First-Order Hennessy-Milner Logic

• First-Order Hennessy-Milner Logic is an umbrella term for modal frameworks enriched with first-order syntax, linking modal reasoning with bisimulation-invariant definability.
• It encompasses diverse readings such as correspondence-language translations, explicit HML extensions for concurrency and smart contracts, and hybrid or nonclassical adaptations.
• Key methodologies include standard translation, saturation principles, and game-theoretic locality to bridge modal behavior with first-order model theory.

“First-Order Hennessy-Milner Logic” does not denote a single universally fixed formalism in the recent literature. Instead, the term covers a family of closely related projects that connect Hennessy–Milner-style modal reasoning with first-order apparatus. In one reading, the point is correspondence: ordinary modal logic is studied through its standard translation into first-order logic, and modal expressivity is identified with bisimulation-invariant first-order definability. In another reading, HML itself is extended with variables, quantification, state names, or structured action terms, yielding explicitly first-order-style modal languages for concurrency, smart contracts, or data-aware systems (Takeda et al., 2 Jul 2026, Baldan et al., 2011).

1. Terminological scope

Across the cited literature, the phrase refers to several non-equivalent but systematically related notions.

Reading of the term	First-order ingredient	Representative source
Correspondence-language reading	Standard translation and bisimulation-invariant FO fragment	(Takeda et al., 2 Jul 2026)
Explicit HML extension	Variables, binders, quantification, structured actions	(Baldan et al., 2011, Bartoletti et al., 15 Apr 2026)
Hybrid/state-reference reading	Nominals, state variables, satisfaction operators, quantification over states	(Leuştean et al., 2020)
Generalized/nonclassical reading	Intuitionistic, inquisitive, probabilistic, path-based, or semilattice variants	(Groot et al., 30 Jun 2026, Meißner et al., 2019, Wild et al., 2018, Figueira et al., 2023, Groot, 2022)

A recurring misconception is that the expression must refer to a first-order modal logic with quantifiers over individuals. One important recent reverse-mathematical study explicitly says that it “does not introduce a separate formalism called ‘first-order Hennessy-Milner logic’,” and that its first-order content enters through the van Benthem characterization theorem and through coding of models, formulas, and provability (Takeda et al., 2 Jul 2026). A closely related intuitionistic study likewise characterizes a propositional modal logic, $IK$, inside intuitionistic first-order logic, rather than presenting a full first-order modal source language (Groot et al., 30 Jun 2026).

This suggests a broad but precise encyclopedia-level understanding: first-order Hennessy-Milner logic is best treated as an umbrella expression for modal frameworks in which Hennessy–Milner phenomena are mediated by first-order syntax, first-order invariance, or first-order-style enrichment.

2. Classical correspondence and bisimulation invariance

The most canonical first-order reading is the correspondence-language one. In the reverse-mathematical analysis of bisimulation, the modal language is propositional, the first-order language has unary predicates $P_0,P_1,\dots$ corresponding to propositional variables and a binary relation symbol $r$, and the bridge is the standard translation

$$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$

The central first-order notion is then not a separate “first-order Hennessy-Milner logic,” but the class of bisimulation-invariant first-order formulas (Takeda et al., 2 Jul 2026).

Within that setting, the paper formalizes the usual Hennessy–Milner theorem for image-finite and modally saturated pointed models: $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$ Its reverse-mathematical result is exact: over $\mathrm{RCA}_0$, the Hennessy–Milner theorem and its modally saturated version are equivalent to $\mathrm{ACA}_0$. The same paper then separates three forms of van Benthem’s theorem. The semantic form is provable in $\mathrm{RCA}_0$, the syntactic form is provable in $\mathrm{PRA}$, and the hybrid form is equivalent over $\mathrm{RCA}_0$ to the Weak Completeness Theorem for first-order logic and to weak $P_0,P_1,\dots$0-separation (Takeda et al., 2 Jul 2026).

From this perspective, first-order Hennessy–Milner logic is fundamentally about modal definability inside first-order logic. A one-free-variable first-order formula $P_0,P_1,\dots$1 belongs to the modal fragment exactly when it is invariant under bisimulation, and the technical burden lies in proving the transfer from invariance to modality. In the same study, that transfer is formalized by Otto’s elementary proof using locality, finite-depth bisimulation, Ehrenfeucht–Fraïssé games, local tree unravellings, and characteristic modal formulas. For a formula of quantifier rank $P_0,P_1,\dots$2, the locality radius is set to $P_0,P_1,\dots$3, and the resulting characteristic modal formula $P_0,P_1,\dots$4 satisfies

$P_0,P_1,\dots$5

The first-order content is therefore exact, but it is correspondence-theoretic rather than a new object language.

3. Explicit first-order-style extensions of HML

A second line of work builds languages that are HML-like in syntax but explicitly first-order in flavor. The clearest example is the logic for true concurrency over prime event structures. Its formulas can bind events to variables and later refer to them: $P_0,P_1,\dots$6 Here $P_0,P_1,\dots$7 binds an $P_0,P_1,\dots$8-labelled future event to $P_0,P_1,\dots$9, requiring causal dependence on the event bound to $r$0 and concurrency with the event bound to $r$1, while $r$2 says that the event bound to $r$3 is enabled and executable. The full logic characterizes hereditary history-preserving bisimilarity, fragments recover step, pomset, history-preserving, and ordinary bisimilarity, and the fixpoint extension $r$4 is explicitly described as a kind of first-order modal $r$5-calculus (Baldan et al., 2011).

A practical instance appears in smart-contract verification. CHML is presented as a specification language “based on first-order Hennessy-Milner Logic,” with types $r$6, $r$7, $r$8, $r$9, and $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$0, quantification $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$1, transaction modalities

$$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$2

and the smart-contract-specific expressions $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$3 and $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$4. Its semantics is over finite sequences of flagged blockchain states, and its implementation compiles contracts and CHML formulas into Lustre for verification by Kind 2 (Bartoletti et al., 15 Apr 2026).

A further applied variant arises in runtime enforcement. There the logic is not presented as full first-order HML, but as the safety fragment of Hennessy-Milner Logic with recursion over value-passing systems and symbolic actions with data variables and conditions. The paper explicitly links its setting to “Hennessy and Liu’s first-order modal logic for message-passing processes” and “Rathke and Hennessy’s first-order modal $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$5-calculus,” while focusing on enforceability of the normalized safety fragment $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$6 via synthesized action-disabling monitors (Aceto et al., 2022).

The choreography literature shows the same movement in a different semantic setting. Global Logic extends an HML-style action modality with existential quantification, equality $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$7, a spatial parallel connective, and an eventuality operator $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$8. It is therefore quantified and data-aware, but it is not presented as a standard FO-HML over labelled transition systems (Carbone et al., 2011). A many-sorted hybrid route reaches similar territory by adding nominals, state variables, satisfaction operators $$\mathrm{ST}_x(p_n)\equiv P_n(x),\qquad \mathrm{ST}_x(\Box\varphi)\equiv \forall y(r(x,y)\rightarrow \mathrm{ST}_y(\varphi)).$$9, and quantification over states $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$0, together with a standard translation into many-sorted first-order logic (Leuştean et al., 2020).

4. Nonclassical and generalized first-order Hennessy-Milner phenomena

Recent work shows that the Hennessy–Milner/first-order pattern persists far beyond classical propositional modal logic.

For intuitionistic modal logic $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$1, the main theorem says that $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$2 is exactly the IK-bisimulation-invariant fragment of intuitionistic first-order logic. The proof develops a custom notion of IK-bisimulation over birelational semantics, proves a Hennessy–Milner theorem on modally saturated birelational models, and uses intuitionistic analogues of Łoś’s theorem, elementary embeddings, and countable saturation to obtain the final characterization (Groot et al., 30 Jun 2026).

For inquisitive modal logic, a two-sorted first-order framework is used to handle information states and inquisitive states. The key enabling device is graded flatness: $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$3 This boundedness makes a first-order standard translation possible. Because proper inquisitive models are not first-order axiomatizable, the analysis proceeds through pseudo-models, and the Hennessy–Milner theorem is proved using $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$4-saturated relational pseudo-models and a stronger state-level notion called bulk equivalence (Meißner et al., 2019).

In probabilistic transition systems, the analogue of Hennessy–Milner logic is quantitative probabilistic modal logic, and the corresponding first-order language is quantitative probabilistic first-order logic with graded truth values in $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$5. The main van Benthem-style result is approximate rather than exact: $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$6 Here invariance becomes non-expansivity with respect to behavioural distance, and exact modal definability is replaced by uniform approximation (Wild et al., 2018).

Path Predicate Modal Logic offers a different generalization. It admits relation symbols of arbitrary arity as atoms, interpreted over the current explored path rather than over a single world. The paper proves a Hennessy–Milner property for bounded-depth PPML, defines a PPML comonad, and shows that PPML formulas translate into first-order logic with exact quantifier-rank preservation; under bounded arity they also lie inside bounded-variable FO (Figueira et al., 2023).

A semilattice-based variant reaches a van Benthem-style theorem through a non-Kripke semantics. Non-distributive positive logic is translated into one-variable first-order formulas over meet-semilattices, a Hennessy–Milner theorem is proved for a traditional simulation notion using meet-compactness, and the final characterization identifies the logic with the fragment of first-order logic preserved by meet-simulations, a relation that connects pairs of source states with single target states (Groot, 2022).

5. Model-theoretic and proof-theoretic mechanisms

Across these variants, several technical devices recur. Saturation is central. In the classical reverse-mathematical analysis, modally saturated models support the reconstruction of bisimulations from modal equivalence (Takeda et al., 2 Jul 2026). In the intuitionistic setting, $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$7-saturated $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$8-structures yield modally saturated birelational models (Groot et al., 30 Jun 2026). In inquisitive modal logic, $$(M,w)\leftrightsquigarrow(M',w') \quad\text{if and only if}\quad (M,w)\,\underline{\xleftrightarrow{}\,(M',w')}.$$9-saturated relational pseudo-models are required because true downward-closed inquisitive models are generally incompatible with $\mathrm{RCA}_0$0-saturation (Meißner et al., 2019). In the semilattice setting, meet-compactness plays the role that modal saturation plays in classical Hennessy–Milner arguments (Groot, 2022).

Game-theoretic locality is equally prominent. The classical first-order correspondence proof in second-order arithmetic uses $\mathrm{RCA}_0$1-round Ehrenfeucht–Fraïssé games, $\mathrm{RCA}_0$2-bisimulations with $\mathrm{RCA}_0$3, and local tree unravellings to show that bisimulation-invariant first-order formulas are local and therefore modal (Takeda et al., 2 Jul 2026). The probabilistic generalization replaces crisp bisimulation games with an up-to-$\mathrm{RCA}_0$4 game and proves locality with radius $\mathrm{RCA}_0$5 for formulas of rank $\mathrm{RCA}_0$6 (Wild et al., 2018). PPML develops resource-bounded simulation and bisimulation games over sequences rather than points, and packages them comonadically (Figueira et al., 2023).

Standard translation remains the most stable bridge to first-order logic. It appears in ordinary Kripke correspondence (Takeda et al., 2 Jul 2026), in intuitionistic birelational semantics (Groot et al., 30 Jun 2026), in probabilistic expectation-based first-order logic (Wild et al., 2018), and in hybrid many-sorted settings where modalities become existential first-order conditions over relation symbols $\mathrm{RCA}_0$7 and state references become equalities or satisfaction operators (Leuştean et al., 2020).

A more abstract perspective is provided by the Galois-connection framework for Hennessy-Milner theorems. That paper does not treat first-order syntax or semantics, but it isolates a uniform lattice-theoretic recipe: derive behaviour functions from a logic, define a compatibility condition, and transfer fixpoint properties through an adjunction. Its relevance to first-order Hennessy-Milner work is therefore methodological rather than direct (Beohar et al., 2022).

6. Scope, limitations, and applications

The literature makes clear that “first-order Hennessy-Milner logic” should not be read as the name of a single mature standard. Several papers explicitly say they are not about first-order modal logic with quantifiers over individuals, and several others offer only a nearest analogue rather than the exact named formalism (Takeda et al., 2 Jul 2026, Wild et al., 2018).

A second limitation concerns exactness. In probabilistic systems, the analogue of van Benthem’s theorem is a density theorem with bounded-rank approximation, not exact equivalence (Wild et al., 2018). In smart-contract verification, CHML is a finite-depth, non-recursive fragment; the paper explicitly contrasts it with HML with recursion and says it does not express general liveness (Bartoletti et al., 15 Apr 2026). In runtime enforcement, the enforceable target is a normalized safety fragment $\mathrm{RCA}_0$8, not an unrestricted first-order HML (Aceto et al., 2022). In choreography theory, the global logic is undecidable with recursion, although the recursion-free fragment admits a sound and complete proof system and decidable proof search (Carbone et al., 2011).

At the same time, the application range is substantial. Event-structured logics use first-order-style binding to distinguish causality from concurrency (Baldan et al., 2011). Smart-contract verification uses quantified transaction modalities, $\mathrm{RCA}_0$9, and $\mathrm{ACA}_0$0 to specify liquidity, reversibility, conservation, and front-running properties (Bartoletti et al., 15 Apr 2026). Runtime enforcement compiles data-aware safety formulas into bidirectional monitors (Aceto et al., 2022). Intuitionistic, inquisitive, probabilistic, path-based, and semilattice semantics all show that the Hennessy–Milner idea survives substantial shifts in semantics once the appropriate first-order correspondence language, invariance notion, and saturation principle have been identified (Groot et al., 30 Jun 2026, Meißner et al., 2019, Figueira et al., 2023, Groot, 2022).

In that broad sense, first-order Hennessy-Milner logic designates a research program rather than a single calculus: the study of how modal behavioral expressivity interacts with first-order definability, first-order model theory, and first-order-style enrichment. The enduring invariant is the same one that motivates the classical Hennessy–Milner theorem itself: logical indistinguishability should coincide, exactly or approximately, with the right structural notion of behavioural sameness.