How the Location of * Influences Complexity in Kleene Algebra with Tests

Abstract

The universal Horn theory of relational Kleene algebra with tests is of practical interest, particularly for program semantics, where Horn formulas can be used to verify correctness of programs or compiler optimizations. Unfortunately, this theory is known to be Π$_{\rm 1}^{\rm 1}$-complete. However, many formulas arising in practice fall into fragments of the theory that are of lower complexity. In this paper, we see that the location of occurrences of the Kleene asterate operator * within a formula has a great impact on complexity. Using syntactic criteria based on the location of *, we give a fragment of the theory that is Σ$_{\rm 1}^{\rm 0}$-complete, and a slightly larger fragment that is Π$_{\rm 2}^{\rm 0}$-complete. We show that the same results hold over *-continuous Kleene algebras with tests. The techniques exhibit a relationship between first-order logic and the Horn theories of relational and *-continuous Kleene algebra, even though the theories are not first-order axiomatizable.