On the computational complexity of ordinary differential equations
Information and Control
Feasible real functions and arithmetic circuits
SIAM Journal on Computing
Complexity theory of real functions
Complexity theory of real functions
A New Characterization of Type-2 Feasibility
SIAM Journal on Computing
The relative complexity of NP search problems
Journal of Computer and System Sciences
Computable analysis: an introduction
Computable analysis: an introduction
Effective Fixed Point Theorem over a Non-computably Separable Metric Space
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
On the Complexity of Real Functions
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The basic feasible functionals in computable analysis
Journal of Complexity
Complexity of Operators on Compact Sets
Electronic Notes in Theoretical Computer Science (ENTCS)
Computational Complexity: A Conceptual Perspective
Computational Complexity: A Conceptual Perspective
Polynomial and abstract subrecursive classes
Journal of Computer and System Sciences
Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete
Computational Complexity - Selected papers from the 24th Annual IEEE Conference on Computational Complexity (CCC 2009)
Computational Geometry: Theory and Applications
Computational complexity of smooth differential equations
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An advantage of using string functions is that we can define their size in a way inspired by higher-type complexity theory. This enables us to talk about computation on string functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. For example, the task of numerical algorithms for solving a certain class of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results only stated how complex the solution can be. We now reformulate them and show that the operator itself is polynomial-space complete.