On the complexity of reasoning in Kleene algebra

We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E→s=t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for *-continuous Kleene algebra, if E contains only commutativity assumptions pq=qp, the problem is II₁⁰-complete; if E contains only monoid equations, the problem is II₂⁰-complete; for arbitrary equations E, the problem is II₁¹-complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994)