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 as follows: for *-continuous Kleene algebra, (i) if E contains only commutativity assumptions pq = qp, the problem is Pi(1)(0)-complete; (ii) if E contains only monoid equations, the problem is Pi(2)(0)-complete; and (iii) 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 the author [D. Kozen, 1994, Inform. and Comput. 110,366-390]. (C) 2002 Elsevier Science (USA).