Extended Nullstellensatz proof systems
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915494415564800 |
|---|---|
| author | Krajicek, Jan |
| author_facet | Krajicek, Jan |
| contents | For a finite set $\cal F$ of polynomials over fixed finite prime field of size $p$ containing all polynomials $x^2 - x$ a Nullstellensatz proof of the unsolvability of the system $$
f = 0\ ,\ \mbox{ all } f \in {\cal F} $$ in the field is a linear combination $\sum_{f \in {\cal F}} \ h_f \cdot f$ that equals to $1$ in the ring of polynomails. The measure of complexity of such a proof is its degree: $\max_f deg(h_f f)$.
We study the problem to establish degree lower bounds for some {\em extended} NS proof systems: these systems prove the unsolvability of $\cal F$ by proving the unsolvability of a bigger set ${\cal F}\cup {\cal E}$, where set $\cal E$ may use new variables $r$ and contains all polynomials $r^p - r$, and satisfies the following soundness condition: -- - Any $0,1$-assignment $\overline a$ to variables $\overline x$ can be appended by an assignment $\overline b$ to variables $\overline r$ such that for all $g \in {\cal E}$ it holds that $g(\overline a, \overline b) = 0$.
We define a notion of pseudo-solutions of $\cal F$ and prove that the existence of pseudo-solutions with suitable parameters implies lower bounds for two extended NS proof systems ENS and UENS defined in Buss et al. (1996/97). Further we give a combinatorial example of $\cal F$ and candidate pseudo-solutions based on the pigeonhole principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10617 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extended Nullstellensatz proof systems Krajicek, Jan Logic Computational Complexity 03F20, 68Q15 F.1.1; F.4.1 For a finite set $\cal F$ of polynomials over fixed finite prime field of size $p$ containing all polynomials $x^2 - x$ a Nullstellensatz proof of the unsolvability of the system $$ f = 0\ ,\ \mbox{ all } f \in {\cal F} $$ in the field is a linear combination $\sum_{f \in {\cal F}} \ h_f \cdot f$ that equals to $1$ in the ring of polynomails. The measure of complexity of such a proof is its degree: $\max_f deg(h_f f)$. We study the problem to establish degree lower bounds for some {\em extended} NS proof systems: these systems prove the unsolvability of $\cal F$ by proving the unsolvability of a bigger set ${\cal F}\cup {\cal E}$, where set $\cal E$ may use new variables $r$ and contains all polynomials $r^p - r$, and satisfies the following soundness condition: -- - Any $0,1$-assignment $\overline a$ to variables $\overline x$ can be appended by an assignment $\overline b$ to variables $\overline r$ such that for all $g \in {\cal E}$ it holds that $g(\overline a, \overline b) = 0$. We define a notion of pseudo-solutions of $\cal F$ and prove that the existence of pseudo-solutions with suitable parameters implies lower bounds for two extended NS proof systems ENS and UENS defined in Buss et al. (1996/97). Further we give a combinatorial example of $\cal F$ and candidate pseudo-solutions based on the pigeonhole principle. |
| title | Extended Nullstellensatz proof systems |
| topic | Logic Computational Complexity 03F20, 68Q15 F.1.1; F.4.1 |
| url | https://arxiv.org/abs/2301.10617 |