Dexter Kozen. On the Complexity of Reasoning in Kleene Algebra. Information and Computation 179, 152-162, 2002.

We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra assumptions; i.e., 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,

1. if E contains only commutativity assumptions pq=qp, the problem is Pi-0-1-complete (co-r.e. complete);
2. if E contains only monoid equations, the problem is Pi-0-2-complete;
3. for arbitrary equations E, the problem is Pi-1-1-complete.

The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of [Kozen, LICS 1991].