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 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.

This paper deals with the decomposition of surface data into several fractal signal based on the parameter estimation by the Mean Likelihood and Importance Sampling (IS) based on the Monte Carlo simulations. The method is applied to the feature extraction of surface data. Assuming the stochastic models for generating the surface, the likelihood function is defined by using wavelet coefficients and the parameter are estimated based on the mean likelihood by using the IS. The approximation of the wavelet coefficients is used for estimation as well as the statistics defined for the variances of wavelet coefficients, and the likelihood function is modified by the approximation. After completing the decomposition of underlying surface data into several fractal surface, the prediction method for the fractal signal is employed based on the scale expansion based on the self-similarity of fractal geometry. After discussing the effect of additive noise, the method is applied to the feature extraction of real distribution of surface data such as the cloud and earthquakes.

In this paper, we will define Kleene-Stone logic functions which are functions F: [0, 1]n[0, 1] including the intuitionistic negation into fuzzy logic functions, and they can easily represent the concepts of necessity and possibility which are important concepts of many-valued logic systems. A set of Kleene-Stone logic functions is one of the models of Kleene-Stone algebra, which is both Kleene algebra and Stone algebra, as same as a set of fuzzy logic functions is one of the models of Kleene algebra. This paper, especially, describes some algebraic properties and representation of Kleene-Stone logic functions.

Delay models for binary logic circuits have been proposed and clarified their mathematical properties. Kleene's ternary logic is one of the simplest delay models to express transient behavior of binary logic circuits. Goto first applied Kleene's ternary logic to hazard detection of binary logic circuits in 1948. Besides Kleene's ternary logic, there are many delay models of binary logic circuits, Lewis's 5-valued logic etc. On the other hand, multiple-valued logic circuits recently play an important role for realizing digital circuits. This is because, for example, they can reduce the size of a chip dramatically. Though multiple-valued logic circuits become more important, there are few discussions on delay models of multiple-valued logic circuits. Then, in this paper, we introduce a delay model of multiple-valued logic circuits, which are constructed by Min, Max, and Literal operations. We then show some of the mathematical properties of our delay model.

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 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.

Kleene-Stone algebra is both Kleene algebra and Stone algebra. The set of Kleene-Stone logic functions discussed in this paper is one of the models of Kleene-Stone algebra, and they can easily represent the concepts of necessity and possibility which are important concepts for many-valued logic systems. Main results of this paper are that the followings are clarified: a necessary and sufficient condition for a function to be a Kleene-Stone logic function and a formula representing the number of n-variable Kleene-Stone logic functions.

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.