Series-Parallel Posets: Algebra, Automata and Languages

In order to model concurrency, we extend automata theory from the usual word languages (sets of labelled linear orders) to sets of labelled series-parallel posets | or, equivalently, to sets of terms in an algebra with two product operations: sequential and parallel. We rst consider languages of posets having bounded width, and characterize them using depth-nilpotent algebras. Next we introduce series-rational expressions , a natural generalization of the usual rational expressions, as well as a notion of branching automaton. We show both a Myhill-Nerode theorem and a Kleene theorem. We also look at generalizations.