A Class of Generalised Quantifiers for k-Variable Logics

We introduce k-quantifier logics – logics with access to k-tuples of elements and very general quantification patterns for transitions between k-tuples. The framework is very expressive and encompasses e.g. the k-variable fragments of first-order logic, modal logic, and monotone neighbourhood semantics. We introduce a corresponding notion of bisimulation and prove variants of the classical Ehrenfeucht-Fraisse and Hennessy-Milner theorem. Finally, we show a Lindstrom-style characterisation for k-quantifier logics that satisfy Los’ theorem by proving that they are the unique maximally expressive logics that satisfy Los’ theorem and are invariant under the associated bisimulation relations.

💡 Research Summary

The paper introduces a new family of logics called k‑quantifier logics, which generalise the usual k‑variable fragments of first‑order logic, modal logics, and neighbourhood semantics by allowing quantification over k‑tuples of elements with highly flexible access patterns.

A k‑quantifier Q is a syntactic symbol equipped with a signature σ_Q and, for every σ_Q‑structure A together with a k‑assignment α, a set Q(α) ⊆ 𝒫(A^k) of “witness sets”. The definition requires that Q be invariant under isomorphisms and under expansions of the underlying structure. The syntax of the resulting logic L_Q consists of atomic relational formulas, Boolean connectives, and formulas of the form Q φ. Semantically, Q φ holds at a point (A,α) iff there exists a witness set s∈Q(α) such that all k‑tuples γ∈s satisfy φ. This captures a wide range of operators: the usual modal box □, counting quantifiers, reachability, infinite branching, and many more can be expressed as 1‑quantifiers; higher‑arity quantifiers allow simultaneous constraints on several elements.

The authors define a k‑bisimulation game that mirrors the classic bisimulation for modal logic but is driven by the witness sets of the active quantifiers. Two pointed structures are k‑bisimilar (∼_k) if Duplicator has a winning strategy in this game. They then adapt the Ehrenfeucht–Fraïssé back‑and‑forth game to the k‑quantifier setting, showing that Spoiler can win a q‑round game exactly when the two structures differ on some L_Q‑formula of quantifier rank ≤ q. Consequently, k‑bisimilarity coincides with elementary equivalence for L_Q (and with q‑equivalence for bounded rank).

A Hennessy–Milner theorem is proved for L_Q: two structures are k‑bisimilar iff they satisfy the same L_Q‑sentences. This extends the classic result from modal logic to the far more expressive setting of arbitrary k‑quantifiers.

The paper then investigates the Łoś property for k‑quantifier logics. It shows that if L_Q is closed under ultraproducts (i.e., satisfies Łoś’s theorem), then it inherits compactness and the Löwenheim–Skolem property from the underlying first‑order framework. This establishes a solid model‑theoretic foundation for the new logics.

Finally, the authors prove a Lindström‑style characterisation: among all abstract logics that (i) satisfy Łoś’s theorem and (ii) are invariant under the associated k‑bisimulation, L_Q is maximal in expressive power. In other words, any logic with these two properties is a fragment of some k‑quantifier logic. This result mirrors Lindström’s original theorem for FO but now applies to a whole spectrum of logics defined by arbitrary k‑quantifiers.

The paper situates its contribution within a broad literature: it contrasts k‑quantifiers with the far more general Lindström quantifiers, explains how neighbourhood semantics correspond to the unary case, and relates the work to recent extensions of Lindström theorems that incorporate game‑based invariance conditions. By providing a unified, game‑theoretic, and model‑theoretic treatment, the work opens avenues for designing new logics tailored to specific access patterns (e.g., database query languages with limited tuple‑wise navigation) and for analysing their expressive limits, decidability, and algorithmic properties. Future directions suggested include extending the framework to functional signatures, studying combinations of quantifiers, and exploring computational complexity of the associated model‑checking problems.