Tight Bounds on the Length of Strongly Monotone Descending Chains over the Natural Numbers

Core Concepts

The length of controlled strongly monotone descending chains of downwards-closed sets over the natural numbers Nd can be bounded by n2O(d), where n is the initial size and d is the dimension.

Abstract

The paper revisits the "ideal view" of the backward coverability algorithm for vector addition systems (VAS) in light of recent breakthroughs on the coverability problem. It shows that the controlled strongly monotone descending chains of downwards-closed sets over Nd that arise from the dual backward coverability algorithm also enjoy a tight n2O(d) upper bound on their length. This bound translates into the same bound on the running time of the backward coverability algorithm for VAS.
The analysis takes place in a more general setting than previous work, allowing to show the same results and improve on the 2EXPSPACE upper bound for the coverability problem in invertible affine nets. The key is a generalization of the notion of "thinness" from prior work, which is shown to be an inherent property of the order ideals appearing in such descending chains, rather than an a priori condition.
The paper first establishes a lemma on the thinness of order ideals in strongly monotone descending chains. It then uses this lemma to prove the main result on the length of such chains. The applications to VAS coverability and invertible affine nets follow from this general result.

Stats

The paper provides the following key metrics and figures:

The length of a (g, n0)-controlled strongly monotone descending chain of downwards-closed sets over Nd is bounded by Ld + 1, where Ld is defined inductively in equations (6-7).
The bounds Ni and Li satisfy Ni+1 = n · (Li + 2) and Li + 4 ≤ n3i·(lg d+1), where n is the initial size and d is the dimension.

Quotes

"The improved upper bound relies on the notion of a thin vector in Nd [30, Def. 3.6] (somewhat reminiscent of the "extractors" of Leroux [33])."
"Whereas thinness was posited a priori in the proof of Künnemann et al. [30, Thm. 3.3] and then shown to indeed allow a suitable decomposition of minimal covering executions and to eventually prove their result, here in the descending chain setting it is an inherent property of all the order ideals appearing in the chain, thereby providing a "natural" explanation for thinness."

Key Insights Distilled From

On the Length of Strongly Monotone Descending Chains over $\mathbb{N}^d$

by Sylv... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2310.02847.pdf

Deeper Inquiries

What other applications or extensions of the thinness lemma and length bound for strongly monotone descending chains could be explored

One potential application of the thinness lemma and length bound for strongly monotone descending chains could be in the analysis of other well-structured transition systems (WSTS) beyond vector addition systems (VAS) and invertible affine nets. For example, the techniques could be applied to coverability problems in systems like Petri nets, pushdown systems, or even more complex models like timed automata or hybrid systems. By establishing controlled strongly monotone descending chains and proving tight upper bounds on their length, it may be possible to improve the complexity analysis of coverability problems in these systems as well. Additionally, exploring the relationship between thinness and other structural properties of these systems could lead to further insights and applications in verification and formal methods.

How do the techniques developed in this paper relate to or differ from other approaches to analyzing the complexity of wqo-based algorithms, such as the use of extractors or the ideal view

The techniques developed in this paper, particularly the thinness lemma and the analysis of strongly monotone descending chains, offer a novel approach to analyzing the complexity of algorithms in the context of well-structured transition systems. While other approaches, such as the use of extractors or the ideal view, also aim to establish tight complexity bounds for wqo-based algorithms, the focus on controlled descending chains and the concept of thinness provides a unique perspective. By considering the inherent properties of order ideals and their relationships within descending chains, the paper offers a more direct and structured method for bounding the length of these chains and, consequently, the running time of algorithms like the backward coverability algorithm. This approach allows for a deeper understanding of the structural properties of the systems under study and how they impact algorithmic complexity.

Are there other classes of well-structured transition systems beyond VAS and invertible affine nets where the techniques in this paper could yield improved complexity bounds for the coverability problem

The techniques presented in this paper could potentially be applied to a variety of other classes of well-structured transition systems where the coverability problem is of interest. For example, extending the analysis to classes like Petri nets with additional features (such as inhibitor arcs or colored tokens), pushdown systems with specific properties, or even more specialized models like timed automata or hybrid systems could yield valuable insights. By adapting the concepts of controlled descending chains and thinness to these different classes of systems, it may be possible to derive improved complexity bounds for the coverability problem and enhance our understanding of their computational properties. This could open up new avenues for research in formal verification and algorithmic analysis of complex systems.