Foundations of computing with continuous data: algorithms versus experiments with physical systems

n物理世界信息和数据用自然数0、1、2、3、..来表示或编码，通常用二进制表示。物理系统可以被构造来执行数字计算，例如查尔斯·巴贝奇的机械差分机或当代的电子计算机。数字计算机对二进制数执行逻辑和算术运算，并可用于通过使用已知的物理定律来预测机械系统的行为。即使是非常复杂和有用的系统，例如模拟飞机周围的气流，也是如此。然而，是否有可能构建一个物理系统，其行为无法由计算机预测？在这里，我们在理论和实践之间划清界限，像天气这样的混沌系统是一个我们可能必须非常准确地知道初始条件的系统，以便预测未来给定时间的事件。我们将考虑这样一个问题，即是否存在表现更差的系统，在这些系统中，没有任何可以想象的初始精度可以通过数字计算机提供合理的预测。这样的系统可能为计算机提供新的更强大的非数字技术。物理系统是用真实的数字来衡量的。真实的数，比如2的平方根，1.41421.是无限长的，难以在有限的数字计算机中处理。因此，就其本质而言，这个问题涉及计算机如何处理真实的数字的理论，以及它们如何控制真实的数字的计算精度。该项目的一个线程是找到新的方法来“精确地”编程与真实的数字，并将这些方法应用到力学系统，如一般相空间的哈密顿力学。这涉及到系统几何形状的可计算描述。它还涉及计算机如何与机械系统交互的精确规范，这里的一个目的是说，如果有某些限制，某些类别的机械系统是可计算的。给定一个遵循一定物理定律的系统，并使用特定的语言来操纵该系统，那么，什么时候实验结果可以由数字计算机预测呢？当这些结果应用于更一般的系统时，它们是如何分解的？另一条线索是找到新的机械系统，在某种意义上，这些系统是不可计算的，并分析它们。一方面，如果一个物理系统从一开始就被内置到它的设计中，那么它可以找到不可计算的量，尽管有时这会以一种非常不明显的方式发生。因此，在关于可计算性的辩论中，有必要包括机器的构造过程。

n the physical world information and data is represented or coded by the natural numbers 0,1,2,3,..., usually in binary notation. Physical systems can be constructed to carry out digital computation, for example Charles Babbage's mechanical Difference Engine or a contemporary electronic computer. Digital computers execute logical and arithmetic operations on binary numbers, and can be used to predict the behaviour of mechanical systems by using the known laws of physics. This is true even of very complicated and useful systems, such as modelling the airflow round an aircraft. However, is it possible to construct a physical system whose behaviour cannot be predicted by a computer? Here we draw a line between theory and practicality, a chaotic system such as the weather is one where we might have to know the initial conditions incredibly accurately in order to predict events a given time in the future. We will consider the question as to whether there are much worse behaved systems, ones in which no conceivable amount of initial accuracy can ever deliver a sensible prediction by a digital computer. Such systems might suggest new more powerful non-digital technologies for computers.Physical systems are measured using real numbers. Now real numbers, such as the square root of 2, 1.41421..., are infinitely long and difficult to handle in a finite digital computer. Therefore by its nature, this problem involves the theory of how computers deal with real numbers, and how they control the accuracy of their computations with real numbers. One thread of the project is to find new methods for programming 'exactly' with real numbers, and to apply these methods to mechanical systems such as Hamiltonian mechanics on general phase spaces. This involves a computable description of the geometry of the system. It also involves precise specification of how a computer can interact with a mechanical system.One aim here is to say that certain classes of mechanical system are computable if certain restrictions apply. Given a system which obeys certain physical laws, and with a specified language for manipulating the system, when are the results of experiments predictable by a digital computer? How do these results break down when applied to more general systems?The other thread is to find novel mechanical systems, which are , in some sense, not computable, and analyse them. One aspect is that a physical system can find non-computable quantities if they are built into its design from the start, though sometimes this occurs in a very non-obvious fashion. It is then necessary to include the process of construction of the machine in the debate about computability.