Turing Machines Can Be Efficiently Simulated by the General Purpose Analog Computer

Abstract

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.

The Multiple Paths to Multiple Life

“…Furthermore, at this point we also distinguish, as others have (Walker and Davies 2013), between two broad classes of computer-analog and digital-which differ with respect to both hardware and software and reflect a fundamental difference of design in their use of continuous versus discrete variables and differential versus discontinuous hardware elements-differences in L1 and L2. Nevertheless, both are able to realize the property of Turing completeness (Bournez et al 2013) the critical feature at L3.…”

Computable analysis with applications to dynamic systems

“…Analog computing devices based around differential equation models are natural to consider when working with real numbers. Their computational power can be shown to be essentially equivalent to that of Turing machines (Bournez et al 2006, 2013. However, Turing computation is some sense more practical, since it can be applied to discrete mathematical structures and those without a direct physical interpretation.…”