From Muller to Parity and Rabin Automata: Optimal Transformations Preserving (History-)Determinism

We study transformations of automata and games using Muller conditions into equivalent ones using parity or Rabin conditions. We present two transformations, one that turns a deterministic Muller automaton into an equivalent deterministic parity automaton, and another that provides an equivalent history-deterministic Rabin automaton. We show a strong optimality result: the obtained automata are minimal amongst those that can be derived from the original automaton by duplication of states. We introduce the notions of locally bijective morphisms and history-deterministic mappings to formalise the correctness and optimality of these transformations. The proposed transformations are based on a novel structure, called the alternating cycle decomposition, inspired by and extending Zielonka trees. In addition to providing optimal transformations of automata, the alternating cycle decomposition offers fundamental information on their structure. We use this information to give crisp characterisations on the possibility of relabelling automata with different acceptance conditions and to perform a systematic study of a normal form for parity automata.