The sheaf condition on a design presheaf over an architectural site is equivalent to pairwise overlap compatibility and yields unique global designs from compatible local ones, with the equivalence machine-verified in Lean 4.
A mathematical theory of co-design.arXiv preprint arXiv:1512.08055(2015)
5 Pith papers cite this work. Polarity classification is still indexing.
5
Pith papers citing it
abstract
One of the challenges of modern engineering, and robotics in particular, is designing complex systems, composed of many subsystems, rigorously and with optimality guarantees. This paper introduces a theory of co-design that describes "design problems", defined as tuples of "functionality space", "implementation space", and "resources space", together with a feasibility relation that relates the three spaces. Design problems can be interconnected together to create "co-design problems", which describe possibly recursive co-design constraints among subsystems. A co-design problem induces a family of optimization problems of the type "find the minimal resources needed to implement a given functionality"; the solution is an antichain (Pareto front) of resources. A special class of co-design problems are Monotone Co-Design Problems (MCDPs), for which functionality and resources are complete partial orders and the feasibility relation is monotone and Scott continuous. The induced optimization problems are multi-objective, nonconvex, nondifferentiable, noncontinuous, and not even defined on continuous spaces; yet, there exists a complete solution. The antichain of minimal resources can be characterized as a least fixed point, and it can be computed using Kleene's algorithm. The computation needed to solve a co-design problem can be bounded by a function of a graph property that quantifies the interdependence of the subproblems. These results make us much more optimistic about the problem of designing complex systems in a rigorous way.
representative citing papers
An elimination-based rejection-sampling algorithm with optimistic evaluators identifies target-feasible antichains in monotone co-design problems and propagates bounds compositionally through multigraphs.
A category theory framework translates biological mechanics into engineered stimulus-response systems via functors and composition, demonstrated on pinecone-inspired actuators.
A compositional framework based on monotone co-design theory enables joint optimization of robot design, fleet composition, and planning for heterogeneous multi-robot systems under task-specific constraints.
Category theory proves prompt-based learning on perfect foundation models works only for representable tasks, fine-tuning solves tasks in the pretext category, and models can represent unseen target-category objects using source-category structure.
citing papers explorer
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On the Power of Foundation Models
Category theory proves prompt-based learning on perfect foundation models works only for representable tasks, fine-tuning solves tasks in the pretext category, and models can represent unseen target-category objects using source-category structure.