A logical model for representing ambiguous states in multiple-valued logicsystems

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 same properties of ternary functions, which can be represented by regular operations. He called such t ernary functions "regular ternary logic functions". Regular ternary logic F unctions are useful for representing and analyzing ambiguities such as tran sient states 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 into r-valued set-valued funct ions, which are defined as mappings on a set of nonempty subsets of the r-v alued set {0, 1,..., r - 1}. First, the paper will show a method by which o perations on the r-valued set {0, 1,..., r - 1} can be expanded into operat ions on the set of nonempty subsets of {0, 1,..., r -1}. These operations w ill be called regular since this method is identical with the way that Klee ne expanded operations of binary logic into his ternary logic. Finally, exp licit expressions of set-valued functions monotonic in subset of will be pr esented.