The wave equation with computable initial data such that its unique solution is not computable
The wave equation with computable initial data such that its unique solution is not computable
复制标题
具有可计算初始数据的波动方程，其唯一解不可计算
DOI：
10.1016/0001-8708(81)90001-3
复制
发表时间：
1981
影响因子：
1.7
通讯作者：
I. Richards
中科院分区：
文献类型：
--
作者：
M. B. Pour;I. Richards
我们考虑三维波动方程。众所周知，解U（X，y，z，t）由两个初始条件唯一确定：u和Au/在时间t= 0时的值。我们的问题是，可计算的初始数据能产生不可计算的解吗？答案是“是的”，并且可能出现两种完全不同的不可计算性。下面的定理1给出了一个例子，其中解U（X，y，z，t）在时空中的可计算点处取不可计算的真实的值。相反，定理2提供了一个例子，其中解将时空中的每个可计算点序列映射成一个可计算序列：然而U（X，y，z，t）不是一个可计算函数。[We顺便指出，本文中使用的“可计算性”是一个数学逻辑学家熟悉的技术术语：精确的定义将在下一节中详细说明。定理1的证明很短。对于定理2来说，这一点要复杂得多。应该指出的是，这些不可计算的波动方程的解是通常被称为“弱解”的类型-即，虽然连续，但它们不是在所有点上的二次可微。弱解描述了折痕、尖点和其他经常出现在波动现象模型中的不可微模式。弱解的使用是不可避免的;我们将证明不存在所需类型的Cz解。为了方便不熟悉弱解的读者，我们在附录中把它们放在一个连贯的框架中，以与波相关的“能量积分”为基础。(Just由于初始条件是可计算的，所以我们例子中的能量积分也是可计算的真实的。最后，证明所有的不可计算的解决方案必须是“弱”型的草图在本附录的末尾。然而，我们注意到，在阅读该文件的主要部分时，可以不参考增编。本文的结果与Kreisel的评论有关。在[4] Kreisel问是否现有的物理理论-例如，经典力学或量子力学-可以预测理论上存在的物理常数，这不是一个递归的真实的。作者以前在这一领域的工作[9]涉及常微分方程：证明了存在一个可计算的，因此是连续的函数F，使得215
We consider the three-dimensional wave equation. It is well known that the solution U (X, y, z, t) is uniquely determined by two initial conditions: the values of u and au/at at time t= 0. Our question is, can computable initial data give rise to noncomputable solutions? The answer is “yes,” and two quite different types of noncomputability can occur. Theorem 1 below gives an example in which the solution U (X, y, z, t) takes a noncomputable real value at a computable point in space-time. By contrast, Theorem 2 provides an example in which the solution maps each computable sequence of points in space-time into a computable sequence: nevertheless U (X, y, z, t) is not a computable function.[We note in passing that “computability,” as used in this paper, is a technical term familiar to mathematical logicians: the precise definitions are spelled out in the next section.] The proof of Theorem 1 is quite short. That for Theorem 2 is considerably more intricate. It should be mentioned that these noncomputable solutions of the wave equation are of the type commonly referred to as “weak solutions”-ie, although continuous, they are not twice differentiable at all points. Weak solutions describe creases, cusps and other non-differentiable patterns which frequently appear in models of wave phenomena. The use of weak solutions is inevitable; we will prove that no Cz solutions of the desired kind exist. For the convenience of readers unfamiliar with weak solutions, we have put them into a coherent framework in an Addendum, basing our presentation on the “energy integral” associated with the wave.(Just as the initial conditions are computable, the energy integral in our examples is also a computable real.) Finally, the proof that all noncomputable solutions must be of “weak” type is sketched at the end of this addendum. We note, however, that the main part of the paper may be read without reference to the addendum. The results in this paper are related to comments of Kreisel. In [4] Kreisel asks whether existing physical theories-eg, classical mechanics or quantum mechanics-can predict theoretically the existence of a physical constant which is not a recursive real. Previous work of the authors in this area [9] was concerned with ordinary differential equations: it was proved that there exists a computable-and hence continuous-function F such that 215