Existential Graded Modal Logic

• Existential graded modal logic is a fragment of modal logic that enhances classical modal operators with explicit cardinality constraints, enabling precise counting of accessible worlds.
• EGML bridges basic and full graded modal logics, with its satisfiability complexity ranging from PSPACE-complete to NEXPTIME-complete depending on frame conditions.
• EGML underpins model-theoretic preservation theorems and parallels the expressiveness of graph neural networks through tree unravelling and homomorphism-count characterizations.

Existential graded modal logic (EGML) is a fragment of modal logic that augments the classical modal operators with explicit cardinality constraints, enabling the expression of statements about the number of accessible worlds that satisfy certain properties. EGML is situated between basic modal logic and full graded modal logic in expressiveness and plays a fundamental role in modal preservation theorems, complexity analysis, and connections to combinatorial invariants and machine learning architectures.

1. Formal Syntax and Semantics

The syntax of existential graded modal logic is based on a countable set of propositional atoms $P = \{p_1, \ldots, p_n\}$ and a single binary relation symbol $R$. The formulas are generated by the grammar: $$\varphi ::= p \quad|\quad \neg p \quad|\quad \varphi \land \varphi \quad|\quad \varphi \lor \varphi \quad|\quad \Diamond^{\geq k}\varphi$$ where $p \in P$, $k \geq 1$, and $\neg p$ is treated as a positive literal (negation is only allowed in front of atoms, not arbitrary subformulas).

• $\Diamond^{\geq k} \varphi$ asserts "there exist at least $k$ $R$-successors satisfying $\varphi$".
• The dual operator $\Box^{\leq k}\varphi$ is defined as $$\Box^{\leq k} \varphi \equiv \neg \Diamond^{\geq k+1} \neg \varphi$$ and asserts "at most $k$ $R$-successors satisfy $\varphi$".

The semantics is given in terms of Kripke structures $M = (W, R, V)$, where $W$ is a nonempty set of worlds, $R \subseteq W \times W$ is the accessibility relation, and $V: P \to \mathcal{P}(W)$ assigns truth values to atoms. For a pointed model $(M, w)$:

• $(M, w) \models p$ iff $w \in V(p)$
• $(M, w) \models \neg p$ iff $w \notin V(p)$
• $(M, w) \models \varphi \land \psi$ iff $(M, w) \models \varphi$ and $(M, w) \models \psi$
• $(M, w) \models \varphi \lor \psi$ iff $(M, w) \models \varphi$ or $(M, w) \models \psi$
• $(M, w) \models \Diamond^{\geq k} \varphi$ iff there exist at least $k$ distinct $v$ with $wRv$ and $(M, v) \models \varphi$
• $(M, w) \models \Box^{\leq k} \varphi$ iff at most $k$ many $R$-successors $v$ of $w$ satisfy $(M, v) \models \varphi$ (0905.3108, Wałęga et al., 2 Feb 2026).

2. Relationship to Ordinary and Graded Modal Logics

Ordinary modal logic can be embedded into EGML by observing that:

• The basic possibility operator $\Diamond\varphi$ abbreviates $\Diamond^{\geq 1}\varphi$.
• The necessity operator $\Box\varphi$ abbreviates $\Box^{\leq 0}\neg\varphi$.

The expressive gap arises because EGML can directly state properties such as "at least $k$ accessible successors satisfy $\varphi$," going beyond the mere existence assertions of standard modal logic. EGML is strictly less expressive than full graded modal logic (which allows unrestricted negation and counting both "at least" and "at most"), but contains formulas like "there exist at least $k$ successors satisfying $p \land \Diamond^{\geq m} q$" that are not definable in the existential positive fragment of basic modal logic (Wałęga et al., 2 Feb 2026).

3. Complexity of Satisfiability

The complexity of the satisfiability problem for EGML is tightly linked to underlying frame properties:

Frame-class conditions	GradeSat($\mathcal{F}$) Complexity
None/only reflexive, serial, symmetric	PSPACE-complete
Euclidean or (symmetric ∧ transitive)	NP-complete
Transitive but not symmetric or Euclidean	NEXPTIME-complete

• If transitivity and Euclideanness are absent, the tree-model property holds, and satisfiability is PSPACE-complete, generalizing Ladner’s result for basic modal logic.
• When Euclidean properties or both symmetry and transitivity are assumed, satisfiability is NP-complete because models can be collapsed to small cone models.
• With (pure) transitive frames lacking symmetry/Euclideanness, EGML fails to have the tree-model property, necessitating exponentially large models and resulting in NEXPTIME-completeness. This reflects the complexity jump directly tied to the failure of usual tableau methods and the need for non-tree-like models (0905.3108).

4. Preservation Theorems and Tree Unravelling

A central model-theoretic result for EGML is its tight connection to embedding invariance and fragmentation by tree-unravelling. For modal depth $L$:

• Any class of finite pointed models $\mathcal{C}$ invariant under $L$-unravelling and preserved under embeddings is definable by some existential graded formula of depth $L$.
• Conversely, every EGML formula of depth $L$ is both $L$-unravelling invariant and embedding-preserved.

The mechanism is as follows: each finite unravelling of height $L$ can be characterized up to isomorphism by a particular EGML formula ("minimal tree" argument), and every such class corresponds to a finite disjunction of these characteristic formulas. Well-quasi-ordering under embedding on bounded-height trees ensures this characterization is finite. The bounded-height condition is essential; without it, infinite antichains prevent such normal forms (Wałęga et al., 2 Feb 2026).

5. Characterization via Homomorphism-Counting

EGML admits a robust Lovász-style model-theoretic invariance characterization:

• For pointed LTSs $(M, a)$ and $(N, b)$, the following are equivalent:
  1. They are indistinguishable by homomorphism counts from all finite rooted trees (with counts in the natural numbers semiring).
  2. Their tree-unravellings are isomorphic.
  3. They satisfy the same graded modal formulas.

Formally, for the class $T_\sigma$ of all finite $\sigma$-trees, $M$ and $N$ satisfy the same graded modal formulas iff $$\text{hom}_{\mathbb{N}}(T_\sigma, M) = \text{hom}_{\mathbb{N}}(T_\sigma, N)$$. This generalizes the classical Lovász theorem from first-order logic to graded modal logic. The proof leverages the correspondence between modal formula satisfaction and positive existential conjunctive queries, and the tree-unravelling invariance of graded modal logic (Comer, 2024).

6. Extensions, Limitations, and Applications

Extensions:

• EGML can be relativized with additional modalities such as backward (predecessor) or global counting. These extensions are handled by encoding the relevant access relations or global quantifiers into the signature and reducing again to tree-count characterizations.
• The expressive power of EGML also provides a precise logical correspondence to classes of graph neural networks (GNNs): node-classification functions of monotonic, aggregation-based GNNs are exactly as expressive as EGML formulas of matching modal depth (Wałęga et al., 2 Feb 2026).

Limitations:

• Homomorphism-count characterizations do not extend to the full basic modal logic (bisimulation equivalence), due to the presence of ultimately periodic counting behavior in non-injective semirings and the inability to distinguish certain infinite families of graphs (e.g., complete graphs) using hom-counts alone (Comer, 2024).

Applications:

• EGML is a key instrument in preservation theorems: a property or class is preserved under embeddings and bounded unravelling if and only if it is definable in EGML.
• In descriptive complexity and GNN theory, EGML formulas act as tight logical bounds for the expressive power of neural architectures and other logics invariant under embeddings (Wałęga et al., 2 Feb 2026).

7. Illustrative Example

Consider the formula $\Diamond^{\geq 2} p$, expressing "there are at least two $p$-successors". In a Kripke model $M = (\{a, b, c\}, R, V)$ with $R = \{(a, b), (a, c)\}$ and $V(p) = \{b, c\}$, $M, a \models \Diamond^{\geq 2} p$ holds. The simplest tree test object $T$ of depth $1$ with two children corresponds to this configuration, and $\text{hom}_{\mathbb{N}}(T, M) = 2$ since there are $2!$ ways to map the children to $b$ and $c$ (Comer, 2024).

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Key References:

• "A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics" (0905.3108)
• "Preservation Theorems for Unravelling-Invariant Classes: A Uniform Approach for Modal Logics and Graph Neural Networks" (Wałęga et al., 2 Feb 2026)
• "Lovász Theorems for Modal Languages" (Comer, 2024)