Modal Definability: Semantic Characterizations

• Modal definability is the problem of determining which semantic classes of structures can be exactly characterized by modal formulas.
• It involves analyzing frame versus model formulations, distinguishing local and global definability, and using methods like bisimulation invariance and preservation theorems.
• Recent advances extend the problem to finite pointed Kripke models, graded modal logics, and applications in graph neural networks and dynamic systems.

Searching arXiv for recent and foundational papers on modal definability to support the article. The modal definability problem asks which semantic classes of structures can be characterized exactly by formulas of a modal language. In its basic form, one fixes a modal logic $\mathcal{L}$, a semantics over Kripke-style structures, and asks for a characterization of those classes $\mathcal{C}$ such that there exists a formula $\varphi \in \mathcal{L}$ with

$$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$

Across the literature, this problem appears in several closely related guises: frame definability, model definability, local versus global definability, definability by fragments of modal logic, preservation-theoretic characterizations, and algorithmic questions concerning decidability and complexity. Recent work has sharpened the problem for finite pointed Kripke models under locality assumptions, yielding exact preservation/definability correspondences for modal fragments and related architectures such as graph neural networks (Wałęga et al., 2 Feb 2026). At the same time, other strands treat definability over ordered temporal-functional frames (Burrieza, 1 Jul 2026), finitely valued modal semantics (Teheux, 2015, Badia et al., 2022), fixpoint logics (Jung et al., 29 Sep 2025), Euclidean frame classes (Balbiani et al., 14 Aug 2025), and broad model-theoretic frameworks for modal first-order fragments (Carreiro, 2010).

1. Historical formulations and core distinctions

The general problem is often stated for classes of frames or pointed models. In the frame-oriented formulation, one asks when a class $C$ of relational structures is of the form

$C=\{F\mid F\models \Phi\}$

for some set $\Phi$ of modal formulas. This formulation is explicit in many-valued work on Łukasiewicz semantics, where the problem is presented as determining which classes of frames can be described purely by modal formulas rather than arbitrary first-order conditions (Teheux, 2015).

A model-oriented formulation instead fixes pointed models $(\mathfrak{M},w)$ and asks when a class $\mathcal{C}$ is definable by a modal formula at the distinguished point. In the preservation-theoretic treatment of finite pointed Kripke models, the problem is stated as characterizing exactly which classes of pointed models are definable in fragments such as $\mathrm{GML}^{\exists}$, $\mathcal{C}$0, or $\mathcal{C}$1 (Wałęga et al., 2 Feb 2026).

A second fundamental distinction is between local and global definability. Local definability concerns satisfaction at a designated world. Global definability concerns validity throughout a model or frame. In modal logic with universal modality, this distinction becomes especially important: model classes definable in $\mathcal{C}$2 admit a characterization via elementaryity and closure under surjective bisimulations, while frame classes require a Goldblatt–Thomason-style closure analysis (Sano et al., 2015).

A third distinction concerns the target language. The question may ask whether a first-order condition is definable in basic modal logic, whether a graded modal formula is definable in a counting-free fragment, or whether a fixpoint formula is definable by a modal formula. For example, the definability problem for fragments of $\mathcal{C}$3 with counting asks whether a formula with counting is equivalent to one without counting; in the graded modal setting this becomes the question whether a $\mathcal{C}$4-formula is equivalent to an $\mathcal{C}$5-formula (Kuijer et al., 29 Apr 2025). In the modal $\mathcal{C}$6-calculus, definability becomes the special case of separability where a formula is separated from its negation by a modal formula (Jung et al., 29 Sep 2025).

2. Structural invariance and preservation-theoretic approaches

A standard route to modal definability is to identify the semantic invariances or closure properties that characterize expressibility. In classical modal model theory, bisimulation invariance plays the central role. More general frameworks abstract this pattern: a modal logic $\mathcal{C}$7 is paired with a suitable notion of $\mathcal{C}$8-simulation, and one asks which first-order formulas are invariant under that relation. In an abstract setting for modal first-order fragments, this yields characterization and definability theorems for adequate pairs $\mathcal{C}$9, where definability by a set or single formula is tied to simulation closure together with ultraproduct and ultrapower conditions on the first-order image of the class (Carreiro, 2010).

Recent work refines this perspective for finite pointed Kripke models by shifting attention from bisimulation to preservation under embeddings, injective homomorphisms, and homomorphisms, under a bounded-unravelling locality assumption (Wałęga et al., 2 Feb 2026). The setting is a class $\varphi \in \mathcal{L}$0 of finite pointed models that is invariant under $\varphi \in \mathcal{L}$1-unravelling: $\varphi \in \mathcal{L}$2 Within this local regime, precise preservation theorems hold:

• preservation under embeddings corresponds to definability in existential graded modal logic $\varphi \in \mathcal{L}$3;
• preservation under injective homomorphisms corresponds to definability in existential-positive graded modal logic $\varphi \in \mathcal{L}$4;
• preservation under homomorphisms corresponds to definability in existential-positive modal logic $\varphi \in \mathcal{L}$5 (Wałęga et al., 2 Feb 2026).

This gives an exact answer to a modal definability problem for local classes of finite pointed Kripke models. The same paper emphasizes that the locality condition is not incidental: it is what makes possible a reduction to bounded-height tree-shaped models, where a structural well-quasi-ordering argument can be applied (Wałęga et al., 2 Feb 2026).

An analogous preservation perspective also appears in work on finitely-valued modal logics. There, the central observation is negative rather than positive: finitely-valued modal semantics over crisp frames do not increase frame-definability power beyond classical modal logic. One result states that a class of frames modally definable in a finitely-valued modal logic is already modally definable in the classical two-valued setting, and for a large family of finite algebras the definable frame classes coincide exactly with the classical ones (Badia et al., 2022). This suggests that, at the level of crisp-frame definability, the preservation behavior remains essentially classical.

3. The finite, locality, and bounded unravelling

The finite case is especially delicate because many first-order transfer theorems fail when one restricts from arbitrary to finite structures. A recent investigation asks systematically which classic modal definability and preservation results survive this restriction (Benthem et al., 12 Mar 2026). One of its central positive conclusions is that several semantic characterizations of modal formula classes do transfer to finite structures, and in particular the Bisimulation Safety Theorem survives in the finite (Benthem et al., 12 Mar 2026). This places the finite preservation results for local graded modal fragments in a broader context.

The bounded-unravelling framework provides one of the sharpest current finite answers. For a pointed model $\varphi \in \mathcal{L}$6, the $\varphi \in \mathcal{L}$7-unravelling $\varphi \in \mathcal{L}$8 is a tree-shaped pointed model of height at most $\varphi \in \mathcal{L}$9, whose worlds are finite paths from $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$0 of length $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$1 (Wałęga et al., 2 Feb 2026). Since modal formulas of depth $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$2 are invariant under $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$3-unravelling, this captures a locality principle: membership in a definable class depends only on bounded-radius information.

The structural reason the preservation theorems go through is a bounded-height well-quasi-ordering. On any class of finite pointed tree-shaped Kripke models of uniformly bounded height $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$4, the embedding relation is a well-quasi-order, hence every subset has only finitely many minimal elements up to isomorphism (Wałęga et al., 2 Feb 2026). The paper further shows that the bounded-height hypothesis is essential: without it, embeddings, injective homomorphisms, and homomorphisms need not be well-quasi-orders, since there are infinite antichains (Wałęga et al., 2 Feb 2026).

This wqo result supports a finite-minimal-tree argument. Given an $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$5-unravelling invariant class $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$6, one obtains finitely many minimal tree representatives and then constructs characteristic formulas whose finite disjunction defines $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$7 (Wałęga et al., 2 Feb 2026). A plausible implication is that the modal definability problem for finite structures becomes tractable precisely when locality converts arbitrary structures into a bounded combinatorial domain admitting finitary basis arguments.

4. Goldblatt–Thomason patterns and frame definability

A central classical answer to the frame-definability problem is the Goldblatt–Thomason theorem, which characterizes elementary frame classes definable by modal formulas via closure under generated subframes, disjoint unions, bounded morphic images, together with reflection of canonical or ultrafilter extensions. Many later works revisit this pattern in nonclassical, restricted, or finite settings.

For modal logic with universal modality restricted positively, $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$8, frame definability has an exact Goldblatt–Thomason-style characterization. An elementary frame class is definable in $$(\mathfrak{M},w)\in\mathcal{C}\quad\text{iff}\quad (\mathfrak{M},w)\models \varphi.$$9 iff it is closed under generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes (Sano et al., 2015). This differs from the classical $C$0 case by replacing the disjoint-union condition with reflection of finitely generated subframes.

The same paper also gives a model-theoretic characterization for model classes: a class of Kripke models is definable in $C$1 iff it is elementary and closed under surjective bisimulations (Sano et al., 2015). This cleanly separates model and frame definability.

In finitely-valued Łukasiewicz modal logics, a Goldblatt–Thomason theorem is obtained for two notions of definability: $C$2-definability for classes of $C$3-frames via $C$4, and definability for richer $C$5-frames via a second validity relation (Teheux, 2015). For classes of plain $C$6-frames, the closure-and-reflection conditions match the classical ones, and for elementary classes the paper proves that $C$7-definability coincides with ordinary classical modal definability (Teheux, 2015). By contrast, over $C$8-frames with additional truth-degree predicates, definability is strictly richer, because modal formulas can constrain where restricted truth-value sets are allowed (Teheux, 2015).

Recent Euclidean-frame work addresses not the abstract closure characterization itself but the computability of the modal definability problem over frame classes $C$9 determined by Euclidean modal logics $C=\{F\mid F\models \Phi\}$0 extending $C=\{F\mid F\models \Phi\}$1 (Balbiani et al., 14 Aug 2025). The main result is a dichotomy: for every Euclidean modal logic $C=\{F\mid F\models \Phi\}$2, the modal definability problem for first-order sentences with respect to $C=\{F\mid F\models \Phi\}$3 is decidable iff a certain associated set $C=\{F\mid F\models \Phi\}$4 is finite; in the decidable case the problem is in EXPSPACE, while infinitude of $C=\{F\mid F\models \Phi\}$5 yields undecidability (Balbiani et al., 14 Aug 2025). This places one significant region of frame-definability theory on a precise algorithmic map.

5. Fragments, counting, interpolation, and separation

Many instances of the modal definability problem concern whether formulas in a stronger language collapse into a weaker fragment. In fragments of $C=\{F\mid F\models \Phi\}$6 with counting and corresponding graded modal logics, the definability problem asks whether a formula with counting quantifiers is equivalent to one in a counting-free fragment (Kuijer et al., 29 Apr 2025). A key positive result is that, unlike separation, definability can often be reduced in polynomial time to validity in the stronger logic (Kuijer et al., 29 Apr 2025). For nominal-free settings, this reduction uses a flattening operation that replaces counting modalities by ordinary modalities and checks whether the original formula is equivalent to its flattening (Kuijer et al., 29 Apr 2025).

By contrast, the more general separation problem is much harder. For $C=\{F\mid F\models \Phi\}$7 and graded modal logic with inverse, nominals, and universal modality, separation is RE-complete; dropping inverse or nominals yields decidable cases with coNExpTime- or $C=\{F\mid F\models \Phi\}$8ExpTime-complete complexity depending on the presence of the universal modality (Kuijer et al., 29 Apr 2025). This sharp gap between definability and separation is one of the recurrent themes in contemporary definability theory.

A related phenomenon appears in first-order modal logic. In one-variable first-order modal logics such as $C=\{F\mid F\models \Phi\}$9 and $\Phi$0, Craig interpolation and projective Beth definability fail even under severe syntactic restrictions (Kurucz et al., 2023). As a consequence, the existence of an interpolant or explicit definition cannot be reduced to ordinary validity. The paper formulates the Explicit Definition Existence Problem and the Interpolant Existence Problem as genuine decision problems and proves them decidable in coN2ExpTime for $\Phi$1 and $\Phi$2, with $\Phi$3ExpTime-hard lower bounds (Kurucz et al., 2023). This is not a modal-definability theorem in the frame-class sense, but it shows how definability questions persist even when standard meta-theorems fail.

In the modal $\Phi$4-calculus, definability by modal logic becomes a special case of separability. A formula $\Phi$5 is $\Phi$6-definable over a class $\Phi$7 iff $\Phi$8 and $\Phi$9 are $(\mathfrak{M},w)$0-separable over $(\mathfrak{M},w)$1 (Jung et al., 29 Sep 2025). The resulting complexity landscape depends strongly on the model class: over words, both definability and separability are PSPACE-complete; over arbitrary models and binary trees, definability is ExpTime-complete; over bounded outdegree $(\mathfrak{M},w)$2, separability rises to TwoExpTime-complete and the induced interpolant-existence problem becomes coNExpTime-complete (Jung et al., 29 Sep 2025). This suggests that model geometry, especially branching degree, can alter definability behavior even when the source logic is fixed.

6. Nonclassical, applied, and domain-specific variants

The modal definability problem extends well beyond ordinary Kripke semantics. Several papers show how the same question reappears, often with modified closure conditions or semantic obstacles, in richer semantic settings.

In the bimodal temporal-functional language $(\mathfrak{M},w)$3, the problem is to determine which functional properties of partial functions between ordered flows are definable modally (Burrieza, 1 Jul 2026). Over arbitrary ordered multiflows, only totality and surjectivity are definable, and only over linear orders; all other studied properties—non-totality, injectivity, monotonicity, strict monotonicity, antitonicity, strict antitonicity, and constancy—are undefinable in that unrestricted setting (Burrieza, 1 Jul 2026). When semantics is restricted to minimal functional frames $(\mathfrak{M},w)$4, or controlled via indexed languages or the Uniform Domain condition, many more properties become definable, and the choice between standard and strict temporal readings becomes crucial (Burrieza, 1 Jul 2026). This suggests that what limits expressivity is not simply the modal vocabulary but uncontrolled functional multiplicity.

In dynamic probability logics, frame definability concerns classes of probabilistic dynamical systems and Markov processes (Chopoghloo et al., 2024). The finitary dynamic probability logic $(\mathfrak{M},w)$5 defines measure-preserving abstract dynamical systems and ergodicity within appropriate ambient classes, while strong mixing requires the infinitary extension $(\mathfrak{M},w)$6 (Chopoghloo et al., 2024). The infinitary probability logic $(\mathfrak{M},w)$7 defines stochastic properties such as stationarity, invariance, irreducibility, and recurrence of Markov processes with initial distribution (Chopoghloo et al., 2024). A plausible implication is that infinitary conjunctions become necessary precisely when the target property is asymptotic or $(\mathfrak{M},w)$8-additive in nature.

In Kripke’s theory of truth, a modal language over the space of fixed points is used to study which semantic classifications of sentences are modally expressible (Walsh, 2024). Groundedness and paradoxicality are modally definable, while intrinsicality is not (Walsh, 2024). The paper also characterizes the modally definable relations and axiomatizes the valid modal principles of the fixed-point semantics (Walsh, 2024). This is a particularly clear example where a natural semantic distinction used informally in philosophical logic resists capture by the chosen modal language.

In intuitionistic modal logic with four modal connectives, the definability question is formulated for elementary classes of frames $(\mathfrak{M},w)$9 and answered via signed formulas and inference rules (Balbiani et al., 5 Jun 2026). Seriality and some confluence conditions are modally definable, while several familiar relational properties such as reflexivity, symmetry, transitivity, Euclideanness, and determinism are shown not to be modally definable relative to certain ambient classes (Balbiani et al., 5 Jun 2026). The paper also proves that the global problems of modal definability, positive definability, and negative definability are undecidable in this setting (Balbiani et al., 5 Jun 2026).

Finally, recent work on definite description operators shows how modal extensions affect definability of frame properties and cardinality conditions (Wałęga et al., 2024). The operator $\mathcal{C}$0, read as truth of $\mathcal{C}$1 at the unique world satisfying $\mathcal{C}$2, is strictly more expressive than hybrid operators but strictly less expressive than counting operators (Wałęga et al., 2024). It can define singleton frames via $\mathcal{C}$3, but cannot define exact cardinality $\mathcal{C}$4, which remains expressible in counting logic (Wałęga et al., 2024). This gives a fine-grained expressivity comparison directly relevant to modal definability by language extension.

7. Algorithmic, meta-theoretic, and methodological perspectives

A recurring theme is that modal definability has both semantic and computational faces. Semantically, one seeks closure conditions, invariances, or preservation properties. Algorithmically, one asks whether definability is decidable and how complex the decision problem is.

Over Euclidean frame classes, the definability problem admits a full decidability/undecidability dichotomy with an EXPSPACE upper bound in the positive cases (Balbiani et al., 14 Aug 2025). Over graded modal fragments, definability is often easier than separation and can be reduced to validity in the stronger logic (Kuijer et al., 29 Apr 2025). Over one-variable first-order modal logics, explicit definition existence and interpolant existence are decidable but substantially harder than plain validity (Kurucz et al., 2023). Over bounded-outdegree trees for the $\mathcal{C}$5-calculus, definability complexity varies sharply with branching degree (Jung et al., 29 Sep 2025).

Methodologically, several proof patterns recur. One is translation: finitely-valued modal definability is compared to classical modal definability via suitable syntactic translations (Badia et al., 2022). Another is normal-form reduction: $\mathcal{C}$6-definability is analyzed through closed disjunctive $\mathcal{C}$7-clauses (Sano et al., 2015). A third is bisimulation- or simulation-based characterization, which underlies both abstract frameworks (Carreiro, 2010) and concrete non-definability arguments (Kurucz et al., 2023, Wałęga et al., 2024). A fourth is finite-basis reasoning via well-quasi-ordering, which is central in the preservation theorem for bounded-unravelling-invariant classes (Wałęga et al., 2 Feb 2026).

One common misconception is that stronger truth-value systems or richer semantics automatically enlarge the class of definable frame properties. The finitely-valued results show that this is false for large families of crisp-frame semantics: no more frame classes are definable than in classical modal logic, and often exactly the same classes are definable (Teheux, 2015, Badia et al., 2022). Another common misconception is that definability and interpolation are interchangeable. The failure of Craig interpolation and Beth definability in first-order modal logic demonstrates that definability may become an independent decision problem rather than a corollary of validity (Kurucz et al., 2023).

Taken together, these developments show that the modal definability problem is not a single theorem but a research program. Its modern form spans frame and model semantics, finite and infinite structures, graded and fixpoint extensions, many-valued and intuitionistic settings, and applied semantic domains such as GNNs, temporal-functional systems, and probabilistic dynamics. The unifying question remains constant: which semantic distinctions can a given modal language capture exactly, and by what structural principles can that expressive boundary be described?