# Distributive Minimization Comprehensions and the Polynomial Hierarchy

## TL;DR

This paper introduces a categorical framework for understanding minimization in subrecursive classes, linking it to the polynomial hierarchy through a tiered structure derived from coalgebraic and comprehension concepts.

## Contribution

It extends symmetric monoidal comprehension to distributive minimization comprehension, providing a new categorical perspective on the polynomial hierarchy.

## Key findings

- Categorical model for minimization in subrecursive classes
- Connection between minimization levels and polynomial hierarchy
- Framework for tiered structure of partial functions

## Abstract

A categorical point of view about minimization in subrecursive classes is presented by extending the concept of Symmetric Monoidal Comprehension to that of Distributive Minimization Comprehension. This is achieved by endowing the former with coproducts and a finality condition for coalgebras over the endofunctor sending X to ${1}\oplus{X}$ to perform a safe minimization operator. By relying on the characterization given by Bellantoni, a tiered structure is presented from which one can obtain the levels of the Polytime Hierarchy as those classes of partial functions obtained after a certain number of minimizations.