A word automaton recognizing a language L is good for games (GFG) if its
composition with any game with winning condition L preserves the game's
winner. While all deterministic automata are GFG, some nondeterministic
automata are not. There are various other properties that are used in the
literature for defining that a nondeterministic automaton is GFG, including
"history-deterministic", "compliant with some letter game", "good for trees",
and "good for composition with other automata". The equivalence of these
properties has not been formally shown.
We generalize all of these definitions to alternating automata and show their
equivalence. We further show that alternating GFG automata are as expressive as
deterministic automata with the same acceptance conditions and indices. We then
show that alternating GFG automata over finite words, and weak automata over
infinite words, are not more succinct than deterministic automata, and that
determinizing B\"uchi and co-B\"uchi alternating GFG automata involves a
2Θ(n) state blow-up. We leave open the question of whether
alternating GFG automata of stronger acceptance conditions allow for
doubly-exponential succinctness compared to deterministic automata.Comment: Full version of a paper of the same name accepted fr publication at
the 30th International Conference on Concurrency Theor