Models of nondeterministic regular expressions

Nondeterminism is a direct outcome of interactions and is, therefore a central ingredient for modelling concurrent systems. Trees are very useful for modelling nondeterministic behaviour. We aim at a tree-based interpretation of regular expressions and study the effect of removing the idempotence law X+X = X and the distribution law X·(Y+Z) = X·Y+X·Z from Kleene algebras. We show that the free model of the new set of axioms is a class of trees labelled over A. We also equip regular expressions with a two-level behavioural semantics. The basic level is described in terms of a class of labelled transition systems that are detailed enough to describe the number of equal actions a system can perform from a given state. The abstract level is based on a so-called resource bisimulation preorder that permits ignoring uninteresting details of transition systems. The three proposed interpretations of regular expressions (algebraic, denotational, and behavioural) are proven to coincide. When dealing with infinite behaviours, we rely on a simple version of the ω-induction and obtain a complete proof system also for the full language of nondeterministic regular expressions.