A Sound and Complete Axiomatization of Majority-$n$ Logic

Manipulating logic functions via majority operators recently drew the attention of researchers in computer science. For example, circuit optimization based on majority operators enables superior results as compared to traditional synthesis tools. Also, the Boolean satisfiability problem finds new solution approaches when described in terms of majority decisions. To support computer logic applications based on majority, a sound and complete set of axioms is required. Most of the recent advances in majority logic deal only with ternary majority (MAJ-3) operators because the axiomatization with solely MAJ-3 and complementation operators is well understood. However, it is of interest extending such axiomatization to
$n$
-ary majority operators (MAJ-
$n$
) from both the theoretical and practical perspective. In this work, we address this issue by introducing a sound and complete axiomatization of MAJ-
$n$
logic. Our axiomatization naturally includes existing MAJ-3 and MAJ-5 axiomatic systems. Based on this general set of axioms, computer applications can now fully exploit the expressive power of majority logic.