Algorithmic complexity for theories of commutative Kleene algebras

Abstract

Kleene algebras are structures with addition, multiplication and constants $0$ and $1$, which form an idempotent semiring, and the Kleene iteration operation. In the particular case of $*$-continuous Kleene algebras, Kleene iteration is defined, in an infinitary way, as the supremum of powers of an element. We obtain results on algorithmic complexity for Horn theories (semantic entailment from finite sets of hypotheses) of commutative $*$-continuous Kleene algebras. Namely, $\Pi_1^1$-completeness for the Horn theory and $\Pi^0_2$-completeness for its fragment, where iteration cannot be used in hypotheses, is proved. These results are commutative counterparts of the corresponding theorems of D. Kozen (2002) for the general (non-commutative) case. Several accompanying results are also obtained.