ACM Transactions on Programming Languages and Systems (TOPLAS)
Greibach Normal Form in Algebraically Complete Semirings
CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
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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, (i) if E contains only commutativity assumptions pq=qp, the problem is Pi-0-1-complete; (ii) if E contains only monoid equations, the problem is Pi-0-2-complete; (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 Kozen (1994).