{"title":"Unifying lower bounds for algebraic machines, semantically","authors":"Thomas Seiller , Luc Pellissier , Ulysse Léchine","doi":"10.1016/j.ic.2024.105232","DOIUrl":null,"url":null,"abstract":"<div><div>We present a new abstract method for proving lower bounds in computational complexity based on the notion of topological and measurable entropy for dynamical systems. It is shown to generalise several previous lower bounds results from the literature in algebraic complexity, thus providing a unifying framework for “topological” proofs of lower bounds. We further use this method to prove that <span>maxflow</span>, a <figure><img></figure> complete problem, is not computable in polylogarithmic time on parallel random access machines (<span>pram</span>s) working with real numbers. This improves on a result of Mulmuley since the class of machines considered extends the class “<span>pram</span>s without bit operations”, making more precise the relationship between Mulmuley's result and similar lower bounds on real <span>pram</span>s.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"301 ","pages":"Article 105232"},"PeriodicalIF":0.8000,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Information and Computation","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S089054012400097X","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 present a new abstract method for proving lower bounds in computational complexity based on the notion of topological and measurable entropy for dynamical systems. It is shown to generalise several previous lower bounds results from the literature in algebraic complexity, thus providing a unifying framework for “topological” proofs of lower bounds. We further use this method to prove that maxflow, a complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves on a result of Mulmuley since the class of machines considered extends the class “prams without bit operations”, making more precise the relationship between Mulmuley's result and similar lower bounds on real prams.
期刊介绍：
Information and Computation welcomes original papers in all areas of theoretical computer science and computational applications of information theory. Survey articles of exceptional quality will also be considered. Particularly welcome are papers contributing new results in active theoretical areas such as
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Computational complexity-
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Concurrency and distributed process theory-
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Decision problems in logic-
Design and analysis of algorithms-
Discrete optimization and mathematical programming-
Inductive inference and learning theory-
Logic & constraint programming-
Program verification & model checking-
Probabilistic & Quantum computation-
Semantics of programming languages-
Symbolic computation, lambda calculus, and rewriting systems-
Types and typechecking
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