Minimization of logical functions by the method of subimplicants (Q5951305)

The \(S_N\)-algebra \((\mathbb{N}\cup \{\omega\};+, (\chi_i)_{i\in \mathbb{N}}\), \((\theta_i)_{i\in \mathbb{N}})\) was introduced by the author [Integrated current-type logic (in Russian), Olis Plyus, L'vov (1998)]. The operation + is the extension of ordinary sum defined by \(x+\omega= \omega+x= \omega\), while \(\chi_i(x) =i-x\) and \(\theta_i(x)= \omega\) for \(x<i\), else \(\chi_i(x)= \theta_i (x)=0\). In this paper the well-known Quine method for the minimization of truth functions is extended to functions defined over the \(S_N\)-algebra. The techniques are explained in great detail in a concrete example.