Token Games and History-Deterministic Quantitative Automata

Overview & Synopsis

A nondeterministic automaton is history-deterministic if its nondeterminism can be resolved by only considering the prefix of the word read so far. Due to their good compositional properties, historydeterministic automata are useful in solving games and synthesis problems. Deciding whether a given nondeterministic automaton is historydeterministic (the HDness problem) is generally a difficult task, which might involve an exponential procedure, or even be undecidable, for example for pushdown automata. Token games provide a PTime solution to the HDness problem of Büchi and coBüchi automata, and it is conjectured that 2-token games characterise HDness for all ω-regular automata. We extend token games to the quantitative setting and analyze their potential to help deciding HDness for quantitative automata. In particular, we show that 1-token games characterise HDness for all quantitative (and Boolean) automata on finite words, as well as discounted-sum (DSum) automata on infinite words, and that 2-token games characterise HDness of LimInf and LimSup automata. Using these characterisations, we provide solutions to the HDness problem of Inf and Sup automata on finite words in PTime, for DSum automata on finite and infinite words in NP∩co-NP, for LimSup automata in quasipolynomial time, and for LimInf automata in exponential time, where the latter two are only polynomial for automata with a logarithmic number of weights.