Local Tabularity: Logic & ML Perspectives

• Local tabularity is a property ensuring finiteness of equivalence classes in finite-variable logical fragments, crucial in modal and superintuitionistic logics.
• It is characterized by algebraic equivalence with locally finite Heyting algebras, finite model properties, and bounded frame heights in various logical systems.
• Recent research extends the term to ML, denoting instance-level feature attribution and conditional independence preservation, distinct from its classical logical meaning.

Local tabularity is a finiteness property that, in its classical logical sense, requires each finite-variable fragment of a logic to contain only finitely many equivalence classes of formulas. In superintuitionistic logic this is stated as: for every finite set of propositional variables $\overline{p} = \{p_1,\dots,p_n\}$, there are only finitely many $L$-provable equivalence classes of formulas over $\overline{p}$; algebraically, this is equivalent to local finiteness of the associated class of Heyting algebras. Recent work has also used the same expression in tabular machine learning for target-local or instance-local structure, especially conditional-independence preservation and feature-level attribution, but that usage is distinct from the established proof-theoretic and algebraic notion (Almeida, 16 Jan 2026, Jiang et al., 15 Sep 2025).

1. Core definitions and algebraic semantics

In modal, intermediate, and related algebraic logics, local tabularity is defined by finiteness of finite-variable fragments. A normal modal logic $L$ is locally tabular if, for every finite $k$, there are only finitely many pairwise nonequivalent formulas in $k$ variables. Equivalently, the $k$-generated free Lindenbaum algebra of $L$ is finite for every $k$, and the corresponding algebraic variety is locally finite. The same pattern appears in superintuitionistic and bi-intermediate settings: local tabularity is equivalent to local finiteness of the associated Heyting or bi-Heyting algebraic semantics (Shapirovsky, 2022, Almeida, 16 Jan 2026, Martins et al., 2024).

This algebraic equivalence is central because it translates a syntactic finiteness condition into a structural one. For a superintuitionistic logic $L \supseteq \mathsf{IPC}$, local tabularity is equivalent to $L$0 being locally finite. For extensions of bi-$L$1, the same condition is stated as finiteness of finitely generated algebras in the corresponding variety. In normal modal logic, local tabularity also carries standard consequences: it implies the finite model property, every extension of a locally tabular logic is locally tabular, and finitely axiomatizable extensions are decidable (Shapirovsky, 2022, Martins et al., 2024).

The notion is often contrasted with weaker finiteness conditions such as $L$2-tabularity or 1-finiteness. Several of the cited works emphasize that these weaker notions do not always collapse to local tabularity. In particular, one paper constructs a modal logic that is 1-tabular but not locally tabular, disproving the conjecture that 1-tabularity and local tabularity always coincide (Shapirovsky, 2018).

2. Finite height, pretransitivity, and cluster structure

A major line of research studies when local tabularity can be characterized by frame-theoretic height restrictions. The classical benchmark is the Segerberg–Maksimova criterion: for unimodal transitive logics, local tabularity is equivalent to finite height. Height-bounding formulas are typically written using

$L$3

or closely related variants, and they restrict Kripke frame height in the transitive setting (Shapirovsky, 2018, Shapirovsky, 22 Sep 2025).

Setting	Finite height and local tabularity
Transitive unimodal logics	Finite height is necessary and sufficient
Intermediate logics	Finite height is sufficient but not necessary
Non-transitive unimodal logics	Finite height is necessary but not sufficient
Polymodal logics	Finite height is necessary but not sufficient

Beyond the transitive unimodal case, finite height alone does not settle the question. For non-transitive unimodal and polymodal logics, locally tabular logics are pretransitive and of finite height, but there exist pretransitive logics of finite height that are not locally tabular. One formulation of pretransitivity uses

$L$4

The point is that bounded global height must be supplemented by local constraints on clusters or fragments (Shapirovsky, 22 Sep 2025).

A useful generalization is the cluster criterion. One paper states that the logic of a class $L$5 of Kripke frames is locally tabular iff $L$6 is of finite skeleton height and the logic of its clusters is locally tabular. The same work connects local tabularity to finite modal depth: if a class of frames has finite height and its clusters have finite modal depth $L$7, then

$L$8

where $L$9 is the pretransitivity index and $\overline{p}$0 is the height. This places local tabularity within a broader hierarchy of bounded-complexity conditions (Shapirovsky, 22 Sep 2025).

The finite-height theme also enters Glivenko-type translations. For pretransitive logics, higher finite-height fragments $\overline{p}$1 admit translations into the base logic $\overline{p}$2, and these reductions depend on $\overline{p}$3-tabularity assumptions for finite-height fragments. This connects local tabularity to transfer principles between base logics and their bounded-height extensions (Shapirovsky, 2018).

3. Uniform local tabularity

Uniform local tabularity strengthens local tabularity by requiring a global bound on implication depth. A superintuitionistic logic $\overline{p}$4 is uniformly locally tabular if there exists $\overline{p}$5 such that every formula is $\overline{p}$6-equivalent to a formula of implication depth $\overline{p}$7. Algebraically, $\overline{p}$8 is $\overline{p}$9-uniform iff every finitely generated subalgebra can be generated using only terms of implication depth $L$0. The inclusion chain recorded in the literature is

$L$1

This makes uniform local tabularity a strict strengthening of local tabularity in the intuitionistic setting (Almeida, 16 Jan 2026).

The model-theoretic characterization uses bisimulations: a logic $L$2 is $L$3-uniformly locally tabular iff, for any two models over variables $L$4, if they are $L$5-bisimilar, then they are fully bisimilar. One consequence is that bounded implication depth becomes a robust semantic invariant, not merely a proof-theoretic convenience (Almeida, 16 Jan 2026).

A notable algebraic result is that, for each fixed $L$6, the class of $L$7-uniformly locally finite Heyting algebras forms a variety, whereas the class of all locally finite Heyting algebras does not. Explicit axiomatizations are given for $L$8. The least 2-uniform logic is

$L$9

equivalently

$k$0

with

$k$1

The same paper shows that $k$2 is locally tabular but not uniformly locally tabular, resolving a question of Shehtman, and that $k$3 is pre-uniformly locally tabular above $k$4 (Almeida, 16 Jan 2026).

4. Preservation phenomena, products, and pre-local tabularity

Local tabularity is not merely an intrinsic property of a logic; it is also studied under semantic and algebraic constructions. For polymodal logics, reflexive closure preserves local tabularity: if the logic of the reflexive closure of a class of frames is locally tabular, then the logic of the original class is locally tabular as well. Sum constructions preserve local tabularity when both the logic of the index frames and the logic of the summands are locally tabular. Lexicographic sums do so as well, and suitably axiomatized fusions can preserve local tabularity provided Kripke completeness is available (Shapirovsky, 2022).

Products are subtler. For Kripke complete consistent logics $k$5 and $k$6, local tabularity of both factors is necessary for local tabularity of $k$7, but it is not sufficient. In particular, $k$8 is not locally tabular. A sharp criterion is available when the factors are already locally tabular: $k$9 is locally tabular iff at least one of the frame classes has the bounded cluster property; equivalently, iff the product of skeletons has the reducible path property; equivalently, iff the product logic is 1-finite. The same line of work notes that a locally tabular product may lack the product finite model property (Shapirovsky et al., 2024).

The border case between local tabularity and its failure is pre-local tabularity. A logic $k$0 is pre-locally tabular if it is not locally tabular but every proper normal extension of $k$1 is locally tabular. In normal extensions of products of finite height above $k$2, exactly four pre-locally tabular logics occur: $k$3, $k$4, $k$5, and $k$6, and every non-locally tabular logic in that family is contained in one of them. In the same setting, local tabularity above $k$7 is characterized by the presence of both a bounded height formula $k$8 and a ramified path formula $k$9 (Shapirovsky et al., 25 Jun 2025).

These results correct several possible misconceptions. Local tabularity is not stable under arbitrary products, finite height does not suffice outside the classical transitive unimodal setting, and maximal failures of local tabularity can still admit precise structural classification.

5. Decidability and effective criteria

The decidability of local tabularity is exceptional rather than automatic. One of the clearest positive results concerns finitely axiomatizable extensions of bi-$k$0: if $k$1 is an extension of bi-$k$2, then

$k$3

where $k$4 is the family of finite combs. The logic $k$5 is the unique prelocally tabular extension of bi-$k$6; it is finitely axiomatizable, has the finite model property, and is decidable. This yields a decision procedure for whether a finitely axiomatizable extension of bi-$k$7 is locally tabular (Martins et al., 2024).

A related effective program appears for monadic $k$8-type systems with Casari’s and Barcan-style axioms. For varieties $k$9, local finiteness is characterized semantically by finite depth together with local finiteness of bottom layers viewed as $L$0-algebras, and syntactically by finite depth plus the reducible path property: $L$1 For $L$2, the criterion simplifies: subvarieties are locally finite iff $L$3 for some $L$4. The same work also shows that these methods do not extend straightforwardly beyond depth $L$5, because a translation of the fusion $L$6 into $L$7 preserves and reflects local finiteness (Meadors, 2024).

Taken together, these results show that local tabularity is most tractable when a logic admits either a forbidden-frame characterization, a finite set of Jankov or subframe formulas, or a reducible-path style axiom system. Absent such structure, even strong necessary conditions may fail to give a complete criterion.

6. Distinct recent uses in tabular machine learning

A distinct recent usage of “local tabularity” has appeared in tabular machine learning. In this literature the expression does not refer to finiteness of formula equivalence classes. Instead, it denotes locality aligned with table structure, such as target-local conditional independences or feature-field-level explanations (Jiang et al., 15 Sep 2025, Chaudhary et al., 22 Apr 2026).

In TabStruct, structural fidelity is divided into global structure and local structure. “Local tabularity” denotes preservation of conditional independences specifically related to a chosen prediction target $L$8, and it is measured by local utility $L$9. The paper contrasts this with global utility,

$k$0

and reports that global utility is highly correlated with true global CI when ground-truth structure is known, with rank correlation $k$1. A central conclusion is that strong local utility does not guarantee preservation of the full global structure (Jiang et al., 15 Sep 2025).

In TabSHAP, “local tabularity” refers to instance-level feature attribution for LLM-based tabular classifiers. Inputs are serialized as

$k$2

masking is performed at the level of whole serialized key:value fields, and feature importance is computed by a sampled-coalition Shapley-style estimator using normalized Jensen–Shannon divergence between full-input and masked-input class distributions. The reported benchmarks are Adult Income, with more than 48k samples and 14 features, and Heart Disease, with 1025 samples and 13 features. TabSHAP is reported to achieve sharper deletion-faithfulness drops than random removal and XGBoost+TreeSHAP baselines, with JSD outperforming KL and $k$3 alternatives (Chaudhary et al., 22 Apr 2026).

A plausible implication is that the phrase “local tabularity” now has two technically unrelated but structurally parallel uses: one in logic, where it concerns finite-variable collapse, and one in tabular learning, where it concerns locality at the level of targets, feature fields, or instance-specific predictive structure.