Fully Linear Fragment in Resource Logic

• Fully linear fragment is a unitless, strictly resource-sensitive subsystem of linear logic that omits weakening, contraction, and unit objects.
• It models logical proofs via promonoidal categories and pseudomonoids, ensuring a precise correspondence with proof nets and linear type systems.
• Its NP-complete proof search and bijective mapping with lambda terms enable efficient automated deduction and resource management.

The fully linear fragment denotes the restriction of linear logic—or, more generally, resource-sensitive formal systems—to those subsystems in which all structural rules that allow weakening and contraction are absent, and in which formulas and proof objects exhibit strict linearity and the exclusion (sometimes by omission, sometimes by explicit syntax) of unit objects. This concept is foundational in the semantic and proof-theoretic analysis of linear logic and related type theories, serving as a locus for both categorical interpretation and algorithmic paper.

1. Formal Definition and Scope

The fully linear fragment is most characteristically exemplified by the unitless (or unit-free) multiplicative fragment of linear logic (often abbreviated $\mathrm{MLL}^-$), as well as certain additive or implicational fragments where linearity is enforced at the type and term level.

• Syntax for multiplicative linear logic (MLL) unitless fragment:

$A ::= p \mid p^\bot \mid (A \otimes A) \mid (A \parr A)$ where $p$ is an atomic formula, $\otimes$ is multiplicative conjunction, $\parr$ is multiplicative disjunction. Units $I$ (for $\otimes$) and $\bot$ (for $\parr$) are omitted entirely (Houston, 2013).

• Further examples:
  • In additive/operational calculi, "fully linear" can refer to systems where linear distributivity (across, e.g., sums in lambda calculi) is enforced strictly, and where no resource-weakening constructs exist (Díaz-Caro et al., 2010).
  • In first- and higher-order frameworks, the fully linear fragment may denote the restriction in which every variable appears exactly once in terms/formulas, and all context splitting is total—i.e., no “leftover” or “silent” assumptions (Tarau et al., 2020, Schack-Nielsen et al., 2010).

2. Categorical Semantics: Promonoidal Categories and Pseudomonoids

The categorical semantics of the fully linear (unitless) fragment requires relaxation of the standard monoidal category structure, which is usually designed to model full MLL with units. The absence of unit objects precludes the use of standard monoidal categories, as the coherence and identity morphisms for unit objects are missing.

• Promonoidal Category:

A promonoidal category $(\mathbb{C}, P, J, \alpha, \lambda, \rho)$ consists of a category $\mathbb{C}$, a multiplication functor $P$, potentially a unit functor $J$, and associated associativity/unit natural isomorphisms. In the fully linear fragment, $J$ and all unit isomorphisms are omitted, yielding a semigroup-like structure (Houston, 2013).

• Pseudomonoids in Monoidal Bicategories:

Promonoidal categories are special instances of pseudomonoids in a monoidal bicategory (notably, the bicategory $\mathbf{Prof}$ of categories, profunctors, and natural transformations). A pseudomonoid without a unit object precisely models the algebraic structure needed for unitless MLL semantics (where associativity is up to coherent isomorphism but there is no "identity" element for the tensor) (Houston, 2013).

| Structure | Unit required | Example modeling fully linear fragment? | |------------------------------|:-------------:|:-----------------------------------------------| | Monoidal category | Yes | No | | Semigroup object | No | Yes | | Promonoidal category | No | Yes |

This categorical correspondence yields direct models of fully linear systems—meaning those without structural rules or with tightly constrained resource behavior.

3. Syntactic and Typing Principles

Formalisms describing the fully linear fragment enforce linearity explicitly:

• Sequent calculus:

Rules include only those for the logical connectives, omitting structural rules such as weakening and contraction. Cut-elimination and the subformula property are preserved (Chudigiewitsch, 2021).

• Typings (lambda calculi, type theories):

Typing systems preclude types such as $(T + R) \to S$ and only allow operations that respect linear resource application (i.e., each variable must be used exactly once; sums or alternations are distributed strictly across all arguments and function positions) (Díaz-Caro et al., 2010, Tarau et al., 2020).

• Substitution patterns:

In higher-order unification or pattern calculi, substitutions are restricted to those that preserve linearity—no variable duplication, omission, or change in usage regime. Pattern substitutions in the fully linear fragment correspond only to identity-on-context, matching all variables one-for-one (Schack-Nielsen et al., 2010).

4. Model-Theoretic and Proof-Theoretic Properties

• Soundness and Completeness:

Promonoidal and appropriately constructed categories (e.g., via double glueing (Schalk et al., 2015)) provide fully complete models: every morphism corresponds exactly to a proof net in the logic, and no more.

• Absence of units:

The lack of a unit is not a deficiency of property, but an absence of structure. Any attempt to "add a unit" is unique only up to coherent isomorphism (Houston, 2013).

• Proof nets:

The set of denotable proofs in the fully linear fragment is restricted to acyclic, connected structures, faithfully modeled by proof nets, with categorical semantics matching exactly—no more, no less (Schalk et al., 2015).

5. Algorithmic and Complexity Aspects

• Provability:

Decision of provability in the fully linear fragment (i.e., MLL without additives, exponentials, or units) is NP-complete, owing to the subformula property, cut-elimination, and the efficient checkability of proof nets (linear/quadratic time for correctness) (Chudigiewitsch, 2021).

• Generation of tautologies and proofs:

Declarative algorithms leveraging the Curry-Howard isomorphism implement size-preserving, bijective enumerations of formulas and normal-form proof terms in the implicational fully linear fragment. This scales to billions of unique theorems/proofs and enables the construction of large evaluation and training datasets (Tarau et al., 2020).

• Pattern unification:

Deterministic unification algorithms for the fully linear fragment (as a subcase of the pattern fragment) always terminate and produce most-general unifiers by structural decomposition and inverse substitution, restricted to strictly resource-preserving cases (Schack-Nielsen et al., 2010).

6. Relationship to Fragments, Expressivity, and Limitations

• Resource control:

The fully linear fragment omits all mechanisms of weakening and contraction, and, in many cases, even units. This strictly enforces the use-once discipline and resource balance.

• Expressivity limits and encodings:

Although seemingly limited, various minimal fragments (even using only a single literal or only units) can simulate the full logic under suitable encodings and translations, preserving computational expressivity up to and including undecidable problems when exponentials are present (Kanovich, 2017).

• Intuitionistic vs. non-intuitionistic forms:

Categorical modeling for the intuitionistic case involves further constraints on the base category (e.g., cartesian closure or appropriate closure structures), but the promonoidal and pseudomonoid frameworks extend naturally (Houston, 2013).

7. Technical and Conceptual Significance

The fully linear fragment is pivotal for both foundational and applied investigations in logic, type theory, and categorical semantics.

• Foundation for resource logics:

Its strict exclusion of non-linear resource operations establishes it as the canonical setting for resource-sensitive computation, concurrency (no duplication, no loss), and quantum logic (mirroring no-cloning/no-deletion).

• Catalyst for categorical innovation:

Its requirements have driven the development of promonoidal categories, the theory of pseudomonoids in bicategories, and advanced tools for handling coherence and higher-dimensional algebra.

• Basis for efficient algorithms:

The tractability (NP-completeness) for proof search, the structure of proof nets, and exact bijections with lambda terms make this fragment suitable for both theoretical characterization and high-performance automated deduction systems.

A plausible implication is that the fully linear fragment, while austere, robustly captures the core computational and semantic content of linear logic, serving as a basis from which more complex behaviors—such as those arising from units, additives, or exponentials—can be systematically and transparently developed.