Token Games and History-Deterministic Quantitative Automata

Abstract

AbstractA 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, history-deterministic automata are useful in solving games and synthesis problems. Deciding whether a given nondeterministic automaton is history-deterministic (the 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 problem of Büchi and coBüchi automata, and it is conjectured that 2-token games characterise for all $$\omega $$ ω -regular automata. We extend token games to the quantitative setting and analyze their potential to help deciding for quantitative automata. In particular, we show that 1-token games characterise for all quantitative (and Boolean) automata on finite words, as well as discounted-sum ($${\mathsf {DSum}}$$ DSum ) automata on infinite words, and that 2-token games characterise of $${\mathsf {LimInf}}$$ LimInf and $${\mathsf {LimSup}}$$ LimSup automata. Using these characterisations, we provide solutions to the problem of $${\mathsf {Inf}}$$ Inf and $${\mathsf {Sup}}$$ Sup automata on finite words in PTime, for $${\mathsf {DSum}}$$ DSum automata on finite and infinite words in NP$$\cap $$ ∩ co-NP, for $${\mathsf {LimSup}}$$ LimSup automata in quasipolynomial time, and for $${\mathsf {LimInf}}$$ LimInf automata in exponential time, where the latter two are only polynomial for automata with a logarithmic number of weights.