Generalized Sahlqvist–van Benthem Algorithm

• The Generalized Sahlqvist–van Benthem Algorithm is a framework that extends classical Sahlqvist theory to a broad range of logical systems including non-normal, substructural, and hybrid logics.
• It employs an ALBA-based procedure through preprocessing, reduction, and Ackermann elimination to systematically convert complex modal formulas into first-order frame conditions.
• This approach ensures key meta-theoretical properties such as elementarity, canonicity, termination, and strong completeness across varied semantic frameworks.

The Generalized Sahlqvist–van Benthem algorithm encompasses a spectrum of algorithmic correspondence techniques unifying and extending Sahlqvist theory from classical modal logic to a broad array of logical systems, including normal and non-normal modal logics, substructural and distributive systems, lattice expansions, hybrid logics with binders, possibility semantics, and second-order propositional modal logic. Central to these generalized frameworks is the formulation of syntactic criteria for Sahlqvist and inductive formulas or inequalities, and the specification of rule-based algorithms (typically forms of the ALBA procedure) to systematically reduce them to first-order frame correspondents, ensuring elementarity, canonicity, and (when available) strong completeness.

1. Syntactic Foundations: Signed Generation Trees and Generalized Sahlqvist Classes

Generalized Sahlqvist theory departs from the classical modal context by abstracting the notion of polarity, critical branches, and permissible syntactic patterns using the machinery of signed generation trees. Formulas (or inequalities, sequents) are parsed into signed trees where sign propagation tracks the monotonicity or antitonicity (e.g., due to implication, negation, or lattice connectives). The classification of nodes—Skeleton (join/meet friendly), PIA (residuated/adjoint friendly), and their interaction—determines which branches are “excellent” or “good,” and hence identifies the Sahlqvist and inductive fragments.

For instance, in non-distributive logics, the Sahlqvist class is defined by “excellent branches”—paths from leaves critical with respect to an order-type $\epsilon$ that traverse first through (adjoint) PIA nodes, then through Skeleton nodes, with no interleaving. Inductive inequalities permit more general “good” branches, incorporating dependency-order conditions required for well-founded elimination (Conradie et al., 2016, Zhao, 2016). In hybrid logics with binder or sabotage modal logic, the same signed tree paradigm governs the extended Sahlqvist classes, accounting for syntactic roles of hybrid or dynamic modalities (Zhao, 2021, Zhao, 2020). In second-order modal logic, Sahlqvist classes are organized hierarchically by quantifier alternation, enabling algorithmic reduction at each stratum (Zhao, 2021).

2. Algorithmic Workflow: Preprocessing, Reduction, and Ackermann Elimination

The core algorithm, typically an ALBA variant, operates in three main stages:

1. Preprocessing:
   • Uniform variable elimination (for uniformly monotone/antitone props).
   • Distribution/splitting of connectives to regularize critical branches and decompose complex (co)conjunctions into simpler inequalities or sequents.
   • In languages with multiple sorts, translation into sorted modal companions with explicit stability and change-of-variable constraints (Chrysafis et al., 16 Mar 2025, Hartonas, 23 Nov 2025).
2. Reduction:
   • Systematic application of approximation rules (introducing nominals, co-nominals, or stability variables) at positions governed by the syntactic criteria.
   • Residuation/adjunction rules, pushing connectives into premises or side conditions.
   • Additional rewriting rules (e.g., for hybrid, sabotage, or order-theoretic variants), ensuring the transformed system preserves frame-validity.
3. Ackermann-Style Elimination:
   • Once all critical propositional variables (or second-order variables) occur only positively or only negatively, the Ackermann principle can be invoked to eliminate them globally, yielding pure quasi-inequalities or FO-conditions (Conradie et al., 2016, Zhao, 2020, Zhao, 2021).

Upon successful elimination, the remaining pure constraints are interpreted (typically via standard, regular-open, or sorted translations) as first-order correspondents for the original input.

3. Semantic Expansion: Frames, Sorted Structures, and Algebraic Dualities

The generalization of Sahlqvist–van Benthem algorithms relies critically on the dual algebraic or sorted relational structures appropriate to each logic:

• Possibility Semantics:

Employs full or filter-descriptive possibility frames, with regular-open closures serving as base points for nominal interpretation. The BAO structure is replaced by completely multiplicative Boolean algebras of regular-open sets (Zhao, 2016).

• Normal Lattice Expansions:

In LE-logics, the underlying frames are two-sorted, reflecting join-irreducible and meet-irreducible elements, and the expanded tense modal language includes both left/right adjoints for all connectives (Conradie et al., 2016, Chrysafis et al., 16 Mar 2025).

• Substructural and Distribution-Free Logics:

Sorted residuated frames with Galois connections and quasi-operators enable a uniform treatment of logics without distribution or with substructural operations (Chrysafis et al., 16 Mar 2025, Hartonas, 23 Nov 2025).

• Hybrid and Dynamic Modal Logics:

Hybrid frames use nominals, state variables, and satisfaction operators; dynamic modalities require additional bookkeeping to reflect edge-deletion or swap operations (Zhao, 2021, Zhao, 2020).

• Second-Order Modal Logic:

Kripke frames with propositional quantifiers interpreted via variable assignments to all subsets of the domain, and an expanded hybrid language for algorithmic manipulation (Zhao, 2021).

4. Main Theorems: Soundness, Correspondence, Canonicity, and Completeness

The generalized Sahlqvist–van Benthem algorithms establish a suite of meta-theorems for the respective logic:

• Correctness & Equivalence:

Each successful run yields first-order frame conditions equivalent to the original axiom/inequality in all perfect (or admissible) models/frames (Conradie et al., 2016, Zhao, 2016, Chrysafis et al., 16 Mar 2025).

• Termination and Decidability:

The algorithms are shown to terminate on their respective Sahlqvist or inductive fragments by well-foundedness measures (e.g., depth of critical branches, count of prop-occurrences, or complexity of quantifier alternation) (Zhao, 2021, Conradie et al., 2016, Zhao, 2021).

• Canonicity:

Whenever the run is "safe" (pivotal approximation only at maximal positions, rules preserve admissibility), Sahlqvist and inductive formulas are preserved in canonical extensions, ensuring their canonicity over the relevant algebras and frames (Conradie et al., 2016, Zhao, 2016).

• Strong Completeness:

For logics such as Lemmon's E2-E5 or distributive modal logics on impossible worlds, the ALBA-adapted algorithm delivers new proofs of strong completeness for the elementary frame classes characterized by the algorithmic correspondents (Palmigiano et al., 2016).

5. Concrete Instantiations and Worked Examples

Each setting provides explicit worked examples illustrating the key steps. For instance:

• In possibility semantics, the reduction of $\Box p \leq p$ yields the FO-expression $\forall x \forall y [Rxx \to x=y]$ (reflexivity), matching the regular-open semantics (Zhao, 2016).
• In distribution-free modal logic, the contraction axiom $(p \to (p \to q)) \vdash p \to q$ is algorithmically reduced to a first-order condition expressing the frame contraction property (Chrysafis et al., 16 Mar 2025).
• In second-order modal logic, the irreflexivity rule $\forall q[\forall p(p\to\Diamond p \lor q)\to q]$ is mapped to $\forall x\neg Rxx$, and more complex $\Pi_2$-Sahlqvist formulas are handled correspondingly (Zhao, 2021).
• For sabotage modal logic, the formula $\Box p \to \Box^{\cdot} p$ yields the condition $\forall x \forall y[R(x, y) \to y = x]$ (R is the identity), extending algorithmic correspondence to dynamic logic (Zhao, 2020).

6. Scope, Limitations, and Future Developments

These algorithmic correspondences generalize Sahlqvist–van Benthem theory to settings without classical distribution, without Boolean negation, or with higher-order quantification. Key innovations include sorted modal companions, Galois-stable/unstable variables, and rule systems for non-normal, non-distributive, or dynamic connectives.

Nonetheless, not all axioms (especially outside the inductive or Sahlqvist classes) admit algorithmic reduction. Some second-order Sahlqvist formulas, notably, fail to be canonical, reflecting the complexity introduced by quantifier alternation (Zhao, 2021).

Current research directions include the extension to substructural Gentzen systems (via sorted translation (Hartonas, 23 Nov 2025, Chrysafis et al., 16 Mar 2025)), completeness and canonicity via translation for broader classes, and the integration of such algorithmic reductions into proof-theoretic frameworks for further non-classical logics. The relationship to Kracht-style definability conditions and the formulation of Goldblatt–Thomason theorems in non-classical or dynamic frames is an open and active area of investigation (Zhao, 2020).