Lower Bounds on Tree-like Size and Space of Resolution Proofs over Linear Equations

Core Concepts

The paper studies characterizations of tree-like size and space of resolution proofs over linear equations (Res(⊕)) using combinatorial games. It introduces a class of extensible formulas and proves lower bounds on their tree-like size and space complexity.

Abstract

The paper considers the proof system Res(⊕), which is an extension of the resolution proof system and operates with disjunctions of linear equations over F2. The authors study characterizations of tree-like size and space of Res(⊕) refutations using combinatorial games.
Key highlights:

The authors introduce a class of extensible formulas and prove tree-like size lower bounds on this class using Prover-Delayer games, as well as space lower bounds. This class includes many classical combinatorial principles like the pigeonhole, ordering, and dense linear ordering principles.
For the pigeonhole principle PHP_m^n, the authors show that the size of any tree-like Res(⊕) refutation is at least 2^(n-1), and the space is at least n-1.
For the ordering principle Ordering_n, the authors prove that the size of any tree-like Res(⊕) refutation is at least 2^(n-2), and the space is at least n-2.
For the dense linear ordering principle DLO_n, the authors prove that it is (n-3)/3-extensible with respect to a certain subset of clauses, and derive corresponding lower bounds on tree-like size and space.
The authors also present the width-space relation for Res(⊕), generalizing the results by Atserias and Dalmau for resolution.

Stats

The paper presents the following key figures and metrics:

The size of any tree-like Res(⊕) refutation of PHP_m^n is at least 2^(n-1).
The space of any Res(⊕) refutation of PHP_m^n is at least n-1.
The size of any tree-like Res(⊕) refutation of Ordering_n is at least 2^(n-2).
The space of any Res(⊕) refutation of Ordering_n is at least n-2.
The dense linear ordering principle DLO_n is (n-3)/3-extensible with respect to a certain subset of clauses.

Quotes

"We introduce a class of extensible formulas and prove tree-like size lower bounds on it using Prover–Delayer games, as well as space lower bounds."
"For the pigeonhole principle PHP_m^n, the size of any tree-like Res(⊕) refutation is at least 2^(n-1), and the space is at least n-1."
"For the ordering principle Ordering_n, the size of any tree-like Res(⊕) refutation is at least 2^(n-2), and the space is at least n-2."
"For the dense linear ordering principle DLO_n, we prove that it is (n-3)/3-extensible with respect to a certain subset of clauses, and derive corresponding lower bounds on tree-like size and space."

Key Insights Distilled From

Resolution Over Linear Equations: Combinatorial Games for Tree-like Size and Space

by Svyatoslav G... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08370.pdf

Deeper Inquiries

What other classes of formulas, beyond the ones considered in this paper, can be shown to be extensible and lead to lower bounds on Res(⊕) proofs

In addition to the classes of extensible formulas discussed in the paper, such as the pigeonhole, ordering, and dense linear ordering principles, there are several other classes that can be shown to be extensible and lead to lower bounds on Res(⊕) proofs. One example is the principle of Ramsey theory, which deals with the emergence of order in large structures. By encoding Ramsey-type statements into CNF formulas and showing their extensibility properties, one can establish lower bounds on Res(⊕) proofs. Additionally, combinatorial principles related to graph theory, such as Turán's theorem or the existence of certain graph colorings, can also be formulated as extensible formulas to analyze the complexity of Res(⊕) proofs.

How do the lower bounds for Res(⊕) compare to those for stronger proof systems like Polynomial Calculus or Cutting Planes

The lower bounds for Res(⊕) proofs, as demonstrated in the paper, are often comparable to or even stronger than those for stronger proof systems like Polynomial Calculus or Cutting Planes. This is because Res(⊕) operates with disjunctions of linear equations over F2, allowing for specific combinatorial principles to be efficiently encoded and analyzed. The extensibility properties of certain classes of formulas in Res(⊕) can lead to significant tree-like size and space lower bounds, showcasing the power and efficiency of this proof system in handling combinatorial problems.

Can the techniques developed in this paper be extended to study the complexity of Res(⊕) proofs for other important combinatorial principles beyond the ones considered here

The techniques developed in the paper can indeed be extended to study the complexity of Res(⊕) proofs for other important combinatorial principles beyond the ones considered. By identifying classes of extensible formulas that capture the essence of various combinatorial principles, researchers can apply similar game-based strategies and analysis to establish lower bounds on Res(⊕) proofs for a wide range of problems. This approach allows for a systematic exploration of the computational complexity of different combinatorial principles within the framework of Res(⊕) and provides insights into the inherent difficulty of proving these principles within the specified proof system.