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

Abstract

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 (time and space) elegant and simple characterization of P. We believe it 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.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 model 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 at the computability and complexity level, 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 at a computability and at a computational complexity level. ACM Subject Classification IntroductionThe outcomes of this paper are twofold, and are concerning a priori not closely related topics. Implicit Complexity:Since the introduction of the P and NP complexity classes, much work has been done to build a well-developed complexity theory based on Turing Machines. In particular, classical computational complexity theory is based on limiting resources used by Turing machines, like time and space. Another approach is implicit computational complexity.The term "implicit" in "implicit computational complexity" can sometimes be understood in various ways, but a common point of these characterizations is that they provide (Turing or equivalent) machine-independent alternative definitions of classical complexity. Implicit characterization theory has gained enormous interest in the last decade. This has led to many alternative characterizations of complexity classes using recursive functions, function algebras, rewriting systems, neural networks, lambda calculus and so on.However, most of -if not all -these models or characterizations are essentially discrete: in particular they are based on underlying discrete time models working on objects which are essentially discrete such as words, terms, etc. that can be considered as being defined in a discrete space.Models of computation working on a continuous space have also been considered: they include Blum Shub Smale machines [4], and in some sense Computable A...

On Chemical Reaction Network Design by a Nested Evolution Algorithm

“…This result relies on previous results on analog computation and complexity showing the Turing completeness of polynomial initial value problems (PIVPs), i.e. numerical integration with arbitrary precision of polynomial ODEs from polynomial initial conditions [3]. More precisely, we showed This form of reactions based solely on catalytic synthesis and degradation by annihilation is given by the proof of Turing completeness but is not very appealing from a biochemical point of view.…”

Robust Real-time Computing with Chemical Reaction Networks

“…Lastly, we remark that sojourn time (arc length) is generally a better measure of runtime for CRNs [4]. In the case of bounded CRNs, however, state time suffices as it is always within a constant factor of sojourn time.…”

“…This means that after n seconds, the concentration of X is within 2 ´n of α, so the CRN gains one bit of accuracy every second. Huang et al also required that all species concentrations be bounded to avoid the so-called Zeno paradox of performing an infinite amount of computation in finite time using a fast-growing catalyst species [4]. When this restriction is lifted, the measure of time is no longer linear but rather a function of arc length.…”

CRN++: Molecular programming language

“…This means that f (t) has a solution f (t) = Ce t , which is unique by the Picard-Lindelöf theorem. Therefore, Ce t = (X gtY −X ltY ) 3 X gtY X ltY < 1/(X gtY X ltY ) since the concentrations are at most 1. This constrains X gtY X ltY < C −1 e −t , which implies that X gtY or X ltY converges to 0 exponentially quickly.…”

“…Deterministic (mass-action) chemical kinetics is Turing universal [9], thus in principle allowing the implementation of arbitrary programs in chemistry. Turing universality was demonstrated by showing that arbitrary computation can be embedded in a class of polynomial ODEs [3], and then implementing these polynomial ODEs with mass-action chemical kinetics. While these results establish a sound theoretical foundation and show the power of chemistry for handling computation tasks in general, translating and performing specific computational tasks can lead to infeasibly large and complex sets of chemical reactions.…”