Controlled Sequent Calculi

• Controlled sequent calculi are sequent-based logical systems that internalize proof search restrictions through focusing, polarity management, and rule constraints.
• They utilize methodologies such as structural rule restrictions, control operators, and lazy evaluation to achieve confluence, decidability, and strong normalization.
• Their modular design supports various logics—from classical to arithmetic—facilitating efficient automated theorem proving and formal verification.

Controlled sequent calculi are sequent-based logical systems in which the inference rules internalize restrictions or strategies—focusing disciplines, external calls, structural rule bans, or evaluation scheduling—that constrain proof search, induce confluence, interface with computational semantics, or enforce logical safety. These calculi serve as modular frameworks for classical, intuitionistic, lattice-negation, or arithmetic logics, and appear prominently in polarised, focused, or restricted sequent frameworks designed for both foundational proof theory and efficient automated reasoning.

1. Types of Control Mechanisms

Control in sequent calculi manifests through an overview of rule design, operational semantics, and proof-search discipline:

• Focusing: Division of rules/connectives by polarity enforces strict alternation of invertible (asynchronous) and non-invertible (synchronous) phases, sharply reducing nondeterminism in proof search. The LKF system and its descendants establish focused sequents and phase discipline as intrinsic control mechanisms (Curien et al., 2010, Farooque et al., 2012, Farooque et al., 2013).
• Structural Rule Restriction: Ban or admissibility-only of contraction, weakening, or cut rules, exemplified in G3SDM and G3DM for semi-De Morgan and De Morgan algebras, yields calculi with the “subformula” property, guaranteeing decidability and interpolation (Ma et al., 2016).
• Control Operators and Continuation Bindings: Calculi such as those arising from the duality-of-computation paradigm (System L, λμ-calculus) use term-level binders (μ, ˜μ) to encode control in both computation and proof search. In dLPA^ω, explicit delimited continuations (μ/#) are mandated to maintain type safety in the presence of dependent types (Curien et al., 2010, Miquey, 2018).
• External Procedure Calls: Extensions like LK(T), LKp(T), LK+(T) integrate decision procedure calls into synchronous phases. The calculus delegates ground reasoning to theory solvers (e.g., SMT, DPLL(T)), ensuring sharp control over branch closure based on domain-specific properties (Farooque et al., 2012, Farooque et al., 2013).
• Lazy Evaluation and Store Discipline: In dependent-type calculi for classical arithmetic, control is implemented via store-based environments and syntactic restrictions (nef fragments and dependency lists), delaying substitutions and co-recursive expansions until safe (Miquey, 2018).

2. Focusing, Polarisation, and Syntax

Focusing partitions connectives and literals by polarity, with phases alternating between invertible decompositions and synchronous choices. In LKF, formulas are declared positive or negative, and the phase structure is encoded in sequents—unfocused (⇑) for up-phases and focused (⇓) for down-phases, with rules strictly enforcing decompositions only appropriate to the focus phase (Farooque et al., 2012). System L further refines this with a term language partitioned into commands, values, contexts, and binders; focusing is not merely a search strategy but is reflected in the operational and typing semantics of each syntax class (Curien et al., 2010).

The LKp(T) and LK+(T) systems introduce additional control by allowing polarisation of literals during proof search ("on the fly"), modulating both the ability of Init/call rules to close branches and the overall provability dynamics under external theory reasoning (Farooque et al., 2013). The distinction between positive/negative connectives and their syntactic encodings yields flexibility in search while preserving modularity with respect to background theories.

3. Structural Rule Restriction and Decidability

Calculi such as G3SDM and G3DM eliminate primitive structural rules (weakening, contraction, cut); these rules are only admissible by induction, not primitive. This restriction enforces a weight/subformula property: every rule either builds or decomposes exactly one logical connective, precluding unbounded branching and ensuring decidability by explicit induction on the complexity of sequents (Ma et al., 2016). In these systems, admissibility of structural rules, cut elimination, and interpolation are established via semantic-free Gentzen-style induction, and cross-embeddings into intuitionistic and classical calculi illuminate the proof-theoretic relationships between algebraic and classical logics.

4. Control Operators and Dependency Management

The duality-of-computation framework provides calculi where control operators (μ, ˜μ) are internalized into the term language as binders equating the act of focus selection with explicit continuation handling. In dLPA^ω, the typing and operational discipline requires that dependent elimination is restricted to negative-elimination-free (nef) arguments, propagating postponed substitutions through explicit lists of dependencies. Delimited and co-delimited continuations (μ/# and their duals) are necessary to maintain type safety, consistency, and normalization in the presence of classical control, coinductive streams, and lazy evaluation (Curien et al., 2010, Miquey, 2018).

Normalization (cut elimination) proofs become highly nontrivial in these settings, necessitating small-step decompositions, stores/environments preserving laziness, and classical realizability techniques to guarantee consistency and strong normalization, especially when integrating choice principles or infinite/coinductive objects.

5. External Theory Integration and Procedure Calls

Sequent calculi LK(T), LKp(T), and LK+(T) embed procedure calls as primitive inference rules. The key rules—Call and Init—discharge branches by delegating sets of ground literals to an external procedure T, which returns unsatisfiability judgments. Control here is attained by alternating pure logical decompositions and theory calls, modularizing the calculus with respect to the properties of T (weakening, contraction, instantiation, consistency). The discipline mimics SAT/DPLL(T) operational semantics via focused search and explicit literal set management (Farooque et al., 2012, Farooque et al., 2013).

Cut elimination in these settings proceeds by a fine-grained analysis: primitive cuts on literals and formulas are reduced by leveraging the properties of T and the invertibility of asynchronous rules, with completeness and modularity guaranteed under suitable conditions on the external procedure. These mechanisms enable simulation of efficient decision procedures (SAT, SMT) within the sequent framework, precisely controlling search explosion and closure points.

6. Metatheoretic Properties and Embeddings

Controlled sequent calculi are constructed to support cut elimination, strong normalization, consistency, interpolation, and decidability, frequently via restricted admissibility or orthogonality arguments. The System L variant employs orthogonality of patterns and counterpatterns to achieve confluence. G3SDM, G3DM, and their embedding theorems (Gödel–Gentzen, Glivenko) illustrate cross-system compatibility and translation properties, ensuring that restricted calculi retain the deductive completeness of their unrestricted counterparts within their respective logical universes (Ma et al., 2016, Curien et al., 2010).

In dependent-type calculi such as dLPA^ω, normalization and soundness are established via classical realizability with Krivine's environment method, explicit pole construction, and adequacy lemmas. These properties are crucial in justifying the safe integration of choice principles, coinductive streams, and delayed evaluation into logical frameworks for arithmetic and analysis (Miquey, 2018).

7. Applications and Significance

Controlled sequent calculi underpin high-performance automated reasoning (SAT/SMT solvers), proof assistants integrating computational semantics, and foundational studies of duality, computation, and algebraic logics. By internalizing control mechanisms, these calculi bridge operational and deductive semantics, providing tractable frameworks for proof search, modular integration of domain-specific procedures, and safe embedding of complex type-theoretic features. The modular nature of control—focusing, external calls, structural restriction, syntactic discipline—enables robust extensibility, efficient reasoning, and faithful logical correspondences within and across logical systems.