{"title":"关于单边测试仿射子空间","authors":"","doi":"10.1016/j.tcs.2024.114745","DOIUrl":null,"url":null,"abstract":"<div><p>We study the query complexity of one-sided <em>ϵ</em>-testing the class of Boolean functions <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where <span><math><mi>F</mi></math></span> is any finite field. We give polynomial-time <em>ϵ</em>-testers that ask <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>)</mo></math></span> queries. This improves the query complexity <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mo>|</mo><mi>F</mi><mo>|</mo><mo>/</mo><mi>ϵ</mi><mo>)</mo></math></span> in <span><span>[14]</span></span>. The almost optimality of the algorithms follows from the lower bound of <span><math><mi>Ω</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>)</mo></math></span> for the query complexity proved by Bshouty and Goldreich <span><span>[3]</span></span>.</p><p>We then show that any one-sided <em>ϵ</em>-tester with proximity parameter <span><math><mi>ϵ</mi><mo><</mo><mn>1</mn><mo>/</mo><mo>|</mo><mi>F</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> for the class of Boolean functions that describe <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo>)</mo></math></span>-dimensional affine subspaces and Boolean functions that describe axis-parallel <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo>)</mo></math></span>-dimensional affine subspaces must make at least <span><math><mi>Ω</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>+</mo><mo>|</mo><mi>F</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>Ω</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>+</mo><mo>|</mo><mi>F</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>n</mi><mo>)</mo></math></span> queries, respectively. This improves the lower bound <span><math><mi>Ω</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> that is proved in <span><span>[14]</span></span> for <span><math><mi>F</mi><mo>=</mo><mrow><mi>GF</mi></mrow><mo>(</mo><mn>2</mn><mo>)</mo></math></span>. We also give testers for those classes with query complexity that almost match the lower bounds.<span><span><sup>1</sup></span></span></p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On one-sided testing affine subspaces\",\"authors\":\"\",\"doi\":\"10.1016/j.tcs.2024.114745\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study the query complexity of one-sided <em>ϵ</em>-testing the class of Boolean functions <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where <span><math><mi>F</mi></math></span> is any finite field. We give polynomial-time <em>ϵ</em>-testers that ask <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>)</mo></math></span> queries. This improves the query complexity <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mo>|</mo><mi>F</mi><mo>|</mo><mo>/</mo><mi>ϵ</mi><mo>)</mo></math></span> in <span><span>[14]</span></span>. The almost optimality of the algorithms follows from the lower bound of <span><math><mi>Ω</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>)</mo></math></span> for the query complexity proved by Bshouty and Goldreich <span><span>[3]</span></span>.</p><p>We then show that any one-sided <em>ϵ</em>-tester with proximity parameter <span><math><mi>ϵ</mi><mo><</mo><mn>1</mn><mo>/</mo><mo>|</mo><mi>F</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> for the class of Boolean functions that describe <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo>)</mo></math></span>-dimensional affine subspaces and Boolean functions that describe axis-parallel <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo>)</mo></math></span>-dimensional affine subspaces must make at least <span><math><mi>Ω</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>+</mo><mo>|</mo><mi>F</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>Ω</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>+</mo><mo>|</mo><mi>F</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>n</mi><mo>)</mo></math></span> queries, respectively. This improves the lower bound <span><math><mi>Ω</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> that is proved in <span><span>[14]</span></span> for <span><math><mi>F</mi><mo>=</mo><mrow><mi>GF</mi></mrow><mo>(</mo><mn>2</mn><mo>)</mo></math></span>. We also give testers for those classes with query complexity that almost match the lower bounds.<span><span><sup>1</sup></span></span></p></div>\",\"PeriodicalId\":49438,\"journal\":{\"name\":\"Theoretical Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304397524003621\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524003621","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
We study the query complexity of one-sided ϵ-testing the class of Boolean functions that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where is any finite field. We give polynomial-time ϵ-testers that ask queries. This improves the query complexity in [14]. The almost optimality of the algorithms follows from the lower bound of for the query complexity proved by Bshouty and Goldreich [3].
We then show that any one-sided ϵ-tester with proximity parameter for the class of Boolean functions that describe -dimensional affine subspaces and Boolean functions that describe axis-parallel -dimensional affine subspaces must make at least and queries, respectively. This improves the lower bound that is proved in [14] for . We also give testers for those classes with query complexity that almost match the lower bounds.1
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.