Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma

TL;DR

This paper introduces a new analysis method for deriving tight complexity upper bounds from termination proofs involving Dickson's Lemma, improving upon previous results in the context of bad sequences over (N^k, ≤).

Contribution

It provides a novel approach to extract complexity bounds from termination proofs using Ackermannian and primitive-recursive bounds, which was not previously well-understood.

Findings

Upper bounds on bad sequences over (N^k, ≤) are improved.

Derived bounds are essentially tight, matching the best possible complexity.

The method enhances understanding of complexity analysis in termination proofs.

Abstract

Dickson's Lemma is a simple yet powerful tool widely used in termination proofs, especially when dealing with counters or related data structures. However, most computer scientists do not know how to derive complexity upper bounds from such termination proofs, and the existing literature is not very helpful in these matters. We propose a new analysis of the length of bad sequences over (N^k,\leq) and explain how one may derive complexity upper bounds from termination proofs. Our upper bounds improve earlier results and are essentially tight.