Token Games and History-Deterministic Quantitative-Automata

TL;DR

This paper extends token game techniques to analyze history-determinism in quantitative automata, providing efficient decision procedures for various classes of automata on finite and infinite words.

Contribution

It introduces token game characterizations for quantitative automata's HDness, enabling polynomial and quasipolynomial decision algorithms for multiple automata types.

Findings

1-token games characterize HDness for finite and certain infinite automata.

2-token games characterize HDness for LimInf, LimSup, and Sup automata.

Decidability results include PTime, NP∩co-NP, quasipolynomial, and exponential algorithms.

Abstract

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, history-deterministic automata are useful in solving games and synthesis problems. Deciding whether a given nondeterministic automaton is history-deterministic (the HDness problem) is generally a difficult task, which can involve an exponential procedure, or even be undecidable, as is the case for example with pushdown automata. Token games provide a PTime solution to the HDness problem of B\"uchi and coB\"uchi automata, and it is conjectured that 2-token games characterise HDness for all $\omega$-regular automata. We extend token games to the quantitative setting and analyse their potential to help deciding HDness of 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), Inf and Reachability automata on infinite words, and that 2-token games characterise HDness of LimInf and LimSup automata, as well as Sup automata on infinite words. Using these characterisations, we provide solutions to the HDness problem of Safety, Reachability, Inf and Sup automata on finite and infinite words in PTime, of DSum automata on finite and infinite words in NP$\cap$co-NP, of LimSup automata in quasipolynomial time, and of LimInf automata in exponential time, where the latter two are only polynomial for automata with a logarithmic number of weights.