Complexity of Commutative Infinitary Action Logic

Abstract

We consider commutative infinitary action logic, that is, the equational theory of commutative *-continuous action lattices, and show that its derivability problem is \(\varPi _1^0\)-complete. Thus, we obtain a commutative version of \(\varPi _1^0\)-completeness for non-commutative infinitary action logic by Buszkowski and Palka (2007). The proof of the upper bound is more or less the same as Palka’s argument. For the lower bound, we encode non-terminating behaviour of two-counter Minsky machines.