Labelled Sequent Calculus

• Labelled sequent calculus is a proof-theoretic framework that internalizes relational semantics by annotating formulas with explicit world labels and relational atoms.
• The approach enables modular rule synthesis and systematic handling of modal, substructural, and non-classical logics, often yielding cut-free and invertible systems.
• It supports effective countermodel construction and automated proof search, with applications ranging from separation logic to cyclic proof systems in contemporary metatheory.

A labelled sequent calculus is a proof-theoretic framework that internalizes the relational semantics of non-classical logics—especially modal, intuitionistic, substructural, and other extended logics—by explicitly annotating formulas in the sequent with world-labels and, where necessary, explicit relational atoms. This enrichment with labels allows the system to capture semantic properties (such as accessibility, resource composition, or agent choice) directly in the calculus, yielding calculi that are modular, often cut-free, and well-behaved in terms of invertibility and admissibility of structural rules. The approach, which has developed into a central methodology in contemporary proof theory for modal and substructural logics, provides a bridge between semantic models and syntactic proof systems, supports effective countermodel construction, and enables systematic extension to a wide range of logics via general semantic-to-syntactic correspondences.

1. Syntax and Core Principles

A labelled sequent consists of multisets of labelled formulas and, potentially, relational atoms. Formally, these sequents can be expressed as: $R, \Gamma \vdash \Delta$ where

• $\Gamma, \Delta$ are multisets (or sequences) of labelled formulas like $x:A$, indicating that formula $A$ is asserted at world (label) $x$,
• $R$ is a multiset of relational atoms, e.g., $x R y$, coding the accessibility or structural relation between labels.

Labels are drawn from a countable set, serving as placeholders for worlds, resources, states, objects, etc., depending on the logic. Relational atoms can encode accessibility (modal logic, e.g., $x R y$), resource combination (separation logic: $(x, y z)$), agent actions (STIT: $R_{[i]}xy$), evidence relations (justification logic: $wRv$), or algebraic relationships (lattice-based logics: $j \leq A$).

The inference rules are generally obtained by internalizing the semantic clause for each connective:

• Modal rules introduce or manipulate relational atoms directly, as in:

$$\frac{R, xRy, \Gamma \vdash y:A, \Delta}{R, \Gamma \vdash x:\Box A, \Delta}$$

• Structural rules correspond to frame or algebraic properties (e.g., transitivity: $xRy, yRz \to xRz$) and can be made explicit as rules manipulating $R$.
• Additional structural or proof-theoretic properties (e.g., weakening, contraction, cut) are usually designed to be admissible or even height-preserving admissible, paralleling the setup in Gentzen-style systems.

2. Semantic Internalization and Modularity

The haLLMark of the labelled sequent approach is its tight correspondence with relational semantics. Each inference rule arises by "translating" a semantic condition into a syntactic operation:

• The modal operator $\Box$ is governed by rules that mirror the condition "A holds at all accessible worlds," justifying the explicit use of labels and accessibility atoms.
• In substructural and separation logics, ternary or higher-arity relations represent resource composition (e.g., $(x, y z)$ reflects $z = x \circ y$), supporting the direct encoding of composition, cancellation, and resource splitting.
• For quantified modal logics and logics with non-rigid designators or evidence terms, labels track variable assignment, domain membership, and term denotation explicitly within sequents.

Modularity is realized by synthesizing additional (often structural) rules from semantic "frame axioms." For example, reflexivity, symmetry, partial-determinism, cancellativity, indivisible unit, disjointness, splittability, and more are all first-order conditions on the frame or algebra, each corresponding to a proof-theoretic rule in the calculus (Hóu et al., 2017, Hou et al., 2013, Hou et al., 2013). This modularity enables labelled sequents to adapt to a wide spectrum of logics systematically by plugging in the required semantic rules.

3. Structural Properties and Proof Theory

Labelled sequent calculi are distinguished by their proof-theoretic properties:

• Cut-elimination: For large classes of logics, labelled systems admit a syntactic cut-elimination procedure, often by adapting structural induction and rank-based decompositions, leading to subformula or labelled subformula properties (Ghari, 2014, Hou et al., 2013, Hóu et al., 2017).
• Invertibility of rules: Most rules are invertible, many in a height-preserving fashion, facilitating proof search and meta-theoretic reasoning.
• Admissibility of weakening and contraction: These are typically height-preserving admissible, with contraction often handled on labelled formulas or relational atoms.
• Termination of proof search: Termination can be guaranteed by bounding the number of labels and subformulas, by suitable complexity measures on sequents (e.g., via labelled subterms, constraint system size, or other proof metrics).

These properties are crucial: they underpin decidability and provide the basis for algorithmic proof search, countermodel extraction, and quantification over formal properties such as interpolation or definability (Kuznets, 2016, Hóu et al., 2017, Hou et al., 2013).

4. Comparison with Alternative Formalisms

Labelled sequents are situated among a range of proof systems:

• Nested sequents form a "middle ground" syntactic formalism that encodes modal structure as tree-like or nested compositions of sequents (e.g., $A_1, [A_2]$), remaining inside the modal language, strictly enforcing the subformula property, and often admitting tight syntactic cut-elimination and proof search termination (1004.1845).
• Display calculi generalize the idea of manipulating structural connectives, allowing any substructure to be "displayed" and operated upon. However, they often introduce more complex structural bureaucracy.
• Standard sequent calculi (without labels) typically lack modularity for modal and substructural logics, and frequently suffer from ineliminable cuts or limitations on automated reasoning.

Labelled sequents stand out for their semantic transparency (as they internalize the Kripke- or algebraic semantics explicitly), their modular design (derive rules for new logical axioms directly from the semantic frame conditions), and their proof-theoretic robustness (invertibility, cut-elimination, admissibility) (1101.5445, Lyon, 2021, Pimentel, 2018).

The principal trade-off is syntactic: labelled sequents often violate the strict subformula property—since relational atoms and labels can proliferate and are not strictly subformulas—leading to more "bureaucratic" proofs. This can impact efficient proof search, an issue addressed by localized structural rule application, free variable calculi, and constraint-based approaches (Hou et al., 2013).

5. Applications and Impact in Logical and Computational Metatheory

Labelled sequent calculi have become the backbone of proof-theoretic investigations and automation in a wide range of logics:

• Modal and temporal logics: The approach applies to normal and non-normal modal logics, temporal logics, STIT logics of agency, and epistemic logics (including dynamic variants like PAL) (Berkel et al., 2019, Wu et al., 2022).
• Justification logic: Labelled calculi for justification logics and modal-justification hybrids internalize evidence semantics, leading to decision procedures, countermodel construction, and correspondence results (Ghari, 2014).
• Separation logic: Modular labelled calculi for abstract, Boolean, and propositional separation logics support extensions for properties like cancellativity, disjointness, splittability, and cross-split (Hóu et al., 2017, Hou et al., 2013, Hou et al., 2013).
• Bi-intuitionistic logic: Direct internalization of Kripke semantics and analysis of standard, nested, and labelled calculi equivalence (1101.5445).
• Lattice-based modal logics, rough logics, and algebraic systems: The methodology extends (via order-theoretic semantics and analytic structural rules) to logics where canonical extensions, nominals, and conominals play a central role in the proof calculus, and where modular extension is achieved seamlessly (Berg et al., 18 Jan 2024, Berg et al., 18 Jan 2024).
• Interpolation theory and correspondence: Labelled calculi provide the structure required to prove the interpolation theorems by constructing interpolants directly from cut-free derivations and mapping frame correspondents to proof rules (Kuznets, 2016, Lyon, 2021).
• Non-wellfounded and cyclic proof systems: For logics with fixed points, iteration, or provability modalities (as in bimodal provability logic, PDL, etc.), labelled systems support cyclic or infinitary proof theory, essential for internalizing second-order frame conditions (e.g., converse wellfoundedness in provability logic) and extracting decision procedures with countermodel construction (Becker, 17 Jun 2025, Docherty et al., 2019).

The explicit structure of labelled sequents provides the grounds for effective proof search, metatheoretic investigations, and computational analysis. Refinement methods—transforming labelled systems into nested calculi—reap the semantic transparency of the former and the tight subformula property of the latter (Lyon, 2021, Lyon, 2019).

6. Methodological Extensions and Future Directions

Recent advances include systematic "structural refinement," whereby a labelled calculus derived from semantics is algorithmically transformed into a more economical (usually nested) system, retaining invertibility, cut-elimination, and proof search suitability while reducing syntactic redundancy (Lyon, 2021, Pimentel, 2018, Lyon, 2019). The methodology applies across grammar logics, first-order intuitionistic logics, and logics of agency.

The integration of cyclic proof theory with labelled sequents marks another development, enabling the tractable treatment of provability, fixed-point, and temporal logics with decision procedures and model extraction (Becker, 17 Jun 2025, Docherty et al., 2019).

Algebraic and lattice-based extensions, leveraging canonical extension semantics, have recently led to the introduction of labelled calculi based on inequalities with nominals/cononminals, switch/adjunction rules, and cut-elimination via display and twin properties (Berg et al., 18 Jan 2024). These approaches demonstrate the versatility and extensibility of labelled sequent calculi as a general method in analytic proof theory for non-classical logics.

7. Summary Table: Core Features of Labelled Sequent Calculus

Feature	Description	Contrast
Semantic Internalization	Explicit relational atoms encode frame conditions	Display/hypersequents use structural connectives
Modularity	Rules synthesized from frame axioms or algebraic conditions	Standard sequents require ad hoc rules
Proof Theory	Cut-elimination, invertibility, localization of structure	Gentzen systems often lack modular cut-elimination
Automation	Supports proof search, countermodel extraction	Display/nested systems trade between modularity and subformula property
Syntactic Overhead	Proliferation of labels and atoms, possible bureaucracy	Nested sequents avoid extraneous structure

Labelled sequent calculi thus form a foundational framework in structural proof theory, supporting modularization, extension, and analytic investigation across a wide class of modal, substructural, non-distributive, and quantified logics, while adapting—via refinement and constraint handling—to match both semantic transparency and syntactic economy.