Bisimulations in second-order arithmetic

Abstract: This paper investigates the logical strength of two theorems in modal propositional logic - the Hennessy-Milner theorem and the van Benthem characterization theorem - within the framework of second-order arithmetic. We demonstrate that the Hennessy-Milner theorem is equivalent to $\mathrm{ACA}_0$ over $\mathrm{RCA}_0$. For the van Benthem characterization theorem, we introduce three variants: the semantic, syntactic, and hybrid forms. We show that the semantic form is provable in $\mathrm{RCA}_0$, the syntactic form is provable in $\mathrm{PRA}$, and the hybrid form is equivalent to the weak completeness theorem for first-order logic over $\mathrm{RCA}_0$.

• The paper establishes that the Hennessy–Milner theorem is equivalent to ACA₀ over RCA₀, revealing deep links between modal equivalence and arithmetic comprehension.
• It demonstrates the semantic van Benthem characterization theorem is provable in RCA₀, contrasting with the stronger subsystem requirements for modal equivalence.
• The work employs explicit Kripke model constructions and finite model techniques to analyze proof strength and computability barriers in modal logic.

Bisimulations in Second-Order Arithmetic: Logical Strength of Modal Characterization Theorems

Introduction and Context

This paper investigates the reverse mathematics of two foundational results in modal logic—the Hennessy–Milner theorem and the van Benthem characterization theorem—within the context of second-order arithmetic. The authors analyze the exact subsystems of second-order arithmetic required for various forms of these theorems, providing fine-grained results about their logical strength and computability content. The base theory throughout is $\mathrm{RCA}_0$, with results calibrated against stronger systems such as $\mathrm{ACA}_0$, $\mathrm{WKL}_0$, and $\mathrm{PRA}$. The work situates modal logic within the tradition of analyzing logical theorems via their equivalence with comprehension or separation principles, drawing concrete connections between modal expressivity and arithmetic hierarchies.

Hennessy–Milner Theorem and Equivalence to $\mathrm{ACA}_0$

The Hennessy–Milner theorem posits that, for image-finite Kripke models, modal equivalence between pointed models entails bisimilarity, and vice versa. The paper establishes that over $\mathrm{RCA}_0$, the Hennessy–Milner theorem is equivalent to the subsystem $\mathrm{ACA}_0$. The proof shows that the construction of full valuations extending atomic valuations on image-finite Kripke frames is not generally feasible in $\mathrm{RCA}_0$ and requires the arithmetic comprehension available only in $\mathrm{ACA}_0$. The contrapositive is realized by encoding the range of an injective function $f : \mathbb{N} \to \mathbb{N}$ into modal equivalence of carefully constructed Kripke models, showing that the existence of such a bisimulation yields comprehension for the range of $\mathrm{ACA}_0$0. The proof is highly constructive, with extensive formalization within $\mathrm{ACA}_0$1 supplemented by an explicit dependency on the arithmetic comprehension axiom.

Pragmatically, the result underscores that modal equivalence and bisimulation in infinite settings—as opposed to the finite case—have a computability-theoretic barrier intimately linked to set existence axioms, aligning the modal-theoretic notion of behavioral equivalence with classical questions of arithmetic definability.

van Benthem Characterization Theorem: Semantic, Syntactic, and Hybrid Forms

The van Benthem theorem characterizes modal logic as the bisimulation-invariant fragment of first-order logic. The paper meticulously analyzes three formulations:

Semantic Form: Every bisimulation-invariant first-order property is definable by a modal formula up to logical equivalence.

• Provable in $\mathrm{ACA}_0$2: The paper leverages Otto’s combinatorial, elementary proof (as opposed to van Benthem's ultrafilter- or compactness-driven approach) and fully formalizes the result in $\mathrm{ACA}_0$3. The construction uses finite model theoretic techniques, including unravelling, local games, and arity-restricted formulas, to show that any bisimulation-invariant property of appropriate quantifier rank has an explicit modal formula equivalent to its standard translation.

Syntactic Form: The property of bisimulation invariance for first-order formulas is arithmetically expressible, and the modal definition can be constructively obtained.

• Provable in $\mathrm{ACA}_0$4: The authors show the translation of bisimulation invariance into $\mathrm{ACA}_0$5 formulas, thus demonstrating that the syntactic form of the theorem can be established already in primitive recursive arithmetic. The argument leverages the syntactic expressibility of bisimulation conditions and the $\mathrm{ACA}_0$6-conservativity of $\mathrm{ACA}_0$7 over $\mathrm{ACA}_0$8.

Hybrid Form: The equivalence between bisimulation invariance and definability by modal formulas, expressed as provability of the corresponding standard translation in the proof system.

• Equivalent to the weak completeness of first-order logic over $\mathrm{ACA}_0$9: The equivalence with the weak completeness theorem for first-order logic pinpoints the exact proof-theoretic strength needed, relating the modal result to a weaker system than $\mathrm{WKL}_0$0, specifically to the weak $\mathrm{WKL}_0$1-separation schema.

Summary of Subsystem Boundaries:

Theorem / Form	Subsystem Required
Hennessy–Milner	$\mathrm{WKL}_0$2 (over $\mathrm{WKL}_0$3)
van Benthem, Semantic	$\mathrm{WKL}_0$4
van Benthem, Syntactic	$\mathrm{WKL}_0$5
van Benthem, Hybrid	Weak completeness for $\mathrm{WKL}_0$6 / weak $\mathrm{WKL}_0$7-separation

These logical calibrations are highly nontrivial. Particularly notable is the unexpected weakness required for the semantic van Benthem theorem, as compared to the strong requirements for the Hennessy–Milner theorem.

Proof-theoretic and Computability Analysis

A major contribution is the deep operationalization of model theory and modal logic within the restricted comprehension and induction available in weak subsystems. The authors adopt explicit definitions of Kripke models, modalities, and bisimulation for $\mathrm{WKL}_0$8, coping with the arithmetization of the language and model domains. Of particular importance are the fine analyses of definability and uniformity in the existence of bisimulations and the modal formulas derived from Ehrenfeucht–Fraïssé games, model-theoretic unravellings, and standard translations.

Key technical features include:

• The construction of extended valuations and bisimulation relations via arithmetically definable sets.
• Careful formalization of finite and $\mathrm{WKL}_0$9-restricted bisimulations and their transfer properties.
• A highly detailed stepwise unravelling argument ensuring that properties of the modal fragment can be captured at low logical strength without appeal to full deductive closure.

Numerical results: The equivalence of the Hennessy–Milner theorem to $\mathrm{PRA}$0 confirms that image-finite models alone do not suffice to reduce the logical content to weak base systems. For the van Benthem characterization, the semantic form's provability in $\mathrm{PRA}$1 sharply separates it from the behavior in the intuitionistic and classical predicate logic context, where strong comprehension is known to be required for completeness as demonstrated in earlier reverse mathematics studies.

Implications and Future Directions

The precise logical strength analysis in this work has several implications:

• Clarifies the boundary between modal-definable and more expressive first-order properties in terms of comprehension and separation, making explicit which syntactic and semantic resources are necessary for different forms of modal characterization and transfer results.
• Informs the computability-theoretic content of modal logic in infinite-state systems, indicating limits to effective construction of modal invariants and behaviors strictly due to set-existence constraints.
• Provides a template for further reverse mathematics investigations of modal and other non-classical logics, including intuitionistic modal logic, generalized correspondence results, and the impact of combinatorial principles like $\mathrm{PRA}$2 or $\mathrm{PRA}$3 on logical expressiveness and completeness.
• Demonstrates that strong classical theorems in model theory may have surprisingly weak provenances in modal logic settings, which may not generalize to predicate modal logics or settings with greater expressive power.
• Suggests further connections to algorithmic metatheorems, such as characterizations of bisimulation-invariant PTIME logic, descriptive complexity, and the automation of modal correspondence theory in bounded arithmetic environments.

Pragmatically, the results on proof-theoretic strength provide tools for calibrating the consistency strength of modal logics and their completeness theorems, which can be applied in metamathematical studies of nonstandard models, computable model theory, and the extraction of computable invariants in finite-state verification.

Conclusion

The paper offers a rigorous and granular reverse mathematics analysis of core theorems relating modal logic to bisimulation and their manifestation in second-order arithmetic. The chief contributions are: the equivalence of the Hennessy–Milner theorem to $\mathrm{PRA}$4, the proof of the semantic van Benthem characterization theorem in $\mathrm{PRA}$5, the formalization of the syntactic form in $\mathrm{PRA}$6, and the hybrid form's equivalence to a canonical weak completeness principle. These results stratify the logical landscape of modal logic in arithmetic contexts and provide a foundation for further investigation of the intersection of modal definability, computability, and reverse mathematics.