IS WAVE PROPAGATION COMPUTABLE OR CAN WAVE COMPUTERS BEAT THE TURING MACHINE?

Abstract

According to the Church-Turing Thesis a number function is computable by the mathematically defined Turing machine if and only if it is computable by a physical machine. In 1983 Pour-El and Richards defined a three-dimensional wave $u(t,x)$ such that the amplitude $u(0,x)$ at time 0 is computable and the amplitude $u(1,x)$ at time 1 is continuous but not computable. Therefore, there might be some kind of wave computer beating the Turing machine. By applying the framework of Type 2 Theory of Effectivity (TTE), in this paper we analyze computability of wave propagation. In particular, we prove that the wave propagator is computable on continuously differentiable waves, where one derivative is lost, and on waves from Sobolev spaces. Finally, we explain why the Pour-El-Richards result probably does not help to design a wave computer which beats the Turing machine.2000 Mathematical Subject Classification: 03D80, 03F60, 35L05, 68Q05.