Turing machines can be efficiently simulated by the General Purpose Analog Computer

TL;DR

This paper demonstrates that Turing machine computations can be efficiently simulated by the General Purpose Analog Computer (GPAC) at a complexity level comparable to classical models, bridging a gap in understanding analog computation.

Contribution

It shows, for the first time, that GPACs can simulate Turing machines efficiently at a complexity level, extending the Church-Turing thesis to analog models.

Findings

GPAC simulations are space-efficient for bounded computations

Turing computations can be polynomially reduced to GPAC computations

Establishes a complexity-level equivalence between GPAC and classical models

Abstract

The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynomial reductions. However, the situation is less clear in what concerns models of computation using real numbers, and no analog of the Church-Turing thesis exists for this case. Recently it was shown that some models of computation with real numbers were equivalent from a computability perspective. In particular it was shown that Shannon's General Purpose Analog Computer (GPAC) is equivalent to Computable Analysis. However, little is known about what happens at a computational complexity level. In this paper we shed some light on the connections between this two models, from a computational complexity level, by showing that, modulo polynomial reductions, computations of Turing machines can be simulated by GPACs, without the need of using more (space) resources than those used in the original Turing computation, as long as we are talking about bounded computations. In other words, computations done by the GPAC are as space-efficient as computations done in the context of Computable Analysis.