  We prove undecidability and pinpoint the place in the arithmetical hierarchy
for commutative action logic, that is, the equational theory of commutative
residuated Kleene lattices (action lattices), and infinitary commutative action
logic, the equational theory of *-continuous action lattices. Namely, we prove
that the former is $\Sigma_1^0$-complete and the latter is $\Pi_1^0$-complete.
Thus, the situation is the same as in the more well-studied non-commutative
case. The methods used, however, are different: we encode infinite and circular
computations of counter (Minsky) machines.
