Composable Uncertainty in Symmetric Monoidal Categories for Design Problems (Extended Version)

Technical Summary: Composable Uncertainty in Symmetric Monoidal Categories for Design Problems

Problem Statement
The design of complex systems often relies on categorical models of open systems, specifically the category of design problems (DP), which is a compact closed symmetric monoidal category (SMC). In DP, morphisms represent feasibility relations between resources and functionalities, modeled as monotone maps. While this framework supports compositional co-design, it assumes systems are fully known and deterministic. In practice, systems involve uncertainty (e.g., material properties following distributions, performance bounds, or imprecise probabilities) and parametric dependencies (e.g., design choices affecting optimal parameters). Current methods for handling uncertainty in co-design, such as using upper and lower bounds, are robust but lack quantitative measures of success probability or the ability to model parametric uncertainties that depend on specific design choices. There is a need for a framework that integrates quantitative uncertainty semantics (like intervals, subsets, or distributions) into the compositional structure of open systems while preserving the underlying SMC properties.

Methodology
The authors propose a general scheme to add parametric uncertainty to any symmetric monoidal category (SMC) by leveraging the change-of-base construction for enriched categories. The methodology proceeds in three main steps:

1. Uncertainty Semantics via Monads: Uncertainty is modeled using symmetric monoidal monads ($M$) on a base monoidal category $V$ (e.g., $\mathbf{Set}$, $\mathbf{Pos}$, $\mathbf{Meas}$). The Kleisli category $\mathbf{Kl}_M$ of such a monad forms a Markov category, capturing various uncertainty types (e.g., the powerset monad for subsets, the Giry monad for probability distributions, or the arrow monad for intervals).
2. Change-of-Base for Enrichment: Given a $V$-category $\mathcal{C}$, the authors replace its hom-objects $\mathcal{C}(X,Y)$ with parametric maps $A \to M\mathcal{C}(X,Y)$, where $A$ is a parameter space. This is achieved by applying the change-of-base construction along a symmetric monoidal functor $N: V \to W$ (specifically, the identity-on-objects functor $L_M: V \to \mathbf{Kl}_M$).
3. Partial Slice Functor for Parametrization: To introduce external parametrization, the authors utilize the partial slice functor $U/(-)$, where $U$ is a strict monoidal subcategory of the target category. This functor maps objects to comma categories, effectively treating arrows as families of morphisms indexed by objects in $U$.

By combining these, the authors construct a new 2-category, denoted $(U/MF)^*\mathcal{C}$, where:

• 0-cells are the objects of the original category $\mathcal{C}$.
• 1-cells are morphisms in the Kleisli category of the monad, representing parametric maps $A \to M\mathcal{C}(X,Y)$.
• 2-cells are reparametrization maps (arrows in the Kleisli category) between parameter spaces.

The composition of 1-cells utilizes the strength of the monad to combine parameter spaces and compose the underlying uncertain processes. The resulting structure is a nearly strict symmetric monoidal 2-category, where the interchange law and symmetry hold up to coherent invertible 2-cells (tensorators).

Key Contributions

• General Construction: The paper establishes a rigorous construction that transforms any symmetric monoidal $V$-category $\mathcal{C}$ into a symmetric monoidal 2-category with parametric uncertainty, preserving the original objects and embedding the original category via a strict symmetric monoidal functor.
• Structure Preservation: The construction proves that compact closed structures, (co)monoidal structures, and supplies (such as copy and delete maps) transfer from the original category to the new 2-category. This ensures that compositional reasoning and feedback loops remain valid in the presence of uncertainty.
• Flexibility of Semantics: The framework is agnostic to the specific type of uncertainty, provided it can be modeled by a symmetric monoidal monad. The authors demonstrate applicability to:
  • Possibilistic uncertainty: Using the powerset monad (subsets) or arrow monad (intervals).
  • Probabilistic uncertainty: Using the Giry monad (distributions) on measurable spaces.
• Application to Design Problems (DP): The authors apply this construction to the category of design problems ($DP$). They show that $DP$ can be viewed as enriched in $\mathbf{Pos}$, $\mathbf{Set}$, or $\mathbf{Meas}$, allowing for the creation of categories of parametrized design problems, parametrized intervals of design problems, and parametrized distributions of design problems.

Results and Examples
The paper demonstrates the utility of the construction through several specific applications to the category of design problems:

• Parametrized Design Problems: The 2-category $(U/\text{Id})^*DP$ models design problems where feasibility depends on external parameters (e.g., environmental conditions or design choices).
• Robust Design: Using the powerset monad, $(U/\text{Pow}_\emptyset)^*DP$ models robust design problems where a parameter maps to a non-empty subset of feasible designs, allowing for worst-case analysis.
• Probabilistic Design: Using the distribution monad, $(U/D_\Sigma)^*DP$ models design problems where components have probabilistic performance characteristics. This enables the modeling of random influences and the use of Bayesian decision theory.
• Learning and Inference: The framework supports learning design problems from data. For instance, in a Bayesian setting, the construction allows for the conditioning of design problem distributions on observed data (e.g., feasible triples of velocity, load, and cost) to update beliefs about unknown parameters.
• Active Learning: The compositional nature of the construction allows for the composition of surrogate models for individual components to predict system-level information and update beliefs, facilitating active learning strategies.

Significance
The paper claims that this construction provides a unified, compositional framework for integrating quantitative uncertainty into the study of open systems and co-design. By preserving the underlying symmetric monoidal and compact closed structures, it allows engineers and researchers to apply established categorical tools (such as functorial queries and diagrammatic reasoning) to uncertain systems. The ability to handle various uncertainty semantics (intervals, subsets, distributions) within a single 2-categorical framework addresses the limitation of current methods that rely solely on deterministic bounds. The construction enables the decomposition of complex uncertain design problems into simpler, solvable components while maintaining a rigorous mathematical foundation for decision-making and learning under uncertainty.