The p-valued-input, q-valued-output threshold logic and the (p, q)-polypheck-like function

We have previously proposed a concept of p-valued-input, q-valued-output threshold logic, namely (p, q)-threshold logic, where 2 </q</p, 3</p, and suggested that p-valued logical networks with costs as low as those of 2-valued logical networks could be obtained, by using the (p, q)-threshold elements with small values of q. In this paper, we describe (1) the condition under which there is a 2-place (p, q)-adic function such that the output-closed set F, generated only from , is (p, q)-logically complete, and (2) the fact that any n-place(p, q)-adic function can be realized using at most O(n) elements in the above F.