Token Games and History-Deterministic Quantitative-Automata

Abstract

A nondeterministic (quantitative) 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 or not a given nondeterministic automaton is history deterministic (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 characterize 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 characterize 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 characterize HDness of LimInf and LimSup automata. Using these characterizations, 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 fixed number of weights.