{"title":"On the complexity of orbit word problems","authors":"Michael Maller","doi":"10.1016/j.jco.2025.101929","DOIUrl":null,"url":null,"abstract":"<div><div>In previous work we defined a computational saddle transition problem which arises in the dynamics of diffeomorphisms of the 2−dimensional torus, and proved this problem is in Oracle <strong>NP</strong>, working in a model of computation appropriate for Turing machine computations on problems defined over the real numbers. In this note we report further work on these problems, studying orbit descriptions represented as finite words in periodic points. We show these Orbit Word Problems are again in Oracle <strong>NP</strong>, in our model. Our methods also reveal structures in the set of realized orbit words, suggesting further applications in complexity.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"88 ","pages":"Article 101929"},"PeriodicalIF":1.8000,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Complexity","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X2500007X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In previous work we defined a computational saddle transition problem which arises in the dynamics of diffeomorphisms of the 2−dimensional torus, and proved this problem is in Oracle NP, working in a model of computation appropriate for Turing machine computations on problems defined over the real numbers. In this note we report further work on these problems, studying orbit descriptions represented as finite words in periodic points. We show these Orbit Word Problems are again in Oracle NP, in our model. Our methods also reveal structures in the set of realized orbit words, suggesting further applications in complexity.
期刊介绍：
The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited.
Areas Include:
• Approximation theory
• Biomedical computing
• Compressed computing and sensing
• Computational finance
• Computational number theory
• Computational stochastics
• Control theory
• Cryptography
• Design of experiments
• Differential equations
• Discrete problems
• Distributed and parallel computation
• High and infinite-dimensional problems
• Information-based complexity
• Inverse and ill-posed problems
• Machine learning
• Markov chain Monte Carlo
• Monte Carlo and quasi-Monte Carlo
• Multivariate integration and approximation
• Noisy data
• Nonlinear and algebraic equations
• Numerical analysis
• Operator equations
• Optimization
• Quantum computing
• Scientific computation
• Tractability of multivariate problems
• Vision and image understanding.
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