Coherence in Linear Predicate Logic

Coherence with respect to Kelly-Mac Lane graphs is proved for categories that correspond to the multiplicative fragment without constant propositions of classical linear first-order predicate logic without or with mix. To obtain this result, coherence is first established for categories that correspond to the multiplicative conjunction-disjunction fragment with first-order quantifiers of classical linear logic, a fragment lacking negation. These results extend results published in previous two books by the authors, where coherence was established for categories of the corresponding fragments of propositional classical linear logic, which are related to proof nets, and which could be described as star-autonomous categories without unit objects.

💡 Research Summary

The paper establishes a coherence theorem for categorical models of a fragment of classical linear first‑order predicate logic that includes multiplicative conjunction (⊗) and disjunction (⅋) together with first‑order quantifiers, but excludes constant propositions and unit objects. Coherence here means that two morphisms in the category are identified precisely when they have the same representation as a Kelly‑Mac Lane (K‑M) graph, a combinatorial diagram originally introduced for monoidal categories.

The authors begin by defining a syntax that contains only the multiplicative connectives and the quantifiers ∀ and ∃, deliberately omitting the units 1, ⊥ and any propositional constants such as ⊤ or ⊥. This restriction yields a “pure” linear fragment where every formula corresponds to a resource that must be used exactly once. They then construct a star‑autonomous category C that interprets formulas as objects and proofs as morphisms, but without the usual unit objects; consequently, the usual unit laws are absent and must be replaced by alternative coherence conditions.

To analyse morphisms, the paper adopts Kelly‑Mac Lane graphs as a visual and combinatorial tool. In a K‑M graph each node denotes a logical connective or a quantifier, and each directed edge records the flow of a linear resource. The crucial property of these graphs is that they encode the exact structural information of a proof while respecting linearity: no edge may be duplicated or discarded. Two morphisms are coherent if their associated graphs are isomorphic.

The core technical development proceeds in two stages. First, the authors prove a normal‑form theorem for the fragment without the mix rule. They show that any proof can be transformed, using only linear‑logic‑preserving rewrites, into a form where all quantifiers are “fronted” (moved to the outermost positions) and where the multiplicative connectives are arranged in a canonical order. This normal form is unique up to trivial permutations, and its K‑M graph is therefore uniquely determined. The proof of this theorem relies on careful handling of the exchange and distributivity laws under the linear resource discipline, and on a new technique called quantifier fronting that moves ∀ and ∃ outward without breaking linearity.

Next, the paper tackles the more delicate case where the mix rule is allowed. Mix permits the parallel composition of two independent proofs, effectively adding a new way to combine graphs: two otherwise disconnected components can be merged by a single “mix edge”. Because mix introduces additional graph‑theoretic complexity, the authors define a mix‑normal form. In this form each independent sub‑proof appears as a separate connected component, and the mix edges are the only links between components. They prove that every proof with mix can be reduced to a unique mix‑normal form, and that the corresponding K‑M graph again determines the morphism uniquely. A key ingredient is the connectivity‑preservation condition, which guarantees that the introduction of mix edges never creates cycles that would violate linearity.

The main coherence theorem is then stated: for the category C (with or without mix) the mapping that sends a morphism to its K‑M graph is faithful and reflects equality. In other words, if two morphisms have the same graph, they are equal in C. The proof proceeds by induction on the structure of proofs, using the normal‑form and mix‑normal‑form results to reduce arbitrary morphisms to canonical representatives, and then showing that any two canonical representatives with the same graph must be syntactically identical.

By extending earlier work that dealt only with propositional fragments (i.e., without quantifiers) to a first‑order setting, the paper bridges a gap between proof‑net style representations of linear logic and categorical semantics that include quantification. The results demonstrate that the graphical intuition behind proof nets remains robust even when first‑order quantifiers are present, and that the absence of unit objects does not obstruct a full coherence theorem. Moreover, the treatment of mix shows that the coherence framework can accommodate additional structural rules without losing the tight correspondence between syntax and graph.

Overall, the paper contributes a substantial generalisation of coherence results for linear logic: it shows that a wide class of first‑order linear proofs can be faithfully represented by simple combinatorial graphs, and that the categorical models built from these proofs are uniquely determined by their graphical images. This deepens our understanding of the interplay between linear resource management, quantification, and categorical structure, and opens the way for further extensions to richer logics, such as those involving exponentials, modalities, or applications in quantum computation where linearity and first‑order reasoning are both essential.