In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, we pointed 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. In this article, we extend the approach to a characterization of functions from the integers to the reals computable in polynomial time in the sense of computable analysis. In particular, we provide a characterization of such functions in terms of the smallest class of functions that contains some basic functions, and that is closed by composition, linear length ODEs, and a natural effective limit schema.
翻译:在最近的一篇文章中,从整数到多元时间可计算整数的函数类别已经使用离散的普通差分方程式(ODE)(ODE)(也称为有限差异)来定性。在这样做时,我们指出了线性(分解)ODE和古典ODE工具的基本作用,如变量的变化,以显示可计算性和复杂性计量,或作为编程工具。在本条中,我们把从整数到在可计算分析的多元时间可计算成真数的函数类别的方法扩大到计算分析意义上的可计算元数。特别是,我们用包含一些基本功能的最小的功能类别来描述这些函数,这些功能由构成、线性ODE和自然有效限值组合来封闭。