Shortest possible axiomatizations for the strict implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.
"Shortest Axiomatizations of Implicational S4 and S5." Notre Dame J. Formal Logic 43 (3) 169 - 179, 2002. https://doi.org/10.1305/ndjfl/1074290715