论文标题
使用离散微分方程在多项式时间内计算的整数上的函数表征
A characterization of functions over the integers computable in polynomial time using discrete differential equations
论文作者
论文摘要
本文研究了离散的普通微分方程（ODE）的表达和计算能力，又称（普通）差异方程。它使用这些方程式作为计算和算法设计的中心工具提出了一个新框架。我们介绍了计算理论的离散ODE的一般理论，我们用算法的各种示例来说明这一点，并提供了复杂性和可计算性类别的几种隐含特征。 提出的框架介绍了关于复杂性和计算类别的原始观点。它统一了一些用于表征这些类别的构造，包括使用限制的递归方案中隐性复杂性的经典方法，以及最近通过连续的普通微分方程的类别对可计算性和复杂性的最新表征。它还有助于理解模拟计算与计算理论的经典离散模型之间的关系。 从更具技术性的角度来看，本文指出了线性（离散）ODES和经典ODE工具的基本作用，例如变量的变化以捕获可计算性和复杂性测量值，或者是编程许多算法的工具。
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes. The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory. At a more technical point of view, this paper points out the fundamental role of linear (discrete) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.
