On the size of good-for-games Rabin automata and its link with the memory in Muller games

Abstract

In this paper, we look at good-for-games Rabin automata that recognise a Muller language (a language that is entirely characterised by the set of letters that appear infinitely often in each word). We establish that minimal such automata are exactly of the same size as the minimal memory required for winning Muller games that have this language as their winning condition. We show how to effectively construct such minimal automata. Finally, we establish that these automata can be exponentially more succinct than equivalent deterministic ones, thus proving as a consequence that chromatic memory for winning a Muller game can be exponentially larger than unconstrained memory.

Infinite Separation between General and Chromatic Memory

“…In a follow-up work, Casares, Cocombet and Lehtinen [6] achieve a larger gap between GenMem(W ) and ChrMem(W ). Namely, they construct a Muller W over n colors such that GenMem(W ) is linear in n and ChrMem(W ) is exponential in n.…”

“…First, Casares [5] showed that ChrMem(W ) equals the minimal size of a deterministic Rabin automaton, recognizing W , for every Muller W . Second, Casares, Colcombet and Lehtinen [6] showed that GenMem(W ) equals the minimal size of a goodfor-games Rabin automaton, recognizing W , for every Muller W . The latter result complements an earlier work by Dziembowski, Jurdzinski and Walukiewicz [9], who characterized GenMem(W ) for Muller W in terms of their Zielonka's trees [14].…”

“…chaotic) memory and chromatic memory. The relationship between them was first addressed in the Ph.D. thesis of Kopczyński [11], followed by several recent works [3,5,6].…”

Half-Positional Objectives Recognized by Deterministic Büchi Automata

“…Memory requirements have been precisely characterized for some classes of ω-regular objectives (not encompassing the class of DBA-recognizable objectives), such as Muller conditions [27,62,17,18] and safety specifications, i.e., objectives that are closed for the Cantor topology [25]. The latter also uses the order of the equivalence classes of the right congruence as part of its characterization.…”