The Wave Propagator Is Turing Computable

Abstract

Pour-El/Richards [PER89] and Pour-El/Zhong [PEZ97] have shown that there is a computable initial condition f for the three dimensional wave equation u_tt = Δu, u(0; x) = f(x); u_t(0; x) = 0; t ε ℝ, x ε ℝ³, such that the unique solution is not computable. This very remarkable result might indicate that the physical process of wave propagation is not computable and possibly disprove Turing’s thesis. In this paper computability of wave propagation is studied in detail. Concepts from TTE, Type-2 theory of effectivity, are used to define adequate computability concepts on the spaces under consideration. It is shown that the solution operator of the Cauchy problem is computable on continuously differentiable initial conditions, where one order of differentiability is lost. The solution operator is also computable on Sobolev spaces. Finally the results are interpreted in a simple physical model.