Bounded-Depth Frege Lower Bounds for Random 3-CNFs via Deterministic Restrictions
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910587848491008 |
|---|---|
| author | Gryaznov, Svyatoslav Talebanfard, Navid |
| author_facet | Gryaznov, Svyatoslav Talebanfard, Navid |
| contents | A major open problem in proof complexity is to demonstrate that random 3-CNFs with a linear number of clauses require super-polynomial size refutations in bounded-depth Frege systems. We take the first step towards addressing this question by establishing a super-linear lower bound: for every $k$, there exists $ε_k > 0$ such that any depth-$k$ Frege refutation of a random $n$-variable 3-CNF with $Θ(n)$ clauses has $Ω(n^{1 + ε_k})$ steps w.h.p. Our proof involves a novel adaptation of the deterministic restriction technique introduced by Chaudhuri and Radhakrishnan (STOC'96).
For a given formula, this technique provides a method to fix a small number of variables in a bottom-up manner, ensuring that every surviving gate has small fan-in. Consequently, the resulting formula depends on a limited number of variables and can be simplified to a constant by a small variable assignment. Adapting this approach to proof complexity requires addressing the usual challenges associated with maintaining the hardness of a given instance. To this end, we introduce the following generalizations of standard proof complexity tools:
- Weak expanders: These bipartite graphs relax the classical notion of expansion by only requiring that small sets have a non-empty boundary, while intermediate-sized sets have a large boundary. This property is sufficient to preserve hardness (e.g., for resolution width) and is easier to maintain as we remove vertices from the graph.
- Formula assignments: To simplify a Frege proof, we consider a generalization of partial restrictions that assign values to formulas instead of just variables. We treat these assignments as new axioms added to our formula, as they generally cannot be expressed as variable substitutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounded-Depth Frege Lower Bounds for Random 3-CNFs via Deterministic Restrictions Gryaznov, Svyatoslav Talebanfard, Navid Computational Complexity A major open problem in proof complexity is to demonstrate that random 3-CNFs with a linear number of clauses require super-polynomial size refutations in bounded-depth Frege systems. We take the first step towards addressing this question by establishing a super-linear lower bound: for every $k$, there exists $ε_k > 0$ such that any depth-$k$ Frege refutation of a random $n$-variable 3-CNF with $Θ(n)$ clauses has $Ω(n^{1 + ε_k})$ steps w.h.p. Our proof involves a novel adaptation of the deterministic restriction technique introduced by Chaudhuri and Radhakrishnan (STOC'96). For a given formula, this technique provides a method to fix a small number of variables in a bottom-up manner, ensuring that every surviving gate has small fan-in. Consequently, the resulting formula depends on a limited number of variables and can be simplified to a constant by a small variable assignment. Adapting this approach to proof complexity requires addressing the usual challenges associated with maintaining the hardness of a given instance. To this end, we introduce the following generalizations of standard proof complexity tools: - Weak expanders: These bipartite graphs relax the classical notion of expansion by only requiring that small sets have a non-empty boundary, while intermediate-sized sets have a large boundary. This property is sufficient to preserve hardness (e.g., for resolution width) and is easier to maintain as we remove vertices from the graph. - Formula assignments: To simplify a Frege proof, we consider a generalization of partial restrictions that assign values to formulas instead of just variables. We treat these assignments as new axioms added to our formula, as they generally cannot be expressed as variable substitutions. |
| title | Bounded-Depth Frege Lower Bounds for Random 3-CNFs via Deterministic Restrictions |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2403.02275 |