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
On the Complexity of Convex Hulls of Subsets of the Two-Dimensional Plane
Electronic Notes in Theoretical Computer Science (ENTCS)
Polynomial and abstract subrecursive classes
Journal of Computer and System Sciences
Zone diagrams in Euclidean spaces and in other normed spaces
Proceedings of the twenty-sixth annual symposium on Computational geometry
Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete
Computational Complexity - Selected papers from the 24th Annual IEEE Conference on Computational Complexity (CCC 2009)
On the Query Complexity of Real Functionals
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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We propose a new framework for discussing 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, which we call regular 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 important advantage of using regular functions is that we can define their size in the way inspired by higher-type complexity theory. This enables us to talk about computation on regular 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. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of them. As an interesting example, the task of numerical algorithms for solving the initial value problem 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 could only state how complex the solution can be. We now reformulate them to show that the operator itself is polynomial-space complete.