{"title":"Kolmogorov complexity and nondeterminism versus determinism for polynomial time computations","authors":"","doi":"10.1016/j.tcs.2024.114747","DOIUrl":null,"url":null,"abstract":"<div><p>We call any consistent and sufficiently powerful formal theory that enables to algorithmically verify whether a text is a proof <strong>algorithmically verifiable mathematics</strong> (av-mathematics). We study the fundamental question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of av-mathematics.</p><p>Our goal is to show strong indications that nondeterminism is more powerful than determinism for polynomial time computations. To do that, we do not consider decision problems only, but also compression algorithms. We show that at least one of the following three claims must be true:</p><ul><li><span>(i)</span><span><p><figure><img></figure></p></span></li><li><span>(ii)</span><span><p>non-determinism is more powerful than determinism for polynomial-time compression</p></span></li><li><span>(iii)</span><span><p>for each polynomial-time compression algorithm there exists another one of the same asymptotic time complexity that compresses infinitely many strings logarithmically stronger</p></span></li></ul><p>Another surprising consequence of P = NP would be that time-bounded Kolmogorov complexity for any polynomial bound can be computed by deterministic algorithms in polynomial time.</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":"https://www.sciencedirect.com/science/article/pii/S0304397524003645/pdfft?md5=cc80c7c87843aa9d2e4febe406dc48b5&pid=1-s2.0-S0304397524003645-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524003645","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}
引用次数: 0
Abstract
We call any consistent and sufficiently powerful formal theory that enables to algorithmically verify whether a text is a proof algorithmically verifiable mathematics (av-mathematics). We study the fundamental question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of av-mathematics.
Our goal is to show strong indications that nondeterminism is more powerful than determinism for polynomial time computations. To do that, we do not consider decision problems only, but also compression algorithms. We show that at least one of the following three claims must be true:
(i)
(ii)
non-determinism is more powerful than determinism for polynomial-time compression
(iii)
for each polynomial-time compression algorithm there exists another one of the same asymptotic time complexity that compresses infinitely many strings logarithmically stronger
Another surprising consequence of P = NP would be that time-bounded Kolmogorov complexity for any polynomial bound can be computed by deterministic algorithms in polynomial time.
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.