Fractal Geometry in Complexity Classes

分形几何方面的复杂性类和计算复杂性的中心问题将研究使用资源有界的维度，完善资源有界的措施的方法。具体主题包括：多项式时间维度将用于理解NP和其他复杂性类的分形几何，重点是复杂性类与其完整集之间的自相似性。假设，如“NP有正p维”，意味着只有平均情况下的硬度将比较资源有限的措施假设，并研究其后果的计算复杂性。零1法律的资源有限的规模的复杂性类将研究无条件和去随机化假设。对比行为的多项式-将进一步研究不同阶标度维数的时间约简，以产生对完备性现象和随机预言假设失败的新见解。超越鞅方法的限制，本文的研究还包括一些超越计算复杂性的相关研究：维数与对数损失预测的等价性将用于将nite-state维数应用于普适预测。VC维数与分形维数的关系将在计算学习理论中得到发展和应用。构造维数与Hausdor的对应原理。维数将开发提供一个新的Kolmogorov复杂性的方法，以简化证明和建立新的结果在经典fractal geometrics.Intellectual merit：作为证明其在线书目，资源有限的措施一直是一个中心的方法计算复杂性与超过100篇论文，60名作者在过去的15年。建议的研究是一个具有挑战性的计划，细化这种方法使用资源有限的尺寸。这将产生对复杂性类别结构的进一步定量洞察，对资源有限测度的局限性的新理解，以及对资源有限测度假设的合理性的更好评估。这项研究还将在计算复杂性、信息论、学习理论和分形几何之间建立新的联系，预期长期效果是增加这些领域之间的跨学科互动。来自以前不同领域的概念和想法将通过资源限制维度的统一概念来应用，以产生新的思维方式。研究结果将作为技术报告和会议出版物广泛提供。将维持所有相关论文的在线书目，以帮助研究人员更好地查阅文献。来自代表性不足群体的研究生将在该项目中发挥重要作用。

Fractal-geometric aspects of complexity classes and central questions in computational complexity will be studied using resource-bounded dimension, refining the resource-bounded measure approach. Specific topics include:Polynomial-time dimension will be used to understand the fractal geometry of NP and other complexity classes, with an emphasis on the self-similarity between a complexity class and its complete sets. Hypotheses such as "NP has positive p-dimension" that imply only average-case hardness will be compared to resource-bounded measure hypotheses and investigated for their consequences in computational complexity.Zero-one laws for the resource-bounded dimensions of complexity classes will be investigated unconditionally and under derandomization assumptions.The contrasting behavior of polynomial-time reductions in different orders of scaled dimension will be further studied to yield new insight into completeness phenomena and the failure of the random oracle hypothesis.Moving beyond limitations of the martingale approach, compressibility will be employed to develop resource-bounded measure and dimension in small complexity classes.The proposed research also includes some related investigations that go beyond computational complexity:The equivalence of dimension and log-loss prediction will be used to apply .nite-state dimension to universal prediction. Relationships between VC-dimension and fractal dimension will be developed and applied in computational learning theory.Correspondence principles for constructive dimension and Hausdor. dimension will be developed to provide a new Kolmogorov-complexity method to simplify proofs and establish new results in classical fractal geometry.Intellectual merit: As evidenced by its online bibliography, resource-bounded measure has been a central approach to computational complexity with over 100 papers by 60 authors during the past 15 years. The proposed research is a challenging program that refines this approach using resource-bounded dimension. This will yield further quantitative insight into the structure of complexity classes, new understanding of the limitations of resource-bounded measure, and better assessment of the reasonableness of resource-bounded measure hypotheses.Broader impacts: This research will also establish new links between computational complexity, information theory, learning theory, and fractal geometry, with the expected long-term effect of increasing interdisciplinary interaction among those fields. Concepts and ideas from previously distinct areas will be applied through the unifying concept of resource-bounded dimension to yield new ways of thinking. The research results will be made widely available as technical reports and conference publications. Online bibliographies of all relevant papers will be maintained to help researchers with better access to the literature. Graduate students from underrepresented groups will play a significant role in the project.