Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma

Abstract

Dickson's Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and model-checking, constraint solving, logic, etc. While Dickson's Lemma is well-known, most computer scientists are not aware of the complexity upper bounds that are entailed by its use. This is mainly because, on this issue, the existing literature is not very accessible.We propose a new analysis of the length of bad sequences over (N k , ≤), improving on earlier results and providing upper bounds that are essentially tight. This analysis is complemented by a "user guide" explaining through practical examples how to easily derive complexity upper bounds from Dickson's Lemma.

Trace inclusion for one-counter nets revisited

“…We will see that this procedure is optimal: The problem is Ackermannian, i.e. it is non-primitive recursive and lies exactly at level ω of the Fast Growing Hierarchy [4].…”

“…We show non-primitive recursiveness by reduction from the control-state reachability problem for incrementing counter machines [3,4].…”

“…For the rest of this section, we recall a recent result from Figueira, Figueira, Schmitz, and Schnoebelen [4], that allows us to provide the exact complexity of the OCN trace universality problem in terms of its level in the Fast-Growing Hierarchy.…”

Lossy Counter Machines Decidability Cheat Sheet

“…In particular, an Ackermann upper bound holds for most decidable problems on lossy counter machines, and this can be refined to primitive-recursive upper bounds at various levels when one restricts attention to machines with a fixed number of counters. We refer to our upcoming paper for more details [23].…”

The Ideal View on Rackoff’s Coverability Technique

“…A striking illustration is provided by vector addition systems (VAS): the complexity bounds offered e.g. by [8] are in Ackermann, whereas coverability in VAS has long been known to be ExpSpace-complete thanks to a lower bound by Lipton [14] and an upper bound by Rackoff [16].…”