The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete

Commutative, integral and bounded GBL-algebras form a subvariety of residuated lattices which provides the algebraic semantics of an interesting common fragment of intuitionistic logic and of several fuzzy logics.It is known that both the equational theory and the quasiequational theory of commutative GBL-algebras are decidable (in contrast to the noncommutative case), but their complexity has not been studied yet. In this paper, we prove that both theories are in , and that the quasiequational theory is -hard.