{"title":"Ramsey's theorem for pairs, collection, and proof size","authors":"L. Kolodziejczyk, Tin Lok Wong, K. Yokoyama","doi":"10.1142/s0219061323500071","DOIUrl":null,"url":null,"abstract":"We prove that any proof of a $\\forall \\Sigma^0_2$ sentence in the theory $\\mathrm{WKL}_0 + \\mathrm{RT}^2_2$ can be translated into a proof in $\\mathrm{RCA}_0$ at the cost of a polynomial increase in size. In fact, the proof in $\\mathrm{RCA}_0$ can be found by a polynomial-time algorithm. On the other hand, $\\mathrm{RT}^2_2$ has non-elementary speedup over the weaker base theory $\\mathrm{RCA}^*_0$ for proofs of $\\Sigma_1$ sentences. \nWe also show that for $n \\ge 0$, proofs of $\\Pi_{n+2}$ sentences in $\\mathrm{B}\\Sigma_{n+1}+\\exp$ can be translated into proofs in $\\mathrm{I}\\Sigma_{n} + \\exp$ at polynomial cost. Moreover, the $\\Pi_{n+2}$-conservativity of $\\mathrm{B}\\Sigma_{n+1} + \\exp$ over $\\mathrm{I}\\Sigma_{n} + \\exp$ can be proved in $\\mathrm{PV}$, a fragment of bounded arithmetic corresponding to polynomial-time computation. For $n \\ge 1$, this answers a question of Clote, Hajek, and Paris.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"1 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219061323500071","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 3
Abstract
We prove that any proof of a $\forall \Sigma^0_2$ sentence in the theory $\mathrm{WKL}_0 + \mathrm{RT}^2_2$ can be translated into a proof in $\mathrm{RCA}_0$ at the cost of a polynomial increase in size. In fact, the proof in $\mathrm{RCA}_0$ can be found by a polynomial-time algorithm. On the other hand, $\mathrm{RT}^2_2$ has non-elementary speedup over the weaker base theory $\mathrm{RCA}^*_0$ for proofs of $\Sigma_1$ sentences.
We also show that for $n \ge 0$, proofs of $\Pi_{n+2}$ sentences in $\mathrm{B}\Sigma_{n+1}+\exp$ can be translated into proofs in $\mathrm{I}\Sigma_{n} + \exp$ at polynomial cost. Moreover, the $\Pi_{n+2}$-conservativity of $\mathrm{B}\Sigma_{n+1} + \exp$ over $\mathrm{I}\Sigma_{n} + \exp$ can be proved in $\mathrm{PV}$, a fragment of bounded arithmetic corresponding to polynomial-time computation. For $n \ge 1$, this answers a question of Clote, Hajek, and Paris.
我们证明了$\mathrm理论中$\ for all \ Sigma ^0_2$句子的任何证明{WKL}_0+\mathrm{RT}^2_2$可以转换为$\mathrm中的证明{RCA}_0以多项式大小增加为代价。事实上,$\mathrm中的证明{RCA}_0$可以通过多项式时间算法找到。另一方面,对于$\Sigma_1$句子的证明,$\mathrm{RT}^2_2$比较弱的基础理论$\mathrm{RCA}^*_0$具有非初等加速。我们还证明了对于$n\ge0$,$\mathrm{B}\Sigma\{n+1}+\exp$中的$\Pi_{n+2}$句子的证明可以以多项式代价转换为$\mathrm{I}\Sigma_{n}+\exp$中的证明。此外,$\mathrm{B}\Sigma\{n+1}+\exp$在$\mathrm{I}\Sigmon\{n}+\exp$上的$\Pi_{n+2}$守恒性可以在$\math rm{PV}$中得到证明,$\math rm{PV}$是一个与多项式时间计算相对应的有界算术片段。对于$n\ge 1$,这回答了Clote、Hajek和Paris的问题。
期刊介绍:
The Journal of Mathematical Logic (JML) provides an important forum for the communication of original contributions in all areas of mathematical logic and its applications. It aims at publishing papers at the highest level of mathematical creativity and sophistication. JML intends to represent the most important and innovative developments in the subject.