Asynchronous wreath product and cascade decompositions for concurrent behaviours

We develop new algebraic tools to reason about concurrent behaviours modelled as languages of Mazurkiewicz traces and asynchronous automata. These reflect the distributed nature underlying causality concurrency between events, can be said support true concurrency. They generalize that have been so efficient in understanding, classifying reasoning word languages. In particular, we introduce an version wreath product operation describe trace recognized by such products (the so-called principle). then propose a decomposition result for recognizable languages, analogous Krohn-Rhodes theorem, prove this special case acyclic architectures. Finally, analyze two automata-theoretic operations. One, local cascade product, is direct implementation operation. The other, global sequences, although conceptually operationally similar translates more complex which uses gossip automaton Mukund Sohoni. This leads interesting applications characterization definable first-order logic: they are accepted restricted 2-state reset automata, also sequence Over alphabets theorem holds, automata sufficient this, turn, identification simple temporal logic expressively complete alphabets.