The Proof Analysis Problem
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916026366558208 |
|---|---|
| author | Arteche, Noel Atserias, Albert de Rezende, Susanna F. Khaniki, Erfan |
| author_facet | Arteche, Noel Atserias, Albert de Rezende, Susanna F. Khaniki, Erfan |
| contents | Atserias and Müller (JACM, 2020) proved that for every unsatisfiable CNF formula $φ$, the formula $\operatorname{Ref}(φ)$, stating "$φ$ has small Resolution refutations", does not have subexponential-size Resolution refutations. Conversely, when $φ$ is satisfiable, Pudlák (TCS, 2003) showed how to construct a polynomial-size Resolution refutation of $\operatorname{Ref}(φ)$ given a satisfying assignment of $φ$. A question that remained open is: do all short Resolution refutations of $\operatorname{Ref}(φ)$ explicitly leak a satisfying assignment of $φ$?
We answer this question affirmatively by giving a polynomial-time algorithm that extracts a satisfying assignment for $φ$ given any short Resolution refutation of $\operatorname{Ref}(φ)$. The algorithm follows from a new feasibly constructive proof of the Atserias-Müller lower bound, formalizable in Cook's theory $\mathsf{PV_1}$ of bounded arithmetic.
Motivated by this, we introduce a computational problem concerning Resolution lower bounds: the Proof Analysis Problem (PAP). For a proof system $Q$, the Proof Analysis Problem for $Q$ asks, given a CNF formula $φ$ and a $Q$-proof of a Resolution lower bound for $φ$, encoded as $\neg \operatorname{Ref}(φ)$, whether $φ$ is satisfiable. In contrast to PAP for Resolution, we prove that PAP for Extended Frege (EF) is NP-complete.
Our results yield new insights into proof complexity: (i) every proof system simulating EF is (weakly) automatable if and only if it is (weakly) automatable on formulas stating Resolution lower bounds; (ii) we provide Ref formulas exponentially hard for bounded-depth Frege systems; and (iii) for every strong enough theory of arithmetic $T$ we construct unsatisfiable CNF formulas exponentially hard for Resolution but for which $T$ cannot prove even a quadratic lower bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Proof Analysis Problem Arteche, Noel Atserias, Albert de Rezende, Susanna F. Khaniki, Erfan Computational Complexity Logic in Computer Science Logic Atserias and Müller (JACM, 2020) proved that for every unsatisfiable CNF formula $φ$, the formula $\operatorname{Ref}(φ)$, stating "$φ$ has small Resolution refutations", does not have subexponential-size Resolution refutations. Conversely, when $φ$ is satisfiable, Pudlák (TCS, 2003) showed how to construct a polynomial-size Resolution refutation of $\operatorname{Ref}(φ)$ given a satisfying assignment of $φ$. A question that remained open is: do all short Resolution refutations of $\operatorname{Ref}(φ)$ explicitly leak a satisfying assignment of $φ$? We answer this question affirmatively by giving a polynomial-time algorithm that extracts a satisfying assignment for $φ$ given any short Resolution refutation of $\operatorname{Ref}(φ)$. The algorithm follows from a new feasibly constructive proof of the Atserias-Müller lower bound, formalizable in Cook's theory $\mathsf{PV_1}$ of bounded arithmetic. Motivated by this, we introduce a computational problem concerning Resolution lower bounds: the Proof Analysis Problem (PAP). For a proof system $Q$, the Proof Analysis Problem for $Q$ asks, given a CNF formula $φ$ and a $Q$-proof of a Resolution lower bound for $φ$, encoded as $\neg \operatorname{Ref}(φ)$, whether $φ$ is satisfiable. In contrast to PAP for Resolution, we prove that PAP for Extended Frege (EF) is NP-complete. Our results yield new insights into proof complexity: (i) every proof system simulating EF is (weakly) automatable if and only if it is (weakly) automatable on formulas stating Resolution lower bounds; (ii) we provide Ref formulas exponentially hard for bounded-depth Frege systems; and (iii) for every strong enough theory of arithmetic $T$ we construct unsatisfiable CNF formulas exponentially hard for Resolution but for which $T$ cannot prove even a quadratic lower bound. |
| title | The Proof Analysis Problem |
| topic | Computational Complexity Logic in Computer Science Logic |
| url | https://arxiv.org/abs/2506.16956 |