Kyoichi Nakashima

Variance-Importance of System Components

The paper defines the variance-importance of a component as the product of a relative sensitivity coefficient of component variance and the variance of component-unreliability estimate. Variance-importance is useful for identifying components that appreciably contribute to the uncertainty of system unreliability.

Optimal Design of a Series-Parallel System with Time-Dependent Reliability

The paper formulates an optimal reliability design problem for a series system made of parallel redundant subsystems. The variables for optimization are the number of redundant units in each subsystem and the reliability of each unit. There is a cost-constraint. The time for which the system reliability exceeds a specified value is to be maximized. Similarly the cost could be minimized for a constraint on the mission time and reliability. A solution method for the formulated problems is presented along with an example.

An Efficient Bottom-up Algorithm for Enumerating Minimal Cut Sets of Fault Trees

The paper improves the conventional bottom-up algorithm for enumerating minimal cut sets of fault tree. It is proved that, when the logical product of two reduced sum-of-product forms is expanded by the distribution rule, one need only check if each resulting term is absorbed by some terms of two original sum-of-product forms. The algorithm for executing this process is presented and illustrated by an example. The entire computer program is given in a supplement and the computational results for several examples are presented to demonstrate the efficiency of the algorithm.

Some fundamental properties of multiple-valued Kleenean functions and determination of their logic formulas

Multiple-valued Kleenean functions that are models of a Kleene algebra and are logic functions expressed by logic formulas composed of variables, constants, and logic operations AND OR, and NOT are discussed. The set of Kleenean functions, is a model with the largest number of logic functions among existing models of a Kleene algebra, such as fuzzy logic functions, regular ternary logic functions, and B-ternary logic functions. Mainly, it is shown that any p-valued Kleenean function is derived from a monotonic ternary input functions and any p-valued unate function is derived from a unate binary input function. The mapping relations between them and the method to determine the logic formula of the Kleenean function and unate function from that of the monotonic ternary input function and unate binary input function, respectively, are classified. 7-or-less-valued Kleenean functions and unate functions of 3-or-fewer variables are enumerated. It is known that the number of p-valued Kleenean functions increases stepwise and that of unate functions increases smoothly as p becomes larger. >

A necessary and sufficient condition for multiple-valued logical functions representable by AND, OR, NOT, constants, variables and determination of their logical formulae

A necessary and sufficient condition is derived for p-valued logical functions representable by logical formulas built from variables, constants 0,1....p-1, and the logical operations AND, OR, and NOT (p-valued LR-functions). A method for determining a logical formula for p-valued LR-functions is presented. since the method deals only with 3/sup n/ input vectors, it gives the logical formulas easily.<<ETX>>

Set-valued functions and regularity

In this paper, we focus on regularity and set-valued functions. The regularity was first introduced by S.C. Kleene (1952) into the propositional connectives of a ternary logic. Then, M. Mukaidono (1986) expanded the regularity of Kleene into n-variable ternary functions, and a ternary function which is regular is called a regular ternary logic function. Some studies expanded regular ternary logic functions into /spl tau/-valued functions, and studied properties of them. In this paper, we propose another extension of the concepts of the regularity in the sense of Kleene and Mukaidono. That is, we introduce regularity into r-valued set-valued functions. Further, we give properties of the set-valued functions with the regularity.

Multi-interval truth valued logic

Many types of fuzzy truth values have been proposed, for example, numerical truth values, interval truth values, triangular truth values and trapezoid truth values and so on. This paper will discuss on algebraic structures on a new type of fuzzy truth values called multi-interval truth values. A multi-interval truth value is defined by the union of some of interval truth values. The operations on multi-interval truth values will be defined by applying the extension principle to the operations AND, OR, and NOT on numerical truth values. Then, the paper will show that an algebraic structure of multi-interval truth values with the operations defined is a de Morgan bisemilattice, which is a bisemilattice with the unit and zero, and with a unary operation that satisfies the involution and de Morgans laws.

Some relationships between multiple-valued Kleenean functions and ternary input multiple-valued output functions

The multiple-valued Kleenean functions discussed are multiple-valued-logic functions represented by multiple-valued AND, OR, NOT, constants, and variables. First, when p=odd, ternary input p-valued output functions (or (3, p)-functions for short) are defined, and when p=even, ternary input (p+1)-valued output functions ((3, p+1)-functions for short) are defined by adding the value (p-1)/2. A derivation rule is proposed as a link between (3, p)-functions (or (3, p+1)-functions and p-valued (or (p+1)-valued) Kleenean functions. For p=odd, the mapping from monotonic (3,p)-functions to p-valued Kleenean functions is a bijection. For p=even, since the mapping from monotonic (3, p+1)-functions to p-valued Kleenean functions is not a bijection, a condition which makes the mapping a bijection is developed. Moreover, Kleenean functions with no constants are derived from B-ternary logic functions by the rule; then the mapping is a bijection.<<ETX>>

Multiple-valued logical functions derived from two-valued input multiple-valued output functions

A derivation rule for deriving p-valued logical functions from two-valued input p-valued output functions ((2, p)-functions) is provided. It is proved that by this rule, one p-valued function is derived from one (2, p)-function uniquely. The relationship between some classes of (2, p) functions and the classes of p-valued functions derived from them is clarified. Especially important is the result that all the p-valued unate functions are derived from monotonic (2, p)-functions, all the p-valued majority functions are derived from (2, p)-threshold functions, and all the p-valued functions derived from nonmonotonic (2, p)-functions can be expressed by logical formulae using only AND, OR, constants, and certain variables.<<ETX>>

On Optimal Redundancy of Multivalue-Output Systems

This paper considers improving the reliability of multivalue-output systems by the use of n-redundant systems in which n copies of systems are used redundantly and the output is determined from the outputs of those copies by the voter. A k-out-of-n redundant system minimizes the mean loss caused by the occurrence of output errors under the condition that the voter can be composed of only two kinds of operators, logical sum and logical product. The optimal k depends on the probability and loss matrices, but it can be specified in some special cases. The mean loss of multivalue-output systems with multichannels can be minimized by adopting k-out-of-n redundancy for each channel. The results provide a powerful guide to the improvement of fail-safe characteristics of many systems and the design of fault-tolerant systems.