General Relational Semantics Framework

• General relational semantics is a foundational framework that interprets languages, logics, and computational systems through relations rather than functions, emphasizing invariance and expressiveness.
• It employs structured models such as sorted frames, polarities, and residuated operations to achieve uniformity, completeness, and modular reasoning across mathematical and logical systems.
• The framework underpins diverse applications in physics, databases, and program logics by aligning semantic models with operational invariance and algebraic, categorical principles.

A general relational semantics framework provides abstract, rigorous, and unifying principles for interpreting languages, logics, and computational systems in terms of relations—rather than functions or sets—across a wide spectrum of mathematical, logical, and program-theoretic domains. Such frameworks achieve not only expressive power (capturing operational invariance, type parametricity, congruences, etc.) but also enable algebraic, categorical, and modal structure that supports completeness, modularity, and uniformity of reasoning.

1. Motivations and Conceptual Shift

Relational semantics replaces the traditional view of interpreting syntax as functions into a "Platonic" set-theoretic universe with a more flexible and often more invariant approach based on binary (or multiary) relations, equivalence classes, or Galois connections. This shift is motivated by the following principles and technical requirements:

• Invariance: In many semantic situations (parametric polymorphism, general-relativistic field theory, non-distributive logics), physical or logical meaning is preserved not by functional but by relational invariants under symmetries (automorphisms, polymorphic instantiations, etc.) (Sterling et al., 2022, François et al., 4 Jul 2024, Sojakova et al., 2018).
• Expressiveness: Relations can encode not just input-output mappings but also mutual dependencies, behavioral equivalences, probabilistic divergences, or bisimulations (Maillard et al., 2019, Sato et al., 2022, Rozplokhas et al., 2020).
• Uniformity: Providing a single semantic recipe that applies across logics (classical, intuitionistic, modal, substructural), algebraic settings (lattices, posets, structures with quasi-operators), and computational models (relational type theory, relational programming) (Hartonas, 23 Nov 2025, Pombo et al., 2021, Fussner, 2020).
• Operational/Physical Compatibility: In physics and database theory, the relational approach aligns semantics with observed or "nameable" values and measurements, not arbitrary set-theoretic elements (François et al., 4 Jul 2024, Kent, 2012, Kelly et al., 2012, Pombo et al., 2021).

2. Core Mathematical Structures

2.1 Sorted/Residuated Frames and Polarity

At the algebraic and logical level, a general relational semantics framework is built around sorted residuated frames or polarities:

• Sorted Frame: A structure $\mathfrak{F} = (s, W, I, (R_j), \sigma)$ where $s$ is a set of sorts (typically $s = \{1, \partial\}$), each $W_t$ is a carrier for sort $t$, the incidence $I$ is a fundamental binary relation (e.g., polarity relation for concept lattices), and each $R_j$ is an additional (possibly multi-place) relation encoding operations or connectives (Hartonas, 23 Nov 2025, Hartonas, 2021).
• Polarity: A two-sorted frame $(X, I, Y)$, with Galois connection between sets, generating complete lattices of stable subsets.
• Residuated Operations: Higher-arity frame relations induce algebraic operators (normal, residuated, or quasi-operators), accounting for logical connectives beyond conjunction/disjunction, including implications and modalities.

2.2 Generalized Model Constructions

• Kripke and Veltman Frames: Standard and generalized Veltman semantics use relational frames with binary (accessibility) and set-valued (witness) components, enabling a semantics for interpretability and other specialized modal logics (Joosten et al., 2020).
• Relational Type Theory and Parametricity: Types are interpreted as relations, often in categorical/fibrational structures reflecting face/degeneracy/naturality, as in the abstract scheme of reflexive-graph categories with isomorphisms (Sojakova et al., 2018, Stump et al., 2021).
• Database and Predicate Calculus Models: Relations are set-theoretic objects parameterized by arbitrary signatures (possibly many-sorted, non-numerically indexed), supporting projection, join, selection, and satisfaction conditions in categorical and logical frameworks (Kent, 2012, Kelly et al., 2012).
• Poset-Indexed/Algebra-Valued Models: Frames with structured "fibers" (each world x carries an algebra $A_x$), and assignment functions constrained by "antichain" or flow conditions, unify temporal and intuitionistic logics and general substructural logics (Fussner, 2020).

3. Satisfaction, Invariance, and Modality

• Automorphism-Invariance: Physical or logical observables are required to be invariant under the automorphism group of the structure (e.g., Aut(P) for general-relativistic gauge field theory), leading to "gauge-invariant" or "relationally-dressed" variables and equations (François et al., 4 Jul 2024).
• Relational Satisfaction: The core semantic clause assigns satisfaction not at points, but in terms of preservation of relations—e.g., for A→B in Kripke/frame semantics: $w \models A \to B$ iff for all accessible $y \geq w$, $y \models A$ implies $y \models B$; for interpretability, one extends the clause to allow witness sets (Fussner, 2020, Joosten et al., 2020).
• Modalities and Relational Liftings: Proof-relevant modalities, logical relations as types, set-valued accessibility, and codensity liftings for monads support advanced forms of relational reasoning (parametricity, bisimulation, differential privacy, quantitative cost, etc.) (Sterling et al., 2022, Sato et al., 2022, Maillard et al., 2019).

Domain	Relational Structure	Satisfaction/Observable
General-Relativistic Gauge Theory	Aut(P)-principal bundle, DFM, bisections	Automorphism-invariant forms on field space (François et al., 4 Jul 2024)
Parametric Type Theory	Reflexive-graph categories with isos	Natural transformation/naturality
Database Theory	Set/category-valued tables, IFF-instances	FO formula satisfaction via tuples
Modal/Substructural Logics	Sorted frames, polarities, Galois maps	Sahlqvist-van Benthem correspondence
Program Logics (Monads)	Graded relational/lifting structure	Soundness via monad/divergence axioms

4. Uniformity, Completeness, and Correspondence

• Uniform Construction: A general recipe, given an algebraic or logical signature, produces a class of relational frames, uniform satisfaction clauses, and completeness theorems. This is realized via canonical extensions (choice-free or polarity-based), categorical derivations, and representation/duality theorems (Hartonas, 23 Nov 2025, Hartonas, 2021, Fussner, 2020).
• Completeness and Filtration: For each class of logic or frame, completeness follows by embedding the algebraic side into the relational (via representation theorems), often using canonical or perfect extensions. Generalized Veltman semantics supports filtration, yielding the finite-model property and decidability for broad classes of interpretability and modal logics (Joosten et al., 2020, Fussner, 2020, Hartonas, 23 Nov 2025).
• Sahlqvist Correspondence and Reductions: Uniform Sahlqvist-van Benthem algorithms, abstracted to multi-sorted or polarity-based settings, provide effective correspondents for axiomatic extensions, bridging modal and frame-level semantics (Conradie et al., 2022, Hartonas, 23 Nov 2025).
• Meta-Theoretical Parametricity: Abstract frameworks characterize precisely which structures (e.g., λ^2-fibrations with face/degeneracy, Beck-Chevalley) yield sound models of System F parametricity and similar typing disciplines (Sojakova et al., 2018, Stump et al., 2021).

5. Illustrative Applications

• Physics and Gauge Theory: The geometric relational framework for general-relativistic gauge field theories demonstrates how "points" and "labels" acquire meaning only through field-theoretic coincidences; relationally-dressed variables and equations express physical invariants and coordinate-free dynamics, applying the dressing field method and relational moduli space constructions (François et al., 4 Jul 2024).
• Database Semantics: Relational semantics in databases is formulated via entity and relation classifications, key functor and tuple natural transformations, and data migration diagrams. This supports both the distinguished (entities distinct from relations) and unified (entities as relations) models, capturing foreign key constraints and functorial data migration (Kent, 2012, Kelly et al., 2012).
• Relational Program Logics: The Next 700 Relational Program Logics and its extensions provide a general method to build relational reasoning tools (RHL, RHTT, acRL, etc.) for arbitrary monadic effects, supporting combinations (state, probabilities, nondeterminism, cost counting, exceptions) via coherent effect observations, relative monads, and codensity liftings (Maillard et al., 2019, Sato et al., 2022).
• Logical Parametricity and Type Theory: Relational Type Theory and general frameworks for relational parametricity provide realizability semantics, extensionality principles, and folding/unfolding of inductive/recursive types using generalized relations, supporting higher parametricity and proof-relevant settings (Stump et al., 2021, Sojakova et al., 2018).

6. Limitations, Extensions, and Future Directions

• Scope and Expressiveness: While general relational semantics frameworks subsume a wide class of logics and computational models (classical, intuitionistic, substructural, multi-valued, etc.), some logical varieties lack the finite-model property, or require subtler preservation under conuclear operations (Fussner, 2020, Hartonas, 2021).
• Technical Subtleties: The interaction of higher-arity relations, residuated/conuclear identities, or degenerate/connection structures in categorical frameworks can be nontrivial; coherence of adjoints and naturality in parametric models demands careful handling (Sojakova et al., 2018, Hartonas, 2021).
• Future Directions: Ongoing work seeks higher-dimensional generalizations (cubical sets for n-parametricity), graph- or hypergraph-based semantics for further logical systems, extensions to full effectful or quantitative (cost, privacy) program logics, and direct algebraic/categorical integration into software engineering tools and modular semantic description languages (Conradie et al., 2022, Sato et al., 2022, Pombo et al., 2021, Maillard et al., 2019).
• Operational-Physical Interpretation: In mathematics, physics, and database theory, there is increasing emphasis on nameable value semantics, where all individuals are terms or measurements present in syntax or physical field data, rather than abstract points of an external domain (François et al., 4 Jul 2024, Pombo et al., 2021, Kent, 2012).

7. Conclusion

General relational semantics frameworks provide a foundational and unifying platform for interpreting logic, type theory, programming languages, modal and substructural logics, gauge field theory, and data management—all in terms of relations that are compatible with underlying symmetries, invariance principles, and operational structures. They support a modular, compositional, and algebraically robust semantics, facilitate proofs of completeness and uniformity, and directly connect abstract theory with practical, computational, and physical applications across disciplines (Hartonas, 23 Nov 2025, Sterling et al., 2022, François et al., 4 Jul 2024, Sojakova et al., 2018, Maillard et al., 2019, Kent, 2012, Hartonas, 2021, Fussner, 2020).