Analyzing the Counting Complexity of the Skolem Problem

Concepts de base

The author explores the #P-hardness and #P-completeness of counting zeros in linear recurrence sequences, focusing on the Skolem Problem.

Résumé

The content delves into the complexity of counting zeros in linear recurrence sequences, particularly focusing on the Skolem Problem. It discusses the NP-hardness of the problem, introduces a computational perspective on the Skolem-Mahler-Lech theorem, and establishes #P-hardness for variants like #Skolemω. The article also addresses related problems like LRSInclusion and provides insights into their computational complexities.
The content highlights key results such as the reduction from Subset Sum to #Skolemω, establishing its #P-hardness. It also presents algorithms for finding primes in arithmetic progressions and discusses open research questions regarding the Skolem Problem's decidability and complexity bounds.

Stats

Currently, decidability is only known for LRS of order at most 4.
The problem of counting zeros of a given LRS is shown to be #P-hard.
Upper bounds on the cardinality of zero sets are known.
For instances generated in their reduction, counting zeros is shown to be #P-complete.

Citations

"The presence of a zero in an integer linear recurrent sequence is NP-hard to decide." - Vincent D. Blondel and Natacha Portier
"Deciding ω-regular properties on linear recurrence sequences." - S. Almagor et al.
"LRSInclusion is ΠP2-hard." - G. Jindal and J. Ouaknine

Idées clés tirées de

On the Counting Complexity of the Skolem Problem

by Gora... à arxiv.org 03-04-2024

https://arxiv.org/pdf/2403.00098.pdf

Questions plus approfondies

Can advancements narrow down the gap between decidability and NP-hardness in the Skolem Problem

Advancements in the field of theoretical computer science and automata theory may potentially narrow down the gap between decidability and NP-hardness in the Skolem Problem. While currently, only LRS of orders up to 4 have known decidability results, further research could focus on developing new techniques or algorithms that can handle higher-order LRS. By exploring more sophisticated mathematical methods or computational approaches, researchers might be able to extend the decidability boundary for the Skolem Problem beyond its current limitations. Additionally, advancements in complexity theory and algorithm design could lead to breakthroughs that bridge the existing gap between decidability and NP-hardness.

Is it possible to show NP-hardness for LRS of constant order to improve understanding

Showing NP-hardness for LRS of constant order is a significant goal that could greatly enhance our understanding of linear recurrence sequences (LRS). Currently, NP-hardness has been established for high-order LRS but not for those of constant order. By proving NP-hardness specifically for LRS with a fixed order, researchers can delve deeper into the complexities inherent in these sequences at different levels of abstraction. This advancement would provide valuable insights into how various parameters such as order impact computational complexity within this domain. Furthermore, establishing NP-hardness for constant-order LRS would contribute to a more comprehensive understanding of their structural properties and algorithmic behavior.

What implications does determining un = 0 have for solving EquSLP efficiently

Determining whether un = 0 efficiently plays a crucial role in solving EquSLP (Equation Satisfiability Problem), which involves checking if certain exponential sums are equal to zero over integers. Efficiently verifying un = 0 given an integer n is essential because it directly relates to solving EquSLP instances where such conditions need to be satisfied. If there were deterministic algorithms capable of quickly confirming whether un equals zero based on specific values of u and n, it would significantly streamline EquSLP problem-solving processes by providing a reliable method for validating key equations efficiently.