A special group of ternary functions, called regular ternary logic functions, are defined. These functions are useful in switching theory, programming languages, algorithm theory, and many other fields—if we are concerned with the indefinite state in such fields. This correspondence describes the fundamental properties and representations of the regular ternary logic functions.

Regular Ternary Logic Functions&#8212;Ternary Logic Functions Suitable for Treating Ambiguity

The concepts of $\textbf{SAT}$-hardness and $\textbf{SAT}$-completeness modulo npolylogn time and linear size reducibility, denoted by $\textbf{SAT}$-hard (npolylogn, n) and $\textbf{SAT}$-complete (npolylogn, n), respectively, are introduced. Regardless of whether $\textbf{P} = \textbf{NP}$ or $\textbf{P} \neq \textbf{NP}$, it is shown that intuitivelyEach $\textbf{SAT}$-hard (npolylogn, n) problem requires essentially at least as much deterministic time as, andEach $\textbf{SAT}$-complete (npolylogn, n) problem requires essentially the same deterministic time as the satisfiability problem for 3CNF formulas.It is proved that the $\leqq$, satisfiability, tautology, unique satisfiability, equivalence, and minimization problems are already $\textbf{SAT}$-complete (npolylogn, n), for very simple Boolean formulas and for very simple systems of Boolean equations. These completeness results are used to characterize the deterministic time complexities of a number of problems for lattices, propositional calculi, ...

The complexity of very simple Boolean formulas with applications

We investigate the free Kleene algebra on n free generators, FK( n ). This algebra is essentially the same as the set of n -variable fuzzy switching functions. The cardinality of FK( n ) has been previously determined only for n = 1, 2, and 3. We present a general method for enumeration problems of this kind. We show FK(4) has 160,297,985,276 elements. The final section contains new upper and lower bounds for the cardinality of FK( n ). We also consider the number of regular functions and show that there are 11,197,129,615, and 430,904,428,717 such functions in 1, 2, 3, and 4 variables.

Enumerating fuzzy switching functions and free kleene algebras

This paper analyzes the main issues involved in the construction of Fuzzy Set Theories. It reviews both standard solutions (based on the well-known triangular norms and conorms) as well as less the conventional proposals that provide alternative views on, for example, the definition of fuzzy connectives or the study of their properties.

An overview on the construction of fuzzy set theories

This paper analyzes some of the main issues involved in the construction of fuzzy set theories. It reviews both standard solutions (based on the well-known t-norms and t-conorms), as well as less conventional proposals that provide alternative views on, e.g. the definition of fuzzy connectives and the study of their properties.

On Fuzzy Set Theories

A set of independent and complete axioms for a fuzzy algebra (Kleene algebra)

Both the steady states and some transient states of switching circuits can be described by B-ternary logic in which the truth values 0, 1 and 1/2 are used respectively to represent false, true and uncertainty. This paper showed the methods of detecting and identifying various kinds of static hazards contained in combinational switching circuits by means of the canonical forms of the B-ternary logic functions realized by the circuits. Particularly, a method was derived which could algebraically detect all logic hazards contained in the circuits. It was also pointed out that there were some dynamic hazards which were detectable by B-ternary logic.

Masao Mukaidono

Papers

A set of independent and complete axioms for a fuzzy algebra (Kleene algebra)

The B-ternary logic and its applications to the detection of hazards in combinational switching circuits