Affiliation:
1. Ecole Polytechnique, LIX
2. Universidade do Algarve and Instituto de Telecomunicaçõe
3. Ecole Polytechnique, LIX and Department of Computer Science, University of Oxford
Abstract
The outcomes of this article are twofold.
Implicit complexity.
We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous elegant and simple characterization of P. We believe it is the first time complexity classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of Computable Analysis.
Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations.
Continuous-Time Models of Computation.
Our results can also be interpreted in terms of analog computers or analog models of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both in terms of computability and complexity, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both in terms of computability and complexity.
Funder
Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicações through the FCT project
DGA Project CALCULS and French National Research Agency (ANR) Project
European Unions Horizon 2020 research
Marie Skłodowska-Curie
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Cited by
21 articles.
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