Graded Feasibility Modality

• Graded feasibility modality is a parameterized operator that tracks resource or cost bounds across logical systems, type theories, and machine learning architectures.
• It employs semirings or lattices to represent a continuum of resource grades using modalities like □k, supporting compositional and fine-grained resource reasoning.
• Applications include enhanced zero-shot learning, multitask neural architectures, and certified optimizations in computational and formal systems.

A graded feasibility modality is a parameterized logical or semantic operator—typically denoted $\Box_k$, $\Box_r$, or $\square_s$—designed to track resource or feasibility bounds within a type system, logic, or machine learning architecture. Unlike classical modalities, which are purely binary, graded feasibility modalities admit a continuum or lattice of levels (grades), supporting fine-grained, compositional reasoning about feasibility, cost, or realism. They provide core infrastructure for resource-sensitive type theories, open-world cognitive systems, and multitask feasibility profiling in machine learning.

1. Mathematical Foundations and Typing Rules

Graded feasibility modalities appear in resource-aware logical systems, such as graded modal dependent type theory (GrTT) (Moon et al., 2020) and resource-bounded type theory (RBTT) (Mannucci et al., 7 Dec 2025). These systems adopt a (commutative) semiring or lattice $(S, 0, +, 1, \cdot)$ of grades, whose elements represent cost, feasibility, or resource bounds.

In GrTT, the graded necessity modality $\square_k A$ is indexed by $k \in S$, where $A$ is a type or proposition. The intended reading is “a value of type $A$ guaranteed to cost at most $k$.” Typing judgments carry usage vectors recording, for each variable, its grade of consumption at the term and type level: $$(\Delta \mid \sigma_s \mid \sigma_r) \odot \Gamma \vdash t : A$$ where, e.g., $\square_k A$ obeys the introduction and elimination rules: $\infer[\Box\text{I}] {(\Delta\mid k+\sigma_s\mid \sigma_r)\odot\Gamma\vdash \square_k t : \square_k A} {(\Delta\mid \sigma_s\mid \sigma_r)\odot\Gamma\vdash t : A & (\Delta\mid \sigma_r\mid 0)\odot\Gamma \vdash A:\mathsf{Type}}$

$\infer[\Box\text{E}] {(\Delta\mid \sigma_1+\sigma_3\mid \sigma_2+\sigma_4)\odot\Gamma\vdash \mathsf{let}\,\square\,x = t_1\,\mathsf{in}\,t_2 : B} {(\Delta\mid \sigma_1\mid \sigma_2)\odot\Gamma\vdash t_1 : \square_{k}A\quad (\Delta,\sigma_2 \mid \sigma_3, k+\sigma_2 \mid \sigma_4,r+\sigma_1)\!\odot\!(\Gamma,x:A)\vdash t_2:B}$

(Moon et al., 2020, Mannucci et al., 7 Dec 2025)

In RBTT, the same pattern is instantiated with an abstract resource lattice $(L, \preceq, \oplus, \sqcup, \bot)$: $\inferrule[(Box)] {\Gamma \vdash_{r; b} t : A \quad b \preceq s} {\Gamma \vdash_{r; b} \mathrm{box}_{s}(t) : \Box_{s}A} \qquad \inferrule[(Unbox)] {\Gamma \vdash_{r; b} t : \Box_{s}A} {\Gamma \vdash_{r; b \oplus \delta_{\mathrm{unbox}}} \mathrm{unbox}(t) : A}$ with a “monotonicity” (weakening) rule: $\inferrule[(Monotone)] {\Gamma \vdash_{r; b} t : \Box_{s_1}A \quad s_1 \preceq s_2} {\Gamma \vdash_{r; b} t : \Box_{s_2}A}$ (Mannucci et al., 7 Dec 2025)

Concrete instantiations include the “tropical semiring” $$S_{\mathit{feas}} = ([0,\infty], \min, \infty, +, 0)$$, where $a + b = \min(a, b)$ models least feasible cost, and $a \cdot b = a + b$ models additive resource accumulation (Moon et al., 2020).

2. Categorical, Algebraic, and Semantic Properties

Categorically, a graded feasibility modality is modeled as a graded comonad or a graded interior operator. For each grade $s$, there is an endofunctor $\square_s : \mathcal{C} \to \mathcal{C}$ with natural transformations:

• Counit: $\varepsilon : \square_{1} \to \mathsf{Id}$
• Comultiplication: $$\delta_{s, r}: \square_{sr} \to \square_s \square_r$$ obeying coherence and monoidal structure (e.g., $$\square_s(A \times B) \cong \square_s A \times \square_s B$$) (Moon et al., 2020).

In the presheaf semantics for resource-bounded type theory, types are interpreted as presheaves over $L$ (the lattice of bounds), and the box modality becomes: $$\llbracket \Box_s A \rrbracket(r) = \{ (\mathrm{box}_s(v), b) \mid (v,b) \in \llbracket A \rrbracket(r), b \preceq s \}$$ with inclusion serving as the counit, and monotonicity realized by index shifting (Mannucci et al., 7 Dec 2025).

3. Graded Feasibility in Statistical and Machine Learning Contexts

A complementary operationalization of graded feasibility arises in compositional zero-shot learning (OW-CZSL) (Kim et al., 16 May 2025). Here, the graded feasibility score $g(s,o)$ for state–object pairs is extracted as the unnormalized logit from an LLM (e.g., Vicuna-13B) prompted with:

Does a/an {s} {o} exist in the real world? (Answer: Yes/No)

This logit is normalized: $$\hat{g}(s, o) = \frac{g(s,o) - \min_{u,v} g(u,v)}{\max_{u,v} g(u,v) - \min_{u,v} g(u,v)}$$ Pairs with $\hat{g}(s,o) \geq \tau$ form the feasible set. These scores function as a continuous $\Box$-modality for reality-judgments, supporting richer ranking, ROC-style threshold tuning, and explicit uncertainty tracking (Kim et al., 16 May 2025).

Empirically, using logit-based graded feasibility gating improves the harmonic mean (H) scores on standard OW-CZSL benchmarks (MIT-States, UT-Zappos, C-GQA) relative to previous GloVe/ConceptNet gates, e.g., on UT-Zappos: CoOp+GloVe $H=34.2$ versus FLM $H=40.6$ (Kim et al., 16 May 2025).

4. Multitask Graded Feasibility in Application Domains

In neural multitask learning, “graded feasibility” refers to empirically stratifying tasks (e.g., segmentation, conversion, bias correction in MRI analysis) by their learnability and joint-optimization compatibility (Eslami et al., 2021). Feasibility is assessed via convergence rates, statistical accuracy (NCC, Dice), and inter-task tradeoffs under single-task and multitask regimes.

For example, bias-field correction and cross-modality conversion are empirically the easiest tasks (rapidly reaching high NCC), segmentation is significantly harder, and multitasking segmentation with conversion is feasible (Dice up from $0.52$ to $0.73$ for U-Net), whereas multitasking with bias correction causes catastrophic degradation (Dice down to $0.13$) (Eslami et al., 2021).

This profiling results in a graded feasibility ranking:

Task Pairing	Feasibility Rank	Multitask Effect
Bias Correction / Conversion	Easiest	No accuracy gain (fast conv.)
Segmentation + Conversion	Moderate	Significant U-Net benefit
Segmentation + Bias Correction	Hardest	Severe loss for segmentation

Such gradation guides practical choice of multitask regimes in application-specific workflows (Eslami et al., 2021).

5. Metatheoretic Properties and Optimization Implications

The metatheoretic foundation of graded feasibility modalities includes subject-reduction, strong normalization, admissibility of substitution/structural rules, and decidability of type checking (Moon et al., 2020, Mannucci et al., 7 Dec 2025). Specifically for feasibility, the cost soundness theorem in RBTT states: $$\emptyset \vdash_{r; b} t : A \implies \exists v, k.\, t \Downarrow_k v \land k \preceq b \preceq r$$ Ensuring that the operational cost is bounded by the grade proves central for certified, compositional reasoning.

In GrTT, quantitative grades enable optimizations: whenever a binder’s subject‐type grade is $0$ (“irrelevant”), type substitution $[t/x]B$ may be omitted, yielding up to $30$–$40\%$ speedup in certain synthetic benchmarks (Moon et al., 2020).

6. Practical and Theoretical Significance

Graded feasibility modalities are the main instrument for encoding resource and feasibility guarantees in both formal and empirical settings. Their presence:

• Enables compositional certification of resource bounds across arbitrary syntactic and semantic domains—e.g., time, gas, or cost in RBTT (Mannucci et al., 7 Dec 2025).
• Supports uncertainty-calibrated, continuous filtering in open-world recognition tasks, improving both ranking and practical accuracy (Kim et al., 16 May 2025).
• Structures the design of multi-output learning systems and the selection of auxiliary tasks according to empirical graded feasibility profiles, leading to robust task architectures (Eslami et al., 2021).

A plausible implication is that further synergies may emerge by combining syntactic graded modalities with learned, data-driven feasibility oracles—e.g., integrating FLM-like scoring into program synthesis or certified AI.

7. Limitations and Outlook

Several limitations are intrinsic to current approaches:

• LLM or data biases can propagate through in-context or empirical feasibility estimates, particularly for rare or out-of-distribution compositions (Kim et al., 16 May 2025).
• The cost of evaluating all graded pairs (e.g., $|\mathrm{States}|\times|\mathrm{Objects}|$ for LLM gating) remains significant (Kim et al., 16 May 2025).
• Type-theoretic modalities require a semiring or lattice structure, restricting the class of feasible gradings deployable in practice (Moon et al., 2020).
• In multitask domains, negative transfer can emerge if auxiliary tasks are misaligned, as seen with segmentation and bias correction (Eslami et al., 2021).

Extensions include broadening to multi-modal and compositional feasibility, elaboration of chain-of-thought explanations as features, and active learning based on low-confidence gradings (Kim et al., 16 May 2025). The theoretical framework remains compatible with a range of resource semantics, from tropical cost calculi to abstract lattices, maintaining a central position in recent developments in resource-sensitive computation and open-world reasoning.