Relational Semantics of Linear Logic

• Relational semantics is a resource-sensitive set-theoretic interpretation of linear logic that maps formulas to sets and proofs to relations.
• It achieves full faithfulness for MELL proof-nets by using k-injective experiments to ensure unique semantic mappings corresponding to cut-elimination.
• Extensions include finite and countable multisets and colored modalities, enabling practical applications in λ-calculus, type theory, and model checking.

The relational semantics of linear logic provides a resource-sensitive, set-theoretic interpretation of linear proofs, foundational for understanding both the operational behavior and the denotational invariants of linear logic. It realizes formulas as sets and proofs as relations, modeling the resource discipline intrinsic to linear logic through the use of structured multisets. This semantics is now central to the development of linear logic, supporting deep results in proof theory, type theory, λ-calculus, and automata theory.

1. Fundamental Structures and Interpretation

The relational semantics is formulated in the symmetric monoidal closed category Rel, whose objects are sets and whose morphisms are binary relations $R \subseteq X \times Y$ between sets $X$ and $Y$. The monoidal (tensor) product interprets the linear tensor $\otimes$ as the cartesian product of sets: $$\llbracket A \otimes B \rrbracket = \llbracket A \rrbracket \times \llbracket B \rrbracket$$. Exponentials are interpreted via finite multisets: for a formula $A$, the exponential $\llbracket !A \rrbracket$ is the set of finite multisets over $\llbracket A \rrbracket$ (Carvalho, 2015, Grellois et al., 2014).

Linear logic proofs, formalized as cut-free proof-nets, are interpreted by constructing, for every such net $\pi$ with conclusions $A_1, \ldots, A_n$, a relation

$$\llbracket \pi \rrbracket \subseteq \llbracket A_1 \rrbracket \times \cdots \times \llbracket A_n \rrbracket$$

comprising all conclusion-point tuples generated by assigning elements ("experiment labels") to the wires in a manner compatible with the local rules of the net (handling multiplicatives, units, boxes, contraction/weakening, and axioms) (Carvalho, 2015).

Table: Correspondence in the Relational Model

Linear Logic Construct	Model Interpretation	Notes
$A \otimes B$ (Tensor)	$$\llbracket A \rrbracket \times \llbracket B \rrbracket$$	Cartesian product
$!A$ (Exponential)	Finite multisets over $\llbracket A \rrbracket$	Resource duplication/weakening
Proof-net $\pi$	Relation $$\llbracket \pi \rrbracket \subseteq \Pi_i \llbracket A_i \rrbracket$$	Set of experiment results

2. Injectivity and Faithfulness

The relational model is injective for multiplicative exponential linear logic (MELL): two cut-free, $\eta$-expanded MELL proof-nets $\pi_1$, $\pi_2$ have the same relational interpretation if and only if they are equivalent under cut-elimination:

$$\llbracket \pi_1 \rrbracket = \llbracket \pi_2 \rrbracket \iff \pi_1 \equiv_{\mathrm{cut}} \pi_2$$

This establishes that Rel is a fully faithful and complete model for proof-net equivalence in MELL. The proof leverages k-injective experiments—numerically annotating box copies—to reconstruct the unique proof structure from any distinguishing semantic tuple (Carvalho, 2015, Carvalho et al., 2010). If two distinct nets have the same semantics, a reconstruction produces both, implying syntactic equivalence.

In the absence of weakenings, this injectivity is complete (even identifying proof-nets up to auxiliary-door assignment for boxes). When weakenings are allowed, semantics loses the ability to distinguish certain syntactic differences, as the empty multiset collapses distinctions (Carvalho et al., 2010).

3. Infinitary and Colored Variants

Expanding the base category to Rel$_\infty$, objects are sets (possibly uncountable), and the exponential modality $!A$ is interpreted as the set of finite or countable multisets $\mathcal{M}_{\leq\omega}(A)$. This infinitary relational model accommodates inductive (least) and coinductive (greatest) fixpoints via Conway operators, realized by concrete run-tree constructions. Two extremal fixpoints are given:

• Inductive $Y^\mu$: collects pairs $(w,a)$ for finite accepting run-trees,
• Coinductive $Y^\nu$: for all accepting (possibly infinite) run-trees.

Coloring enrichments—via a comonad $♦$ assigning a finite color to every element—permit the modeling of parity games and parity-acceptance conditions in fixpoint computations. The combination of these comonads ($\diamond = ! \circ ♦$) yields a model with a mixed inductive/coinductive fixpoint operator, enabling parity-dependent semantics critical for higher-order model checking and verification (Grellois et al., 2014, Grellois et al., 2015).

4. Applications: Type Theory, λ-Calculus, and Model Checking

Within this semantics:

• λ-Calculus Interpretation: The simply-typed λ-calculus is embedded via Church encodings, with the Kleisli category of the comonad $!$/($! \circ ♦$ for colors) providing a cartesian closed structure for interpreting terms with fixpoints, recursion, and parametric iteration (Grellois et al., 2014, Grellois et al., 2015).
• Intersection Types and Normalization: Points in the relational semantics correspond to (non-idempotent) intersection types, supporting type-theoretic characterization of strong normalization and resource usage (Carvalho, 2015, Carvalho et al., 2013). Strong normalizability of nets is equivalent to the non-emptiness of their (decorated) semantics.
• Higher-Order Model Checking: The colored relational model interprets higher-order recursion schemes (HORS) and alternating tree automata (ATA/APT) as dual relations; their semantic interaction captures tree acceptance and leads directly to decidability results for monadic second-order logic (MSO) over infinite hyper-trees (Grellois et al., 2015, Grellois et al., 2015).

Table: Relational Model Applications

Area	Mechanism	Reference
Type-based normalization	Relational interpretation $\longleftrightarrow$ intersection types	(Carvalho et al., 2013, Carvalho, 2015)
λ-calculus with recursion, fixpoint	Kleisli category, parametric Y-combinator	(Grellois et al., 2014)
Higher-order model checking (MSO/higher-order)	Colored exponentials, parity acceptance, fixpoints	(Grellois et al., 2015, Grellois et al., 2015)

5. Extensions and Algebraic Foundations

The relational semantics unifies with linear algebraic models through the theory of modules over $\Sigma$-semirings (Tsukada et al., 2022). In this abstraction, sets and relations correspond to free modules and join-preserving maps over the Boolean semiring, respectively. Tensor, hom, and exponential structures in Rel emerge as specializations of the corresponding notions in the module category, casting classical, probabilistic, and weighted relational models into a common algebraic framework.

The semantics extends to fragments with idempotent exponential modalities (IdLL), utilizing totality spaces as the relational model. Here, all iterates of the exponential coincide (proof-theoretically and semantically), yielding fewer cut-free proofs and a canonical collapse of iterated duplication (Slavnov, 2014).

Geometric/topological invariants have also been assigned to relational interpretations by down-closing the set of proofs and extracting abstract simplicial complexes. Homology computations on these complexes capture geometric structures underlying the relational semantics (Barbarossa, 2024).

6. Decidability, Algorithmic Aspects, and Quantitative Refinements

Relational semantics supports algorithmic analyses:

• Type-checking: Deciding whether a given tuple belongs to the semantics of a proof-structure is decidable (and linear-time for the multiplicative fragment), via symbolic token-based algorithms corresponding to the propagation of experiments (Guerrieri et al., 2016).
• Resource Quantification: The semantics can be refined quantitatively to capture the number of interactions or resources used, especially in models such as the checkers calculus. Fine-grained resource sensitivity, tracked via colored typings and interaction counts, refines extensional equivalence and relates directly to contextual preorders and the execution cost of $\lambda$-terms (Lancelot et al., 4 Jan 2026).

7. Significance and Broader Impact

Relational semantics anchors a range of theoretical advances:

• It provides a canonical, complete, and injective denotational model for proof-nets in core fragments of linear logic, surpassing earlier coherence-based models in faithfulness (Carvalho, 2015, Carvalho et al., 2010).
• It underpins strong normalization results, decidability in higher-order model-checking, and type-based resource analysis by linking relational points to semantic/operational phenomena (Carvalho et al., 2013, Grellois et al., 2015, Grellois et al., 2015).
• Its algebraic and topological generalizations unify distinct linear logic models and expose new connections to abstract algebra and topology (Tsukada et al., 2022, Barbarossa, 2024).

Through its clarity in modeling resource-sensitive behavior and its compatibility with proof-net syntax and cut-elimination, the relational semantics of linear logic continues to structure developments across proof theory, programming languages, and verification.