A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

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

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ 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 $\operatorname{PTIME}$. 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 $\operatorname{PTIME}$, 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

NP-Logic Systems and Model-Equivalence Reductions

In this paper we investigate the existence of model-equivalence reduction between NP-logic systems which are logic systems with model existence problem in NP. It is shown that among all NP-systems with model checking problem in NP, the existentially quantified propositional logic (\exists PF) is maximal with respect to poly-time model-equivalent reduction. However, \exists PF seems not a maximal NP-system in general because there exits a NP-system with model checking problem D^P-complete