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

Abstract

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\mathrm{PTIME}$ of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple characterization of $\mathrm{PTIME}$. This is the first time such 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. This extends to deterministic complexity classes above polynomial time. This may provide a new perspective on classical complexity, by giving a way to define complexity classes, like $\mathrm{PTIME}$, 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, i.e.~by using the framework of analysis.

References in corpus (3)

• Computational complexity of solving polynomial differential equations over unbounded domains
• Computing with Polynomial Ordinary Differential Equations
• Computational complexity of solving polynomial differential equations over unbounded domains with non-rational coefficients

Cited by in corpus (5)

• Rate-Independent Computation in Continuous Chemical Reaction Networks
• Rate-independent continuous inhibitory chemical reaction networks are Turing-universal
• CRN++: Molecular Programming Language
• Recursion schemes, discrete differential equations and characterization of polynomial time computation
• Continuous Ordinary Differential Equations and Transfinite Computations