On the Average Size of Glushkov and Equation Automata for KAT Expressions

Kleene algebra with tests (KAT) is an equational system that extends Kleene algebra, the algebra of regular expressions, and that is specially suited to capture and verify properties of simple imperative programs. In this paper we study two constructions of automata from KAT expressions: the Glushkov automaton ($\mathcal{A}_{\mathsf{pos}}$), and a new construction based on the notion of prebase (equation automata, $\mathcal{A}_{\mathsf{eq}}$). Contrary to other automata constructions from KAT expressions, these two constructions enjoy the same descriptional complexity behaviour as their counterparts for regular expressions, both in the worst-case as well as in the average-case. In particular, our main result is to show that, asymptotically and on average the number of transitions of the $\mathcal{A}_{{\mathsf{pos}}}$ is linear in the size of the KAT expression.

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