f are called separable in f.We examine the maximal separable subsets of variables and their conjugates, introduced in the paper, proving that each such set has at least one conjugate. The essential arity gap gap(f) of the function f is the minimal number of essential variables in f which become fictive when identifying distinct essential variables in f. We also investigate separable sets of variables in functions with non-trivial arity gap. This allows us to solve several important algebraic, computational and combinatorial problems about the finite-valued functions.

Asymptotically Optimal in Reliability Circuits in Two Bases Under Failures of 0 (k − 1) Type at the Outputs of Elements

We consider a problem of the realization of k-valued logics functions (k ≥ 3) by circuits in two bases: in the Rosser–Turkett basis and in its dual basis. We assume that the basis gates are exposed to faults at outputs: only of type 0 or only of type k − 1, and they pass into faulty states independently of each other. We describe a constructive method for the synthesis of an asymptotically optimal reliable circuit for almost any function of k-valued logic, we found the upper and lower bounds of circuits unreliability and the class of functions for which the lower bounds are true.

On a theory of the b-function in positive characteristic

We present a theory of the b-function (or Bernstein–Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic p>0. Its b-function b_f is defined to be an ideal of the algebra of continuous k-valued functions on {mathbb {Z}}_p. The zero-locus of the b-function is thus naturally interpreted as a subset of {mathbb {Z}}_p, which we call the set of roots of b_f. We prove that b_f has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Mustaţă and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of b_f and relate it to the test ideals of f.

Existence of Universal Functions for the Class of Linear k-Valued Functions with Moderate k

The article describes the construction of discrete functions which, by some of their values, specify (generate) arbitrary linear functions. The cases of prime and sufficiently large composite k have been considered previously. In the present study we finally solve the problem of existence of such functions for almost all k and n variables. The proof of the probabilistic upper bound and the general approach are due to A. A. Voronenko. The proof for small k has been developed by N. K. Voronova. The proof for k from 21 to 48 is the result of indispensable cooperation of V. P. Il'yutko and A. A. Voronenko.

Recognizing Parameters of the Information Security Unit Implemented by the Threshold K-Valued Function

This article discusses some approaches to the recognition of the parameters of the threshold k-valued functions, which can be used for building information processing and security units. The main focus is put on the issue of proving k-valued function belonging to the threshold class. For solving this problem it is proposed to use the input coefficients of expansion and increase. With the help of the latter, the coefficients of linear forms of the k-valued threshold function are procedurally approximated. Along with the proposed analytical approach, the article discusses an algorithmic method based on reducing the problem of finding a threshold representation of k-valued functions to the system of linear inequalities, for the solution of which the ellipsoid method, modified by Khachiyan, is applied. The comparative analysis of the proposed methods is carried out based on experiments.

On the outcome of false images of linear k-valued functions for composite numbers k when the number of variables increases

The problem of constructing discrete functions such that parts of their value sets determine (generate) arbitrary linear functions is considered. A case in which k is a prime number was considered earlier by the author. It is proved that the existence of such partial functions wshen the number of independent variables is no less then two implies they exists for any arbitrary greater number of independent variables. Upper estimates linear with respect to the number of independent variables are proved for the size of the domain of universal functions. The existence of two-variable universal functions is proved for sufficiently large k.

Function of One Regular Separable Relation Set Decided for the Minimal Covering in Multiple Valued Logic

Multiple-valued logic is an important branch of the computer science and technology. Multiple-valued logic studies the theory, multiple-valued circuit & multiple-valued system, and the applications of multiple-valued logic included.In the theory of multiple-valued logic, one primary and important problem is the completeness of function sets, which can be solved depending on the decision for all the precomplete sets(also called maximal closed sets) of K-valued function sets noted by P K * , and another is the decision for Sheffer function, which can be totally solved by picking out all of the minimal covering of the precomplete sets. In the function structure theory of multi-logic, decision on Sheffer function is an important role. It contains structure and decision of full multi-logic and partial multi-logic. Its decision is closely related to decision of completeness of function which can be done by deciding the minimal covering of full multi-logic and partial-logic. By theory of completeness of partial multi-logic, we prove that function of one regular separable relation is not minimal covering of P K * under the condition of m = 2, σ = e.

Smooth representations and Hecke modules in characteristicp

Let G be a p-adic Lie group and I ⊆ G be a compact open subgroup which is a torsionfree pro-p-group. Working over a coefficient field k of characteristic p we introduce a differential graded Hecke algebra for the pair (G, I) and show that the derived category of smooth representations of G in k-vector spaces is naturally equivalent to the derived category of differential graded modules over this Hecke DGA. 1 Background and motivation Let G be a d-dimensional p-adic Lie group, and let k be any field. We denote by Modk(G) the category of smooth G-representations in k-vector spaces. It obviously has arbitrary direct sums. We fix a compact open subgroup I ⊆ G. In Modk(G) we then have the representation indI (1) := all k-valued functions with finite support on G/I with G acting by left translations. Being generated by a single element, which is the characteristic function of the trivial coset, indI (1) is a compact object in Modk(G). It generates the full subcategory Modk(G) of all representations V in Modk(G) which are generated by their I-fixed vectors V I . In general Modk(G) is not an abelian category. The Hecke algebra of I by definition is the endomorphism ring HI := EndModk(G)(ind G I (1)) op . We let Mod(HI) denote the category of left unital HI -modules. There is the pair of adjoint functors H : Modk(G) −→ Mod(HI) V 7−→ V I = HomModk(G)(ind G I (1), V ) , and T0 : Mod(HI) −→ Modk(G) ⊆ Modk(G) M 7−→ indI (1)⊗HI M .

On the Generation of False Images of Linear k-Valued Functions

The article examines the construction of discrete functions such that a part of their value set specifies (generates) arbitrary linear functions. For prime k greater than 5, we prove the existence of such partial functions with no fewer than two variables and derive linear upper and lower bounds in the number of variables for the size of their definition domain.

О надструктуре класса квазиоднородных k-значных функций

О надструктуре класса квазиоднородных k-значных функций

Bi-decomposition of multi-valued logical functions and its applications

The bi-decomposition of multi-valued logical (MVL) functions, including disjoint and non-disjoint cases, is considered. Using a semi-tensor product, an MVL function can be expressed in its algebraic form. Based on this form, straightforward verifiable necessary and sufficient conditions are provided for each case, respectively. The constructive proofs also lead to constructing corresponding decompositions. Using these results, the implicit function theorem (IFT) of k-valued functions, as a special bi-decomposition, is obtained. Finally, as an application, the normalization of dynamic–algebraic (D–A) Boolean networks is investigated using IFT of k-valued functions.

Finite Symmetric Functions with Non-Trivial Arity Gap

Given an n-ary k-valued function f, gap(f) denotes the essential arity gap of f which is the minimal number of essential variables in f which become fictive when identifying any two distinct essential variables in f. In the present paper we study the properties of the symmetric function with non-trivial arity gap (2 ≤ gap(f)). We prove several results concerning decomposition of the symmetric functions with non-trivial arity gap with its minors or subfunctions. We show that all non-empty sets of essential variables in symmetric functions with non-trivial arity gap are separable. ACM Computing Classification System (1998): G.2.0.