Undecidability conjecture for the two-variable, two-unary-predicate fragment of QS5

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Let QS5\mathbf{QS5} be the quantified modal logic QS5, and consider its fragment using two individual variables and two unary predicate letters. Undecidability conjecture. The fragment of QS5\mathbf{QS5} in the language with two individual variables and two unary predicate letters is algorithmically undecidable. This would settle one of the remaining decidability questions for small-variable, small-signature fragments of quantified modal logics; the paper notes that the corresponding fragment with three individual variables and one unary predicate letter, and with two individual variables and three unary predicate letters, is already undecidable.

References

Primary source

Mikhail Rybakov, “Recursive inseparability of classical theories of a binary predicate and non-classical logics of a unary predicate”, arXiv:2505.00524 (2025).

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