Combining implementation-level composition with optimization in quantale-enriched co-design

Determine a principled method to integrate implementation-level composition modeled in the powerset quantale P(I) with optimization evaluated in the cost quantale Cost = ([0, ∞], ≥, +, 0) within the quantale-enriched co-design framework; specifically, identify a valid change-of-base or alternative construction that maps feasible implementation sets S ⊆ I to costs while preserving the laxity requirements necessary for composition, given that the intuitive map S ↦ inf_{i∈S} c(i) is not a lax function from P(I) to Cost.

Background

Implementation-aware design problems in this framework are composed using the powerset quantale P(I) of feasible implementations, which allows components to return sets of valid implementation choices and to compose them via series, parallel, and feedback operations.

A natural goal is to optimize over these feasible sets by moving from P(I) to the cost quantale Cost through a change of base. The most direct mapping that sends a feasible set S to the cost of its cheapest element, inf_{i∈S} c(i), fails to be a lax function, which prevents its use for valid change-of-base and breaks the compatibility required for composition.

Consequently, a mechanism is needed to connect implementation-level composition and quantitative optimization without violating the structural (laxity) conditions that ensure enriched categories and profunctors remain well-defined under change of base.