Kuznetsov-Muravitsky Isomorphism in Modal Logic

• The paper establishes that mutually inverse, meet- and join-preserving translation maps create a complete lattice isomorphism between normal extensions of KM and GL.
• Kuznetsov-Muravitsky Isomorphism is defined by translating intuitionistic modal formulas into classical ones, ensuring preservation of properties like Kripke completeness and the finite model property.
• The framework employs KM algebras, duality between modal Heyting algebras and Esakia spaces, and pre-filtration techniques to underpin robust algebraic and rule-rewriting strategies.

The Kuznetsov-Muravitsky Isomorphism establishes a precise lattice-theoretic and algebraic correspondence between normal extensions of intuitionistic modal logic $\mathsf{KM}$ and those of classical provability logic $\mathsf{GL}$. At its core, the isomorphism reveals that the landscape of strong modal logics over intuitionistic and classical frameworks is tightly connected through specific translation procedures and algebraic enrichments. Its significance is further amplified by canonical rule systems, duality theory for modal (Heyting) algebras, and its ability to preserve key model-theoretic properties such as Kripke completeness and the finite model property (Bezhanishvili et al., 13 Nov 2025, Jibladze et al., 22 May 2024).

1. Precisely Formulated Isomorphism and Companion Translations

The primary result, the Kuznetsov-Muravitsky Isomorphism, asserts the existence of mutually inverse, meet- and join-preserving maps

$$\sigma\colon \mathsf{NExt}(\mathsf{KM}) \longrightarrow \mathsf{NExt}(\mathsf{GL}), \qquad \rho\colon \mathsf{NExt}(\mathsf{GL}) \longrightarrow \mathsf{NExt}(\mathsf{KM})$$

that yield a complete lattice isomorphism between the normal extension lattices of $\mathsf{KM}$ and $\mathsf{GL}$: $$\sigma = \rho^{-1}: \mathsf{NExt}(\mathsf{KM}) \cong \mathsf{NExt}(\mathsf{GL}).$$ In concrete terms, the translation $T$ maps each intuitionistic modal formula to a classical one by sending $\boxtimes$ to $\square$ and intuitionistic implication to a composed modal operator $\square^+$. The maps

$$L \mapsto \mathrm{Th}\{\,T(\varphi): \varphi\in L\,\}, \qquad M \mapsto \{\varphi: T(\varphi)\in M\}$$

establish the explicit isomorphism at the level of normal logic extensions (Bezhanishvili et al., 13 Nov 2025).

2. Algebraic Foundations: Kuznetsov-Muravitsky Algebras

A Kuznetsov-Muravitsky (KM) algebra consists of a Heyting algebra $(H, \wedge, \vee, \to, 0, 1)$ equipped with an additional unary operation $A(-)$ subject to:

• (KM1) $x \leq A(x)$,
• (KM2) $A(x)\rightarrow x = x$,
• (KM3) $$A(x) \wedge ((y \rightarrow x) \rightarrow x) = ((A(x) \rightarrow y) \rightarrow (A(x) \rightarrow x))$$, for all $x, y \in H$. Equivalently, for every $a\in H$, the filter $D_a(H) = \{ d\in H \mid a\leq d,\, d\to a = a \}$ is principal, and $A(a)$ is its least element (Jibladze et al., 22 May 2024). These axioms characterize the enrichment of Heyting algebras for intuitionistic modal logics.

The one-step enrichment $H \to H(A(a))$, defined for any Heyting algebra $H$ and $a\in H$, produces a new algebra containing a canonical element $A(f(a))$ and a homomorphic image $f[H]$ of $H$ that is isomorphic to $H$. Repeated application of the enrichment across all elements yields a KM-algebra $HKM$ into which $H$ embeds, and $HKM$ lies in the variety generated by $H$ (Jibladze et al., 22 May 2024).

3. Pre-filtration and Pre-stable Canonical Rules

Pre-filtration provides an algebraic mechanism for extracting finite models and rules from possibly infinite structures. Given a fronton algebra $\mathfrak H$ (algebraic semantics for $\mathsf{KM}$), a valuation $V$, and a finite subformula-closed set $\Theta$, a pre-filtration is a finite fronton $\mathfrak K$ (with associated valuation $V'$) such that:

• The $\{\wedge, \vee\}$-reduct of $\mathfrak K$ is isomorphic to the finite distributive sublattice generated by $V[\Theta]$,
• $V'(p)=V(p)$ for $p\in \Theta$,
• The inclusion $i: \mathfrak K \to \mathfrak H$ is a pre-stable embedding for implications and modalities on $\Theta$.

Associated with such filtrations are pre-stable canonical rules. Given a finite fronton $\mathfrak A$ and domains $D^\to\subseteq A\times A$, $D^{\boxtimes}\subseteq A$, the canonical rule encodes the algebraic behavior of $\mathfrak A$ and its modal operations. These are central in deriving the finite basis of the logic and in providing the means to translate between intuitionistic and classical rule systems (Bezhanishvili et al., 13 Nov 2025).

4. Duality Between Modal Heyting Algebras and Order-topological Spaces

The duality underpinning the isomorphism generalizes Stone duality to modal-logical settings:

• A frontal Heyting algebra (for $\mathsf{mHC}$ and $\mathsf{KM}$) is dual to a modalized Esakia space: a Stone space with relations $\leq,\sqsubset$ satisfying specific continuity and clone-Esakia conditions.
• Normal extensions of $\mathsf{K4}$ or $\mathsf{GL}$ correspond via duality to (possibly universal) classes of $\mathsf{K4}$- or Magari-algebras and are dual to modal spaces (Stone spaces with a single binary relation $R$).

Homomorphisms between algebras correspond to continuous bounded morphisms between spaces. The geometric back-and-forth (bounded-domain) conditions translate directly to the boundedness properties of pre-stable embeddings and surjections, ensuring the duality is fully categorical (Bezhanishvili et al., 13 Nov 2025).

5. Proof Strategy and the Role of Rule-rewriting

The proof of the isomorphism proceeds through several key steps:

• Rule-rewriting: Any rule in the $\mathsf{KM}$ language is equivalent, over $\mathsf{KM}$, to finitely many pre-stable canonical rules that arise from finite frontons via pre-filtration.
• Duality and refutation: By Stone–Esakia duality, refutation of a canonical rule in a fronton algebra corresponds to the existence of a surjective bounded morphism onto its dual, respecting the necessary back-and-forth constraints.
• Monomodal Companion and Translation: Translation $T$ ensures that refutability of intuitionistic rules corresponds exactly to refutability of their classical companions, due to the interpretation of implication as the composed modal operator $\square^+$.
• Skeletal Generation: Every universal class of $\mathsf{K4.Grz}$–algebras is generated by algebras of the form $\sigma H$, where $H$ is a frontal Heyting algebra. This structural step guarantees that every extension of $\mathsf{K4.Grz}$ arises via intersection with the image of $\sigma$.
• Assembly: The embedding provided by $L\mapsto\mathrm{Th}\{T(\varphi) : \varphi\in L\}$ is complete at the lattice level, and the argument via inverse translation ensures the correspondence is indeed an isomorphism (Bezhanishvili et al., 13 Nov 2025).

6. Preservation Properties and Universal Consequences

The isomorphism preserves important logical and model-theoretic properties, as well as providing algebraic insights:

Property	Preserved under Isomorphism (Yes/No)	Notes
Kripke completeness	Yes	$L$ is Kripke complete $\Leftrightarrow$ $\sigma L$ is complete for transitive frames
Finite model property (FMP)	Yes	FMP over intuitionistic frames maps to FMP over finite transitive classical frames
Conservative extension	Yes	Each $\mathsf{KM}$ extension is a conservative extension of its superintuitionistic base

Additionally, every Heyting algebra embeds into a KM-algebra generating the same variety, yielding a “canonical and functorial” enrichment and underpinning the statement that every variety of Heyting algebras is generated by reducts of KM-algebras (Jibladze et al., 22 May 2024). This relationship is central for universal algebraic and logical classification.

7. Central Lemmas and Theorems

The theoretical framework supporting the isomorphism is built on the following results:

• Pre-stable duality: Homomorphisms and their duals correspond precisely under the bounded-domain/back-and-forth paradigm.
• Rule-rewriting theorem: Every single-conclusion $\mathsf{KM}$ rule is equivalent to a finite pre-stable canonical rule set.
• Translation and refutation lemma: For classicizable pre-stable canonical rules, refutability is preserved under the companion translation $T$.
• Skeletal generation theorem: Every universal class of $\mathsf{K4.Grz}$-algebras is generated by images of frontal Heyting algebras.

Together these show that the companion translations $\sigma$ and $\rho$ are mutual inverses and that the isomorphism is robust across both logical and algebraic formalisms (Bezhanishvili et al., 13 Nov 2025).