Abstract
Good-For-Games (GFG) automata are nondeterministic automata that can resolve their nondeterministic choices based on the past. The fact that the synthesis problem can be reduced to solving a game on top of a GFG automaton for the specification (that is, no determinization is needed) has made them the subject of extensive research in the last years. GFG automata are defined for general alphabets, whereas in the synthesis problem, the specification is over an alphabet \(2^{I \cup O}\), for sets I and O of input and output signals, respectively. We introduce and study (I/O)-aware GFG automata, which distinguish between nondeterminism due to I and O: both should be resolved in a way that depends only on the past; but while nondeterminism in I is hostile, and all I-futures should be accepted, nondeterminism in O is cooperative, and a single O-future may be accepted. We show that (I/O)-aware GFG automata can be used for synthesis, study their properties, special cases and variants, and argue for their usefulness. In particular, (I/O)-aware GFG automata are unboundedly more succinct than deterministic and even GFG automata, using them circumvents determinization, and their study leads to new and interesting insights about hostile vs. collaborative nondeterminism, as well as the theoretical bound for realizing systems.
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Notes
- 1.
GFGness is also used in [7] in the framework of cost functions under the name “history-determinism”.
- 2.
In fact, DBP automata were the only examples known for GFG automata when the latter were introduced in [11]. As explained there, however, even DBP automata are useful in practice, as their transition relation is simpler than the one of the embodied deterministic automaton and it can be defined symbolically.
- 3.
Note that the definition is different than open implication in [10], where \(\mathcal {A}'\) open implies \(\mathcal {A}\) if every (I/O)-transducer that (I/O)-realizes \(\mathcal {A}'\) also (I/O)-realizes \(\mathcal {A}'\). For example, an empty \(\mathcal {A}'\) open implies every unrealizable \(\mathcal {A}\), yet need not (I/O)-cover it.
- 4.
A more precise definition of the dual setting adds to the realizability notation the parameter of “who moves first". Then, \(\mathcal {A}_\varphi \) is (I/O)-realizable with the environment moving first iff \(\mathcal {A}_{\lnot \varphi }\) is not (O/I)-realizable with the system (that is, the player that generates signals in O) moving first. Adding this parameter is easy, yet makes the writing more cumbersome, so we give it up.
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Faran, R., Kupferman, O. (2020). On (I/O)-Aware Good-For-Games Automata. In: Hung, D.V., Sokolsky, O. (eds) Automated Technology for Verification and Analysis. ATVA 2020. Lecture Notes in Computer Science(), vol 12302. Springer, Cham. https://doi.org/10.1007/978-3-030-59152-6_9
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