On tiered small jump operators

TL;DR

This paper refines predicative analysis of recursion by using a ramified Ackermann construction, creating a hierarchy of functions that precisely characterizes polynomial time computable functions and extends beyond.

Contribution

It introduces a strict ramification principle and a lambda-calculus representation to analyze and extend complexity class boundaries.

Findings

Hierarchy of functions characterizing O(n^k) time

Diagonalization to obtain exponential functions

Representation via dependent typed lambda-calculus

Abstract

Predicative analysis of recursion schema is a method to characterize complexity classes like the class FPTIME of polynomial time computable functions. This analysis comes from the works of Bellantoni and Cook, and Leivant by data tiering. Here, we refine predicative analysis by using a ramified Ackermann's construction of a non-primitive recursive function. We obtain a hierarchy of functions which characterizes exactly functions, which are computed in O(n^k) time over register machine model of computation. For this, we introduce a strict ramification principle. Then, we show how to diagonalize in order to obtain an exponential function and to jump outside deterministic polynomial time. Lastly, we suggest a dependent typed lambda-calculus to represent this construction.