[Paper Review] A Survey on Continuous Time Computations

Olivier Bournez, Manuel L. Campagnolo|ArXiv.org|Jul 17, 2009

Network Time Synchronization Technologies4 citations

TL;DR

This survey provides a comprehensive overview of continuous time computation theories, examining their computational power and complexity through models like polynomial ODEs, GPAC, and real recursive functions. It establishes that analytic continuous time systems can simulate Turing machines, suggesting equivalent computability to digital computation, and highlights emerging unification in models despite open challenges in complexity and robustness.

ABSTRACT

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.

Motivation & Objective

• To synthesize and update the state of research on continuous time computation theories, building on Orponen's 1997 survey.
• To clarify the computational power of continuous time models, particularly in relation to classical computability and the Church-Turing thesis.
• To investigate the feasibility of a unified complexity theory for continuous time systems, given the lack of consensus on time and input size definitions.
• To examine robustness and noise tolerance in analog systems, especially in the context of undecidability and verification of hybrid systems.
• To identify open problems and research directions, particularly in extending computable analysis to complexity and robustness.

Proposed method

• Surveying and categorizing models of continuous time computation, including analog machines, hybrid systems, and ODE-based systems.
• Analyzing the equivalence between functions computable via polynomial ODEs, GPAC, and real computable functions under recursive analysis.
• Applying computable analysis as a foundational framework to study complexity and computability of continuous time systems.
• Reviewing results on undecidability and robustness, particularly in systems with noise or bounded perturbations.
• Comparing continuous time models with classical digital computation and other non-classical models (e.g., neural networks, quantum, optical).
• Synthesizing recent advances in equivalence results (e.g., [Graça et al., 2005], [Bournez et al., 2007]) to suggest a path toward a unified theory.

Experimental results

Research questions

• RQ1 Can continuous time systems, particularly analytic ODEs, achieve the same computational power as Turing machines?
• RQ2 To what extent are different models of continuous time computation—such as GPAC, polynomial ODEs, and real recursive functions—computationally equivalent?
• RQ3 What are the fundamental obstacles to developing a robust complexity theory for continuous time systems, and how can computable analysis help?
• RQ4 How does noise or imprecision affect the decidability and computational power of continuous time systems?
• RQ5 Under what conditions does undecidability in continuous time systems persist under robust or perturbed dynamics?

Key findings

• Polynomial ODEs, GPAC, and real computable functions under recursive analysis are computationally equivalent, suggesting a potential unification of continuous time computation models.
• Analytic continuous time systems can simulate Turing machines in unbounded state spaces, implying that analog and digital computation are equally powerful in terms of computability.
• Despite progress, a consensus on complexity measures—such as computation time and input size—remains elusive for continuous time models.
• The Strong Turing Thesis may hold for systems of analytic ODEs, but upper bounds are still unknown for general Lipschitzian ODEs and Hopfield networks.
• Robustness to noise remains poorly understood, with open questions on whether undecidability persists under bounded or non-deterministic noise.
• There is growing evidence that a unified framework for continuous time computation—akin to classical computability theory—may be achievable, especially through analytic ODEs and computable analysis.