Quantale-Enriched Co-Design: Toward a Framework for Quantitative Heterogeneous System Design

Abstract: Monotone co-design enables compositional engineering design by modeling components through feasibility relations between required resources and provided functionalities. However, its standard boolean formulation cannot natively represent quantitative criteria such as cost, confidence, or implementation choice. In practice, these quantities are often introduced through ad hoc scalarization or by augmenting the resource space, which obscures system structure and increases computational burden. We address this limitation by developing a quantale-enriched theory of co-design. We model resources and functionalities as quantale-enriched categories and design problems as quantale-enriched profunctors, thereby lifting co-design from boolean feasibility to general quantitative evaluation. We show that the fundamental operations of series, parallel, and feedback composition remain valid over arbitrary commutative quantales. We further introduce heterogeneous composition through change-of-base maps between quantales, enabling different subsystems to be evaluated in different local semantics and then composed in a common framework. The resulting theory unifies feasibility-, cost-, confidence-, and implementation-aware co-design within one compositional formalism. Numerical examples on a target-tracking system and a UAV delivery problem demonstrate the framework and highlight how native quantitative enrichment can avoid the architectural and computational drawbacks of boolean-only formulations.

• The paper extends traditional boolean co-design to quantitatively evaluate resources and functionalities using quantale-enriched categories.
• It introduces heterogeneous composition via change-of-base, uniting distinct metrics such as cost, confidence, and implementation sets.
• Case studies in target tracking and UAV systems validate the framework for multi-objective optimization and modular system integration.

Quantale-Enriched Co-Design: A Framework for Quantitative Heterogeneous System Design

Introduction

The paper "Quantale-Enriched Co-Design: Toward a Framework for Quantitative Heterogeneous System Design" (2603.29921) addresses fundamental limitations of monotone co-design frameworks, specifically their inability to natively represent quantitative criteria such as cost, confidence, or implementation selection. Monotone co-design traditionally models system components based on boolean feasibility relations between resources and functionalities, but this approach does not support compositional quantitative analyses and leads to ad hoc system augmentations and increased computational complexity.

The authors propose a quantale-enriched generalization of co-design, wherein resources and functionalities are modeled as categories enriched over quantales and co-design problems are formalized as quantale-enriched profunctors. Through this generalization, the framework systematically extends compositional design operations to arbitrary quantitative metrics, supports heterogeneous interfacing by means of change-of-base between quantales, and unifies feasibility- and cost-aware co-design along with confidence and implementation-aware systems in a single categorical formalism.

Quantale-Enrichment and Categorical Framework

The central construct of the framework is the quantale—an ordered set equipped with an associative, commutative multiplication and a complete join structure. Key examples demonstrated include:

• Booleans: $$(\{\mathtt{false}, \mathtt{true}\},\leq,\wedge,\mathtt{true})$$ for feasibility analysis,
• Additive Cost: $([0,\infty],\ge,+,0)$ for resource cost evaluation,
• Fuzzy Confidence: $([0,1],\leq,\odot,1)$ for degree-of-belief metrics,
• Product Quantales: Enables tracking of vector-valued multi-criteria metrics or feasible implementation sets.

Systems are described by quantale-enriched categories (Q-categories), where morphisms generalize the hom-objects of classical categories to take values in quantales. Co-design problems are encoded as quantale-enriched profunctors, which assign a quantale value—interpreted as cost, confidence, or feasibility degree—to each resource-functionality pair, subject to natural monotonicity and compositionality constraints.

Figure 1: Uncurrying in monotone co-design (top) versus succinct enrichment through a quantale (bottom), highlighting how the latter preserves the compositional structure and avoids resource space blowup.

This construction establishes series, parallel, and feedback (trace) compositions for co-design problems, and the authors prove that these operations are valid in the enriched setting for any commutative quantale (Theorem 1). The structure enables rigorous, graphical manipulation of systems via string diagrams, aligning with the categorical semantics of traced monoidal categories.

Heterogeneous Composition via Change-of-Base

A notable feature of the framework is its support for heterogeneous composition. In realistic system architectures, components may be evaluated under incomparable metrics (e.g., some in cost, others in feasibility, others in implementation sets). The paper introduces the use of lax quantale homomorphisms (change-of-base maps) to canonically lift/translate between disparate quantale-enriched submodels, enabling compositionality across these semantic boundaries.

For example, mapping cost-quantales to booleans allows thresholding, while embedding booleans into cost quantales uses infinity as a penalty for infeasibility. These operations are precisely characterized by categorical properties, ensuring the resulting systems remain amenable to composition.

Numerical Case Studies

Quantitative Target Tracking

The authors present a quantitative co-design example motivated by target tracking systems, where sensors and processing elements are co-designed for minimum cost under varying power and quality-of-service constraints. The formulation uses cost-enriched profunctors over boolean resource/functionality spaces. Series composition via infimal convolution identifies optimal allocation strategies, as confirmed analytically and by consistency with mcdp, a computational co-design solver.

UAV Delivery System

A more complex example involves a UAV system co-designed with actuator and battery choices under task and resource constraints. Here, the model is flexibly instantiated in different quantales to answer distinct engineering queries: (i) in $Cost$, for minimum lifetime cost as a function of payload and fulfilled deliveries; and (ii) in the powerset of implementations, for enumerating all actuator-battery technology pairs feasible within a given cost and payload.

Figure 2: The compositional Cost-enriched co-design diagram for the UAV highlighting resource and implementation flows.

Figure 3: Lifetime cost curves for varying payloads, showing minimum cost as a function of satisfied deliveries.

Heterogeneous construction allows for queries such as feasible implementation regions (actuator–battery pairs) under strict cost and payload budgets, with the set-theoretic structure naturally handled in the categorical composition.

Figure 4: The compositional model of the UAV co-design problem in the powerset quantale, illustrating the integration of implementation-awareness.

Figure 5: Feasible regions in the payload-lifetime cost space for actuator and battery technology pairs, enabling structured exploration of trade-offs.

Theoretical and Practical Implications

This framework:

• Unifies Feasibility and Quantitative Evaluation: The use of quantale-enrichment captures cost, confidence, and implementation choices natively and compositionally, preserving system structure and enabling richer optimization queries.
• Facilitates Multi-Criteria, Multi-Level System Integration: Product quantales and change-of-base support multi-objective co-design and the interconnection of subsystems under distinct metrics, mitigating the need for ad hoc scalarization or resource augmentation.
• Maintains Efficient Compositionality: By parametrizing the aggregation and feedback operations through the quantale, architectural and computational drawbacks of previous methods are avoided, as evidenced in the comparative case studies.

The categorical unification opens potential connections to compositional convex and parametric optimization (e.g., via linkages to infimal convolution and bifunctions), dagger categories for undirected optimization, and hypergraph categories for objective-algorithm integration.

Limitations and Open Problems

While the framework resolves key structural and computational limitations, the mapping from implementation sets to optimal costs (e.g., selecting lowest-cost elements from powersets) is not always a lax quantale homomorphism, limiting direct compositionality. Future research is necessary to harmonize implementation-level composition and cost-aware optimization.

Conclusion

The quantale-enriched co-design framework extends classical co-design into a compositional, categorical model supporting quantitative, heterogeneous, and implementation-aware system design. By leveraging structured enrichment and change-of-base, the approach provides robust primitives for modular engineering analysis, optimization, and system integration. The theoretical foundations suggest further connections between category theory and modern compositional optimization, and the practical methodology points to efficacy for large-scale, multi-criteria design challenges in robotics, cyberphysical systems, and beyond.

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References:

See (2603.29921) and its bibliography for precise definitions and technical developments.