In this paper, we focus on regularity and set-valued functions. Regularity was first introduced by S. C. Kleene in the propositional operations of his ternary logic. Then, M. Mukaidono investigated some properties of ternary functions, which can be represented by regular operations. He called such ternary functions "regular ternary logic functions". Regular ternary logic functions are useful for representing and analyzing ambiguities such as transient states or initial states in binary logic circuits that Boolean functions cannot cope with. Furthermore, they are also applied to studies of fail-safe systems for binary logic circuits. In this paper, we will discuss an extension of regular ternary logic functions into r-valued set-valued functions, which are defined as mappings on a set of nonempty subsets of the r-valued set {0, 1, . . . , r-1}. First, the paper will show a method by which operations on the r-valued set {0, 1, . . . , r-1} can be expanded into operations on the set of nonempty subsets of {0, 1, . . . , r-1}. These operations will be called regular since this method is identical with the way that Kleene expanded operations of binary logic into his ternary logic. Finally, explicit expressions of set-valued functions monotonic in subset will be presented.

The paper considers multiple-valued logic systems having the property that the ambiguity of the system increases as the ambiguity of each component increases. The partial-ordering relation with respect to ambiguity with the greatest element 1/2 and minimal elements 0, 1 or simply the ambiguity relation is introduced in the set of truth values V {0, 1/ (p1), , 1/2, , (p2) / (p1), 1}. A-monotonic p-valued logic functions are defined as p-valued logic functions monotonic with respect to the ambiguity relation. A necessary and sufficient condition for A-monotonic p-valued logic functions is presented along with the proofs, and their logic formulae using unary operators defined in the ambiguity relation are given. Some discussions on the extension of theories to other partial-ordering relations are also given.

The purpose of this paper is to establish an access control system by using only fingerprint identification. In order to minimize the identification time, we propose a new fingerprint classification suitable for a personal computer, and the real machine by using the classification is introduced. Our classification is implemented by only cores which are one of the features on fingerprint pattern. Therefore, it classifies all fingerprints into one of 11 classes rapidly on a personal computer. In the machine, an input fingerprint is classified and compared with ones registered in the same class. If both the input fingerprint and the registered one match, the person is allowed entry to the restricted area. Simulation results show that 443 fingerprint patterns (45 persons) are classified completely and rapidly. And the machine is effective and useful as identifier for home and room security.

The interconnection problem of binary circuits becomes seriously as the exponential growth of the circuits complexity has been driven by a combination of down scaling of the device size and up scaling of the chip size. Motivated by the problem, there is much research of circuits based on multiple-valued logic. On the other hand, caused by the signal propagation delay, there exist hazards in binary logic circuits. To analyze hazards in binary logic circuits, many multiple-valued logics have been proposed, and studied on their mathematical properties. The paper will discuss on a multiple-valued logic which is suitable for treating static hazards in multiple-valued logic circuits. Then, the paper will show that the prime implicants expressions of r-valued logic functions realize static hazards free r-valued logic circuits.

If there is high uncertainty in the reliability of a system, it is required to know for which components to collect further failure data for making the uncertainty lower. This paper provides some solutions for such a requirement. First, the variance importance of a component is defined as a measure of component's contribution to the variance of the distribution of system unreliability. For reducing the system-variance efficiently, it would be suitable to collect failure data of comopnents having greater values of variance importance. Next, in the situation that we will perform life tests of components for making the variance of the distribution of system failure-rate smaller than its prior value, some problems for determinig both the number of samples to be tested and the test time for each component are formulated and solved.

The paper deals with Kleenean functions defined as fuzzy logic functions with constants. Kleenean functions provide a means of handling conditions of indeterminate truth value (ambiguous states) which ordinary classical logic (binary logic) cannot cope with. This paper clarifies a necessary and sufficient condition for a function to be a Kleenean function. The condition is provided with a set of two conditions, and it will be shown that they are independent of each other.

For systems in which the probability that an incorrect output is observed differs with input values, we adopt the redundant usage of n copies of identical systems which we call the n-redundant system. This paper presents a method to find the optimal redundancy of systems for minimizing the probability of dangerous errors. First, it is proved that a k-out-of-n redundancy or a mixture of two kinds of k-out-of-n redundancies minimizes the probability of D-errors under the condition that the probability of output errors including both dangerous errors and safe errors is below a specified value. Next, an algorithm is given to find the optimal series-parallel redundancy of systems by using the properties of the distance between two structure functions.

This letter shows that an m-valued majority function is realized by m-1 two-valued threshold functions with common weight vector. A necessary and sufficient condition for an m-valued logical function to be a majority function is provided by using the concept of asummability.

Recently, fuzzy logic which is a kind of infinite multiple-valued logic has been studied to treat certain ambiguities, and its algebraic properties have been studied by the name of fuzzy logic functions. In order to treat modality (necessity, possibility) in fuzzy logic, which is an important concept of multiple-valued logic, the intuitionistic logical negation is required in addition to operations of fuzzy logic. Infinite multiple-valued logic functions introducing the intuitionistic logical negation into fuzzy logic functions are called Kleene-Stone logic functions, and they enable us to treat modality. The domain of modality in which Kleene-Stone logic functions can handle, however, is too limited. We will define α-KS logic functions as infinite multiple-valued logic functions using a unary operation instead of the intuitionistic logical negation of Kleene-Stone logic functions. In α-KS logic functions, modality is closer to our feelings. In this paper we will show some algebraic properties of α-KS logic functions. In particular we prove that any n-variable α-KS logic function is determined uniquely by all inputs of 7 values which are 7 specific truth values of the original infinite truth values. This means that there is a bijection between the set of α-KS logic functions and the set of 7-valued α-KS logic functions which are restriction of α-KS logic functions to 7 specific truth values. Finally, we show a necessary and sufficient condition for a 7-valued logic function to be a 7-valued α-KS logic function.

This paper is concerned with the synthesis of ternary logic function based on NOR-type polypheck. The system of fundamental operators consists of cyclic, inverse cyclic, logical sum and logical product. It is shown that any ternary logic function is represented by the logical sum of at most 2n-order terms. The simplification method in the binary logic system is extendedly applied to the ternary logic system. In synthesizing a ternary logic function based on NOR-type polypheck, the efficient tool, size of term, is newly-introduced. Size of term is defined by the total arithmetic sum of truth values of a term over all assignments of variable values. It has such characteristic that the decreased order of a term results in the increase of the size of the term. The utility of NOR-type polypheck is also discussed. The proposed method is applied to the ternary half-adder function as an example.

The reliability analysis of multi-state systems is needed for treating multi-modal failures, failure interdependency and degraded states. This paper defines some types of multi-state monotone systems from the positive relation between two states of components and presents their properties with emphasis on properties of Boolean structure functions with restricted 0-1 variables. Positive-directionally (or negative-directionally) monotone system is defined as the concept of monotone system wider than the former one, and its deterministic and probabilistic properties useful for the reliability analysis are given. Extended definitions of series, parallel and series-parallel systems are also given.

This paper presents a method for implementing the testing and realization of three-valued majority functions by using properties of 02-complete monotonicity which is an extended concept of complete monotonicity in binary logic. It is shown that reduced functions of three-valued majority functions are 02-completely monotonic, and all 7 or less variable three-valued logical functions satisfying the M(1) majority condition are three-valued majority functions if two-valued input three-valued output functions obtained by taking out only output values for 02-input vectors are 02-completely monotonic. For the realization of majority functions, m-variable inequalities are defined from 02-complete monotonicity. The weight vector is determined by solving weight inequalities derived from m-variable inequalities, and then thresholds are obtained. The overall algorithm of the method is given along with an example.

This paper presents some fundamental properties of multiple-valued logic functions monotonic in a partial-ordering relation which is introduced in the set of truth values and does not necessarily have the greatest or least element. Two kinds of necessary and sufficient conditions for monotonic p-valued functions are given with the proofs. Their logic formulas using unary operators defined in the partial-ordering relation and a simplification method for those logic formulas are also given. These results include as their special cases our former results for p-valued functions monotonic in the ambiguity relation which is a partial-ordering relation with the greatest element.

In this paper, the synthesis method of incompletely specified function based on NOR-type polypheck is presented. In synthesizing a function, the size of term is applied. The expanded simplification method in binary logic system is also applied.

The paper considers the fault tree evaluation using the list processing technique. A tree sequence is newly introduced as a representation of the fault tree; it represents completely the branching structure of the tree. The properties and algorithms concerning the tree sequence are indicated. A method for computing the probability of occurrence of the top event is presented, which is featured by the recursive implementation of partition and reduction of the fault tree, through the combined use of the reverse Polish sequence and the tree sequence. This method enables us to evaluate the fault tree containing not only common primary events but also mutually exclusive primary events. By this method we can obtain the results in reasonable computation time for a fairly large scale fault tree having a few hundred primary events. The method can also be used for sensitivity analysis.