## Paper: On the Complexity of Reasoning in Kleene Algebra (at LICS 1997)

**Dexter C. Kozen**

### 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 --> 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).

### BibTeX

@InProceedings{Kozen-OntheComplexityofRe, author = {Dexter C. Kozen}, title = {On the Complexity of Reasoning in Kleene Algebra}, booktitle = {Proceedings of the Twelfth Annual IEEE Symposium on Logic in Computer Science (LICS 1997)}, year = {1997}, month = {June}, pages = {195--202}, location = {Warsaw, Poland}, publisher = {IEEE Computer Society Press} }