Karoliina Lehtinen

Udi Boker, Thomas A. Henzinger, Karoliina Lehtinen et al.

An automaton is history-deterministic if its nondeterminism can be resolved on the fly, only using the prefix of the word read so far. This mild form of nondeterminism has attracted particular attention for its applications in synthesis problems. An automaton $A$ is guidable with respect to a class $C$ of automata if it can fairly simulate every automaton in $C$ whose language is contained in that of $A$. In other words, guidable automata are those for which inclusion and simulation coincide, making them particularly interesting for model-checking. We study the connection between these two notions, and specifically the question of when they coincide. For classes of automata on which they do, deciding guidability, an otherwise challenging decision problem, reduces to deciding history-determinism, a problem that is starting to be well-understood for many classes. We provide a selection of sufficient criteria for a class of automata to guarantee the coincidence of the notions, and use them to show that the notions coincide for the most common automata classes, among which are $ω$-regular automata and many infinite-state automata with safety and reachability acceptance conditions, including vector addition systems with states, one-counter nets, pushdown-, Parikh-, and timed-automata. We also demonstrate that history-determinism and guidability do not always coincide, for example, for the classes of timed automata with a fixed number of clocks.

Karoliina Lehtinen, Keya Prakash

History-determinism is a restricted notion of nondeterminism in automata, where the nondeterminism can be successfully resolved based solely on the prefix read so far. History-deterministic automata still allow for exponential succinctness in automata over infinite words compared to deterministic automata (Kuperberg and Skrzypczak, 2015), allow for canonical forms unlike deterministic automata (Abu Radi and Kupferman, 2019 and 2020; Ehlers and Schewe, 2022), and retain some of the algorithmic properties of deterministic automata, for example for reactive synthesis (Henzinger and Piterman, 2006; Ehlers and Khalimov, 2024). Despite the topic of history-determinism having received a lot of attention over the last decade, the complexity of deciding whether a parity automaton is history-deterministic has, up till now, remained open. We show that history-determinism for a parity automaton with a fixed parity index can be checked in PTIME, thus improving upon the naive EXPTIME upper bound of Henzinger and Piterman that has stood since 2006. More precisely, we show that the so-called 2-token game, which can be solved in PTIME for parity automata with a fixed parity index, characterises history-determinism for parity automata. This game was introduced by Bagnol and Kuperberg in 2018, who showed that to decide if a Büchi automaton is history-determinism, it suffices to find the winner of the 2-token game on it. They conjectured that this 2-token game based characterisation extends to parity automata. Boker, Kuperberg, Lehtinen, and Skrzypcak showed in 2020 that this conjecture holds for coBüchi automata as well. We prove Bagnol and Kuperberg's conjecture that the winner of the 2-token game characterises history-determinism on parity automata.

Ayaan Bedi, Karoliina Lehtinen

We study explorability, a measure of nondeterminism in pushdown automata, which generalises history-determinism. An automaton is k-explorable if, while reading the input, it suffices to follow k concurrent runs, built step-by-step based only on the input seen so far, to construct an accepting one, if it exists. We show that the class of explorable PDAs lies strictly between history-deterministic and fully nondeterministic PDAs in terms of both expressiveness and succinctness. In fact increasing explorability induces an infinite hierarchy: each level k defines a strictly more expressive class than level k-1, yet the entire class remains less expressive than general nondeterministic PDAs. We then introduce a parameterized notion of explorability, where the number of runs may depend on input length, and show that exponential explorability precisely captures the context-free languages. Finally, we prove that explorable PDAs can be doubly exponentially more succinct than history-deterministic ones, and that the succinctness gap between deterministic and 2-explorable PDAs is not recursively enumerable. These results position explorability as a robust and operationally meaningful measure of nondeterminism for pushdown systems.