[Proof Complexity] A new preprint
Azza Gaysin
azza.gaysin at gmail.com
Tue Mar 12 09:13:23 CET 2024
Dear colleagues,
My new preprint "Proof complexity of universal algebra in a CSP dichotomy
proof" is available through ArXiv: https://arxiv.org/abs/2403.06704.
The paper is based on my thesis, which can be found here:
https://nevporyadke.github.io/PDF/Proof_complexity_of_CSP_AG.pdf.
Any comments are very welcome.
Sincerely, Azza Gaysin
Abstract:
The constraint satisfaction problem (CSP) can be formulated as a
homomorphism problem between relational structures: given a structure A,
for any structure X, whether there exists a homomorphism from X to A. For
years, it has been conjectured that all problems of this type are divided
into polynomial-time and NP-complete problems, and the conjecture was
proved in 2017 separately by Zhuk (2017) and Bulatov (2017). Zhuk's
algorithm solves tractable CSPs in polynomial time. The algorithm is partly
based on universal algebra theorems: informally, they state that after
reducing some domain of an instance to its strong subuniverses, a
satisfiable instance maintains a solution.
In this paper, we present the formalization of the proofs of these theorems
in the bounded arithmetic W^1_1 introduced by Skelley (2004). The
formalization, together with our previous results (2022), shows that W^1_1
proves the soundness of Zhuk's algorithm, where by soundness we mean that
any rejection of the algorithm is correct. From the known relation of the
theory to propositional calculus G, it follows that tautologies, expressing
the non-existence of a solution for unsatisfiable instances, have short
proofs in G.
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