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NP-completeness conjecture and structural certificate for history-determinism

Prove that deciding whether a nondeterministic parity automaton with an unfixed parity index is history-deterministic is NP-complete; more specifically, construct for every history-deterministic parity automaton H an equivalent history-deterministic parity automaton H' with at most as many states that enjoys additional structural properties enabling a polynomial-time verifiable certificate of history-determinism, thereby establishing NP membership.

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Background

The authors close the recognition problem for fixed parity index but leave open the general-case complexity, conjecturing NP-completeness. They further posit a structural normalization: every history-deterministic automaton should be equivalent to a size-non-increasing form that facilitates polynomial-time verification of history-determinism, which would yield an NP upper bound when combined with a suitable certificate and simulation strategies.

References

We conjecture that the problem is NP-complete. More specifically, we conjecture that for every history-deterministic automaton H, there is another equivalent history-deterministic automaton H' with as many or fewer states which satisfies certain properties that makes it easier to verify history-determinism of H'.

— The 2-Token Theorem: Recognising History-Deterministic Parity Automata Efficiently (2503.24244 - Lehtinen et al., 31 Mar 2025) in Section 4 (Applications and Concluding Remarks)