Infinitary action logic axiomatises the equational theory of \(*\) -continuous action lattices. Natural examples of these algebraic structures are algebras of formal languages and algebras of binary relations. Infinitary action logic itself is known to be \(\varPi ^0_1\) -complete (Buszkowski, Palka 2007). For the equational theories of the aforementioned specific classes of action lattices, Buszkowski (2006) proved \(\varPi ^0_1\) lower complexity bounds and left the exact complexity estimation an open question. In this paper, we prove that these theories have significantly higher algorithmic complexity, namely, that they are \(\varPi ^1_1\) -complete. This is done by using novel constructions with constants “zero” and “one,” which allow simulating entailment from finite sets of hypotheses. After that, we strengthen the method of Kozen (2002) to prove \(\varPi ^1_1\) -hardness for entailment in the product-free fragment, which is strongly complete w.r.t. relational and language action lattices. This strengthening is done by an infinitary version of a technique by Buszkowski (1982).