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Hydrodynamic and symbolic models of computation with advice

Robert Cardona

TL;DR

The work demonstrates that ideal fluids, modeled by the stationary Euler equations on compact 3D domains, can implement polynomial-time Turing machines with polynomial advice ($P/poly$), establishing a lower bound on the computational capacity of such continuous systems. It develops a rigorous time-simulation framework for conservative ODEs and leverages a Reeb-flow construction to realize these computations as stationary Euler flows on manifolds like $S^3$. In parallel, the authors introduce countable generalized shifts, a broad symbolic model that extends generalized shifts and realizes $P/poly$ in real-time, offering a real-time symbolic alternative to analog-dynamical computation. Together, the results illuminate the computational reach of both continuous hydrodynamic models and symbolic dynamical systems and provide a bridge between analog computation and classical complexity with advice.

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.

Hydrodynamic and symbolic models of computation with advice

TL;DR

The work demonstrates that ideal fluids, modeled by the stationary Euler equations on compact 3D domains, can implement polynomial-time Turing machines with polynomial advice (), establishing a lower bound on the computational capacity of such continuous systems. It develops a rigorous time-simulation framework for conservative ODEs and leverages a Reeb-flow construction to realize these computations as stationary Euler flows on manifolds like . In parallel, the authors introduce countable generalized shifts, a broad symbolic model that extends generalized shifts and realizes in real-time, offering a real-time symbolic alternative to analog-dynamical computation. Together, the results illuminate the computational reach of both continuous hydrodynamic models and symbolic dynamical systems and provide a bridge between analog computation and classical complexity with advice.

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 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.
Paper Structure (18 sections, 12 theorems, 60 equations)

This paper contains 18 sections, 12 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Symbolic dynamics
  3. Turing machines with polynomial advice
  4. Generalized shifts and analog shifts
  5. Alternative notation.
  6. The analog shift map
  7. Simulation with area-preserving diffeomorphisms
  8. Computation with advice in ideal fluid motion
  9. Simulation and complexity with conservative ODEs
  10. Stationary Euler flows computing $P/poly$
  11. Variations of the model
  12. Time complexity via orbit length
  13. Other hydrodynamical systems
  14. Countable generalized shifts and $P/poly$
  15. Countable generalized shifts
  16. ...and 3 more sections

Key Result

Theorem 1

Given a polynomial-time Turing machine with polynomial advice $(T,a)$, there exists a three-dimensional toroidal domain $U$ equipped with some Riemannian metric $g$, and a stationary solution to the Euler equations in $(U,g)$ that simulates $T$ in polynomial time.

Theorems & Definitions (30)

  • Theorem 1
  • Lemma 2
  • proof
  • Definition 3
  • Theorem 4
  • proof
  • Corollary 5
  • Definition 6
  • Remark 7
  • Remark 8
  • ...and 20 more