{"title":"Iterated multiplication in \\( VTC ^0\\)","authors":"Emil Jeřábek","doi":"10.1007/s00153-021-00810-6","DOIUrl":null,"url":null,"abstract":"<div><p>We show that <span>\\( VTC ^0\\)</span>, the basic theory of bounded arithmetic corresponding to the complexity class <span>\\(\\mathrm {TC}^0\\)</span>, proves the <span>\\( IMUL \\)</span> axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the <span>\\(\\mathrm {TC}^0\\)</span> iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, <span>\\( VTC ^0\\)</span> can also prove the integer division axiom, and (by our previous results) the <span>\\( RSUV \\)</span>-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories <span>\\(\\Delta ^b_1\\text{- } CR \\)</span> and <span>\\(C^0_2\\)</span>. As a side result, we also prove that there is a well-behaved <span>\\(\\Delta _0\\)</span> definition of modular powering in <span>\\(I\\Delta _0+ WPHP (\\Delta _0)\\)</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-021-00810-6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 3
Abstract
We show that \( VTC ^0\), the basic theory of bounded arithmetic corresponding to the complexity class \(\mathrm {TC}^0\), proves the \( IMUL \) axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the \(\mathrm {TC}^0\) iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, \( VTC ^0\) can also prove the integer division axiom, and (by our previous results) the \( RSUV \)-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories \(\Delta ^b_1\text{- } CR \) and \(C^0_2\). As a side result, we also prove that there is a well-behaved \(\Delta _0\) definition of modular powering in \(I\Delta _0+ WPHP (\Delta _0)\).
期刊介绍：
The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
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