On the Complexity of Reasoning in Kleene Algebra

Abstract

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 $E\imp s=t$, 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, \begin{itemize} \item if $E$ contains only commutativity assumptions $pq=qp$, the problem is $\Pi_1^0$-complete; \item if $E$ contains only monoid equations, the problem is $\Pi_2^0$-complete; \item for arbitrary equations $E$, the problem is $\Pi_1^1$-complete. \end{itemize} The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994).