A proof complexity conjecture and the Incompleteness theorem
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917270972792832 |
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| author | Krajicek, Jan |
| author_facet | Krajicek, Jan |
| contents | Given a sound first-order p-time theory $T$ capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that $T$ must be incomplete. We leave it as an open problem whether for some $T$ the range of $g_T$ intersects all infinite NP sets (i.e. whether it is a proof complexity generator hard for all proof systems).
A propositional version of the construction shows that at least one of the following three statements is true:
- there is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm),
- $E \not\subseteq P/poly$,
- there exists function $h$ that stretches all inputs by one bit, is computable in sub-exponential time and its range $Rng(h)$ intersects all infinite NP sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_10637 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A proof complexity conjecture and the Incompleteness theorem Krajicek, Jan Logic in Computer Science Logic 03F20, 03F40, 68Q15 F.1.3; F.4.1 Given a sound first-order p-time theory $T$ capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that $T$ must be incomplete. We leave it as an open problem whether for some $T$ the range of $g_T$ intersects all infinite NP sets (i.e. whether it is a proof complexity generator hard for all proof systems). A propositional version of the construction shows that at least one of the following three statements is true: - there is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm), - $E \not\subseteq P/poly$, - there exists function $h$ that stretches all inputs by one bit, is computable in sub-exponential time and its range $Rng(h)$ intersects all infinite NP sets. |
| title | A proof complexity conjecture and the Incompleteness theorem |
| topic | Logic in Computer Science Logic 03F20, 03F40, 68Q15 F.1.3; F.4.1 |
| url | https://arxiv.org/abs/2303.10637 |