The Complexity of Computing in Continuous Time: Space Complexity is Precision

Alapfogalmak

Space complexity corresponds to precision in continuous time computing models.

Kivonat

The content discusses the equivalence of discrete and continuous-time models, focusing on space complexity and precision. It explores algebraic characterizations and simulations of ODEs for computational analysis.
The authors present a method to encode Turing machine configurations into real numbers for simulation purposes. The discussion delves into the complexities of solving ODEs efficiently over compact domains and the challenges with non-compact domains.
Key points include the relationship between time complexity and length of solution curves, robustness concepts, and tractability in dynamical systems. The article highlights recent progress in characterizing functions over integers and reals using ODEs.
Overall, the content provides insights into the intricate nature of continuous time computing models and their implications for computational theory.

Statisztikák

Recent significant progress has linked classical time complexity to the length of solution curves in polynomial ODEs.
Polynomial space functions over integers and reals have been characterized using continuous ODEs.
Functions computable in polynomial time or space have been associated with specific classes of ODEs.
Encoding Turing machine configurations as real numbers enables simulations using dynamical systems.

Idézetek

"Space complexity corresponds to precision."
"Recent papers proposed algebraic characterisations of polynomial-time and -space complexity classes over the reals."
"A major consequence is that space complexity is provably related to numerical stability."
"The authors relate time with the length of solution curve of an ODE."
"The question remains whether we can give a simple equivalent defining space-complexity."

Főbb Kivonatok

The complexity of computing in continuous time

by Manon Blanc,... : arxiv.org 03-06-2024

https://arxiv.org/pdf/2403.02499.pdf

Mélyebb kérdések

How do continuous-time models impact traditional discrete computation theory

Continuous-time models have a significant impact on traditional discrete computation theory by providing a unifying mathematical model through ordinary differential equations (ODEs). These continuous-time models offer a clearer and more natural way to describe computations compared to discrete-time models. They allow for the simulation of complex systems with dynamics that evolve continuously over time, offering a different perspective on problem-solving and analysis. Continuous-time models also provide insights into the behavior of systems in real-world scenarios where time is considered as a continuous variable.

What are the implications of encoding Turing machine configurations as real numbers

Encoding Turing machine configurations as real numbers has several implications. By representing configurations using real numbers, we can map the infinite tape of a Turing machine onto an interval or set of intervals in the real number line. This encoding allows us to apply concepts from real analysis and dynamical systems theory to analyze the behavior and properties of Turing machines computationally. It enables us to simulate the execution of Turing machines using continuous ODEs, bridging the gap between discrete computational processes and continuous mathematical frameworks.

How does numerical stability affect space complexity in solving ODEs

Numerical stability plays a crucial role in determining space complexity when solving ODEs. In the context of ODE solvers, numerical stability refers to how well an algorithm maintains accuracy and precision over time while approximating solutions. A numerically stable ODE solver ensures that small errors introduced during computation do not propagate uncontrollably, leading to inaccurate results or divergent behaviors. When considering space complexity in solving ODEs, ensuring numerical stability is essential for maintaining efficiency and accuracy while managing memory usage effectively.