Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

Olivier Bournez, Daniel S. Graça, Amaury Pouly
2017 Journal of the ACM  
The outcomes of this paper 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
more » ... nary 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.
doi:10.1145/3127496 fatcat:zlpvyqydozfuxipjuyj76oet7y