Three-valued Functions Research Articles

Defining rough sets as core–support pairs of three-valued functions

We answer the question what properties a collection F of three-valued functions on a set U must fulfil so that there exists a quasiorder ≤ on U such that the rough sets determined by ≤ coincide with the core–support pairs of the functions in F. Applying this characterization, we give a new representation of rough sets determined by equivalences in terms of three-valued Łukasiewicz algebras of three-valued functions.

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.

Parameterized digital hardware design of pulse-coupled phase oscillator networks

This paper proposes a parameterized digital hardware design of pulse-coupled phase oscillator networks that exhibit various synchronization phenomena. We introduce a discretization model with three-valued coupled functions to construct simplified pulse-coupled phase oscillators for parameterized design. The designed digital phase oscillator circuit can change the calculation accuracy and the coupling function shape of oscillators by parameters. One- and two-dimensional pulse-coupled phase oscillator networks are emulated using the designed phase oscillators by setting parameters for the desired network. Experimental results show that the proposed circuit emulates in-/anti-phase and different (out-of-phase) synchronization in two pulse-coupled phase oscillators effectively in an analog large scale integration implementation, and also various synchronization phenomena are observed in large-scale one- and two-dimensional networks.

Approximate emergent synchrony in spatially coupled spiking neurons with discrete interaction.

Models for perceptual grouping and contour integration are presented. Connection weights depend on distances and angle differences, while neurons evolve following a spiking dynamics (Izhikevich's model in most of the considered cases). Although the studied synapses depend on discrete three-valued functions, simulations display the emergence of approximate synchrony, making these cognitive tasks possible. Noise effects are examined, and the possibility of achieving similar results with a different neuron model is discussed.

A map of dependencies among three-valued logics

Three-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value.

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 The Computational Consequences of Independence in Propositional Logic

Sandu and Pietarinen [Partiality and Games: Propositional Logic. Logic J. IGPL 9 (2001) 101] study independence friendly propositional logics. That is, traditional propositional logic extended by means of syntax that allow connectives to be independent of each other, although the one may be subordinate to the other. Sandu and Pietarinen observe that the IF propositional logics have exotic properties, like functional completeness for three-valued functions. In this paper we focus on one of their IF propositional logics and study its properties, by means of notions from computational complexity. This approach enables us to compare propositional logic before and after the IF make-over. We observe that all but one of the best-known decision problems experience a complexity jump, provided that the complexity classes at hand are not equal. Our results concern every discipline that incorporates some notion of independence such as computer science, natural language semantics, and game theory. A corollary of one of our theorems illustrates this claim with respect to the latter discipline.

Fuzzy Three-valued Switching Functions Using Choquet Integral

Fuzzy three-valued switching functions have true, false and unknown inputs and outputs values. The functions are calculated by Choquet integral whose fuzzy measure values are assigned from normal three-valued switching functions. The functions are extension of normal three-valued functions and satisfy the monotonicity with respect to vagueness.

Classification of Three-Valued Logical Functions

Classification of Three-Valued Logical Functions

Code-division multiplex technique using bridge functions

A code-division multiplexing technique using a set of orthogonal functions known as bridge functions as carrier codes has been investigated. These functions are three-valued functions which take the values +1, 0 and −1. The results of a computer simulation on the performance of the system in additive white Gausian noise are presented, and the system is compared with conventional time-division multiplexing.

The blocking effect for direct and inverse mass ratios

Abstract The features of the blocking effect for inverse mass ratio are analysed. It is demonstrated that the projectile laboratory differential cross-sections and energies are three-valued functions of the ejection angle in some angular regions. The situations are analysed when the three-valued dependences degenerate and the particles scattered at some angles have a wide energy spectrum. It is shown that for a divergent primary beam there is a limiting angle of recoil ejection smaller than π/2 and recoil cross-sections and energies are two-valued functions of the ejection angle.

Energy and spatial redistributions of the particle flux resulting from its interaction with the scattering center

Energy and intensity distributions of both projectiles and recoil atoms on the plane behind the scattering center at finite distance are calculated for different projectile-to-target mass ratio and different projectile energies. The projectile energy and intensity have been found to be double-valued or three-valued functions of the distance from the collision epicenter. At the same time, the recoil energy and intensity have proved to single-valued and double-valued functions of the distance from the collision epicenter for parallel and divergent projectile fluxes, respectively.

Some remarks on the tensorial linearization of general and linearly separable boolean functions.

The coefficients of the tensorial linearized expression of any boolean (in particular, linearly separable) function are exhibited as solutions of simple algebraic equations. A nontrivial connection with three-valued functions is shown to provide a tool which might serve to express a linearly separable function in terms of its Chow parameters. This formalism is meant for applications in neural net theory and in the study of statistical and critical phenomena of various kinds.

Ternary Threshold Logic

A new logical algebra, ternary threshold logic, is defined and developed. The system is shown to be capable of representing all three-valued functions, and two methods of synthesizing these functions from their truth tables are given. One of the methods produces a normal form that is analogous to the disjunctive normal form of Boolean algebra. A list of single-threshold-operator equivalents of certain normal forms is given. It is pointed out that magnetic film parametrons may be made to exhibit the logical properties of the operators of the system. Designs for ternary adder and comparator stages are given that could be constructed out of magnetic film parametrons.