Discrete interval truth values logic and its application

In this paper, we focus on functions defined on a special subset of the pow er set of {0, 1,..., r - 1} (the elements in the subset will be called disc rete interval truth values) and operations on the truth values. The operati ons discussed in this paper will be called regular because they are one of the extensions of the regularity, which was first introduced by Kleene in h is ternary logic. Mukaidono investigated some properties of ternary functio ns which can be represented by the regular operations. He called such terna ry functions "regular ternary logic functions." Regular ternary logic funct ions are useful for representing and analyzing ambiguities such as transien t states and/or initial states in binary logic circuits that Boolean functi ons 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 to functions on the discrete i nterval truth values. First, we will suggest an extension of the regularity , in the sense of Kleene, into operations on the discrete interval truth va lues. We will then present some mathematical properties of functions on the discrete interval truth values consisting of regular operations and one ap plication of these functions.