Uniform Local Tabularity in Logic

• Uniform local tabularity is a stratified property ensuring every formula can be equivalently rewritten with a bounded implication depth, integrating local tabularity with finite algebraic generation.
• Its algebraic counterpart in Heyting algebras characterizes n-uniform finite properties, enabling explicit axiomatizations and bounded-depth term generation.
• This property distinguishes systems such as wPL from other modal logics, critically impacting decidability and structural insights in non-classical logic research.

Uniform local tabularity is a stratified property in intermediate and modal logics encoding the requirement that all formulas can be rewritten, up to logical equivalence, as formulas of bounded implication depth. This notion generalizes local tabularity and yields robust algebraic and model-theoretic consequences. Its formalization in intuitionistic logic involves both combinatorial depth concepts and algebraic finiteness through Heyting algebras and plays a central role in the structure theory of superintuitionistic and pretransitive modal logics (Almeida, 16 Jan 2026, Shapirovsky, 2018).

1. Formal Definition and Depth Stratification

Let $L = (\wedge, \vee, \rightarrow, \bot, \top)$ be the usual language of intuitionistic propositional calculus (IPC). Each formula $\varphi$ is assigned an implication depth $d(\varphi)$ recursively:

• $d(p) = d(\bot) = d(\top) = 0$ for variables and constants,
• $$d(\varphi \wedge \psi) = d(\varphi \vee \psi) = \max(d(\varphi), d(\psi))$$,
• $$d(\varphi \rightarrow \psi) = \max(d(\varphi), d(\psi)) + 1$$.

An intermediate logic $L \supseteq \mathsf{IPC}$ is locally tabular if for every finite set of propositional variables, there are only finitely many $L$-equivalence classes of formulas. It is $n$-uniform (of uniform depth $n$) if for every formula $\varphi$ there is a logically equivalent $\psi$ with $d(\psi) \leq n$ such that $L \vdash \varphi \leftrightarrow \psi$. The class of $n$-uniform logics is denoted $\mathrm{ULTab}_n$, and a logic is uniformly locally tabular if it is $n$-uniform for some $n$.

2. Algebraic Counterpart: Heyting Algebras and Uniformity

Superintuitionistic logics correspond to varieties of Heyting algebras. A Heyting algebra $H$ is locally finite if every finitely generated subalgebra is finite, mirroring local tabularity in the logic. The $n$-uniformly finite property is defined such that every finitely generated subalgebra $A \subseteq H$ is generated by its generators under all terms of implication-depth $\leq n$. Thus, $L$ is $n$-uniform iff $\operatorname{Alg}(L)$ is $n$-uniformly finite.

Let $\mathrm{HA}_{\operatorname{lf}}^n$ denote the class of all $n$-uniformly finite Heyting algebras. The algebraic interpretation of uniform local tabularity asserts that formulas collapse to those of bounded depth, and subalgebras are generated by bounded-depth terms.

3. Structural Varieties and Explicit Axiomatizations

Unlike locally finite Heyting algebras (not a variety due to breakdown under coproducts), for each $n \in \mathbb{N}$ the class $\mathrm{HA}_{\operatorname{lf}}^n$ is closed under subalgebras, homomorphic images, and arbitrary products, thus forming a variety (Almeida, 16 Jan 2026). For each $n$, there exists a least intermediate logic $\mathrm{Unif}_n$ characterized by these algebras.

Axiomatization for small $n$ proceeds explicitly:

• $n=0$: The sole 0-uniform logic is classical propositional calculus (CPC), since all formulas are equivalent to Boolean combinations of variables.
• $n=1$: The only 1-uniform intermediate logics are CPC and two-element chain logic Sme (also called Sm).
• $n=2$: The least 2-uniform logic (2Uni) sits strictly between the weak Peirce-law calculus (wPL) and Gödel–Dummett logic (LC). It is axiomatized by IPC plus five Yankov–Fine formulas $\mathcal{I}(Q_i)$, each forbidding a rooted frame $Q_i$, specifically:

$$2Uni = wPL \oplus \mathcal{I}(Q_4) \oplus \mathcal{I}(Q_5)$$

4. Distinction from Local Tabularity: Negative Resolution of Shehtman's Question

While local tabularity has strong ties to Kripke semantics (finite height frames in modal logic, e.g., $\mathsf{S4}$ (Shapirovsky, 2018)), it does not imply uniform local tabularity. The weak Peirce-law calculus (wPL):

$${wPL} := \mathsf{IPC} \oplus \big((q \rightarrow p) \vee ((p \rightarrow q) \rightarrow p) \rightarrow p\big)$$

is locally tabular but fails $n$-uniformity for any finite $n$. For every $n,k$ with $k+1 < n$, finite rooted models $M_{n,k}$ and $N_{n,k-1}$ are constructed such that:

• $M_{n,k}$ and $N_{n,k-1}$ are $k$-bisimilar but not $(k+1)$-bisimilar,
• both validate the axioms of wPL.

By bisimulation-characterization (Corollary 2.7 (Almeida, 16 Jan 2026)), wPL does not admit a uniform depth collapse, refuting Shehtman's conjecture that local tabularity implies uniform local tabularity above IPC. Local tabularity and uniform local tabularity thus diverge below modal logic $\mathsf{S4}$.

5. Pre-Uniform Logics and Decidability

A pre-uniformly locally tabular logic is one that is not uniform itself, but all proper extensions are. The unique example above the Kuznetsov–Gerciu base logic (KG) is Box, generated by stacked cascade frames with a maximal point and axiomatized by:

$$\mathrm{Box} = wPL \oplus bw_2 \oplus \neg(p \vee \neg\neg p)$$

where $bw_2$ bounds the antichain-width to 2. Every extension of Box either acquires finite depth or bounded stack-depth, collapsing uniformity to a finite bound.

For $L \supseteq \mathrm{KG}$, $L$ fails to be pre-uniformly locally tabular if and only if $L \subseteq \mathrm{Box}$. Since Box is finitely axiomatizable, uniform local tabularity is decidable in the interval $[\mathrm{KG},\mathsf{IPC}]$ (Almeida, 16 Jan 2026).

6. Uniform Bounds, Kripke Semantics, and the Modal Analogue

Finite height in the corresponding Kripke models yields strong uniform bounds on the number of inequivalent formulas. For unimodal transitive logics (such as $\mathsf{S4}$), Segerberg–Maksimova's theorem asserts local tabularity iff there is finite height. For a logic of height $h$, with $k$ variables, the canonical frame splits into $h+1$ levels, each carrying at most $2^k$ Boolean types. Hence the total number of $k$-formulas is bounded by $2^{(h+1) 2^k}$ (Shapirovsky, 2018). This provides a uniform bound and a decision procedure for each $L[h]$ whenever $L$ is decidable.

Explicit axiomatic translations (generalized Glivenko’s theorem) link logics of height $h+1$ to those of height $h$. For each variable fragment, there exist depth detectors $B_{i,k}(p_0,\dots,p_{k-1})$, producing the equivalence:

$$L[h+1] \vdash \varphi \iff L \vdash \bigwedge_{i=0}^h \left( \Box^*(\Box^* B_{i,k} \rightarrow B_{i,k}) \right) \rightarrow \varphi$$

and

$$L[h+1] = L \oplus \{ \Box^*(\Box^* B_{i,k} \to B_{i,k}) : i=0,\dots,h \}$$

This approach yields effective uniform bounds for logics with finite height (Shapirovsky, 2018).

Table: Stratification of Uniform Local Tabularity in Logic and Algebra

$n$-Uniform Logic	Heyting-Algebra Variety	Least Logic/Axiomatization
$n=0$	$\mathrm{HA}_{\operatorname{lf}}^0$ (Boolean)	CPC
$n=1$	$\mathrm{HA}_{\operatorname{lf}}^1$	CPC, Sme (Sm)
$n=2$	$\mathrm{HA}_{\operatorname{lf}}^2$	2Uni (explicit ax.)

The classification underscores the categorical structure and explicit axiomatizability for low $n$, with broader uniform bounds available for general modal or intuitionistic depth parameters.

7. Significance and Research Directions

Uniform local tabularity provides a refinement of structural finiteness beyond mere local tabularity and is intimately tied to the algebraic generation and model-theoretic stratification of logical systems. The 2026 analysis by Almeida defines, axiomatizes, and constructs counterexamples and decision procedures for this property, addressing longstanding open problems in intermediate logic (notably Shehtman's question) (Almeida, 16 Jan 2026).

A plausible implication is the extension of uniform tabularity methods to other non-classical logics (e.g., modal, many-valued) and the systematic study of pre-uniformly locally tabular and hybrid logics, where depth bounds interact with frame-theoretic structure. Effective bounds, variety classification, and decision algorithms underpin applications in automated deduction and finite model theory.

Uniform local tabularity thus serves as a focal point in the synthesis of algebraic logic, Kripke semantics, and decidable fragments of non-classical proof theory.