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 PTIME 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 PTIME. 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 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. 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.
It its true that one can sometimes convert the multiple use of operators solving ordinary differential equations into a single application @cite_27 , but this happens only in very specific cases, which do not seem to include the classes @math and @math . In particular, the application of nested continuous recursion (i.e. nested use of solving ordinary differential equations) may be needed using their constructions, whereas we define @math using only a simple notion of acceptance and only system of ordinary differential equations.
@cite_27
2014107656
In this paper we show that Shannon's general purpose analog computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleene's classical recursive function theory.We first consider the GPAC and several of its extensions to show that all these models have drawbacks and we introduce an alternative continuous-time model of computation that solves these problems. We also show that this new model preserves all the significant relations involving the previous models (namely, the equivalence with the differentially algebraic functions).We then continue with the topic of recursive functions over the reals, and we show full connections between functions generated by the model introduced so far and a particular class of recursive functions over the reals.