Asynchronous cellular automata for pomsets

Abstract

This paper extends to pomsets without auto-concurrency the fundamental notion of asynchronous cellular automata (ACA) which was originally introduced for traces by Zielonka. We generalize to pomsets the notion of asynchronous mapping introduced by Cori, Métivier and Zielonka and we show how to construct a deterministic ACA~from an asynchronous mapping. Then we investigate the relation between the expressiveness of monadic second-order logic, nondeterministic ACAs and deterministic ACAs. We can generalize Büchi's theorem for finite words to a class of pomsets without auto-concurrency which satisfy a natural axiom. This axiom ensures that an asynchronous cellular automaton works on the pomset as a concurrent read and exclusive owner write machine. More precisely, in this class nondeterministic ACAs, deterministic ACAs and monadic second-order logic have the same expressive power. Then we consider a class where deterministic ACAs are strictly weaker than nondeterministic ones. But in this class nondeterministic ACAs still capture monadic second-order logic. Finally, it is shown that even this equivalence does not hold in the class of all pomsets since there the class of recognizable pomset languages is not closed under complementation.