Hydrodynamic and symbolic models of computation with advice

TL;DR

This paper demonstrates that ideal fluid models and certain symbolic systems can simulate complex computational processes, including Turing machines with advice, revealing new links between physical models and computational complexity.

Contribution

It introduces a novel connection between Euler fluid dynamics and $P/poly$ complexity class, and proposes a new symbolic system capable of real-time Turing machine simulation with advice.

Findings

Ideal fluids can simulate $P/poly$ class computations.

New symbolic systems can simulate Turing machines with advice in real-time.

Physical and symbolic models exhibit non-computable phenomena.

Abstract

Dynamical systems and physical models defined on idealized continuous phase spaces are known to exhibit non-computable phenomena, examples include the wave equation, recurrent neural networks, or Julia sets in holomorphic dynamics. Inspired by the works of Moore and Siegelmann, we show that ideal fluids, modeled by the Euler equations, are capable of simulating poly-time Turing machines with polynomial advice on compact three-dimensional domains. This is precisely the complexity class $P/poly$ considered by Siegelmann in her study of analog recurrent neural networks. In addition, we introduce a new class of symbolic systems, related to countably piecewise linear transformations of the unit square, that is capable of simulating Turing machines with advice in real-time, contrary to previously known models.