The p-valued-input, q-valued-output threshold logic and its application to the synthesis of p-valued logical networks

We propose a concept of p-valued-input, q-valued-output threshold logic, namely (p,q) threshold logic, where 2 ⩽ q ⩽ p, 3 ⩽ p. The idea stems from the consideration that in threshold logic the extension of the input value to the many values is quite easy, while a similar extension concerning outputs is very difficult. It is shown that in a restricted situation, the optimum value q∗ of q can be determined to construct the minimum-cost p-valued network using the (p, q)-threshold elements. Next, as a result of the consideration of (p, q)-logical completeness, a condition for completeness, which is almost equivalent to the q-valued logical completeness, is found.