Complexity theory for operators in analysis

Abstract

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 by 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 highertype 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 PSPACEcompleteness 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. Categories and Subject Descriptors General TermsTheory This is a preprint.

Analytical properties of resource-bounded real functionals

“…Indeed, points of a given size necessarily live in a compact subset so if the size of each point is a natural number then the space must be a countable union of compact sets. Recently Kawamura and Cook [11] developed a framework applicable to the space C[0, 1] (which is not σ-compact), using higher-order complexity theory and in particular second-order polynomials. In particular their theory enables them to prove uniform versions of older results about the complexity of solving differential equations, as well as new results [12,13].…”

Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

“…A typical feature of such results is that they assert the existence of computable objects, but the computational complexity of these objects is unbounded. This is also the case here: using similar techniques as in [41], we can strengthen Theorem 5.1 (ii) to assert for every nonempty co-semi-decidable and convex A ⊆ K the existence of a polynomial-time computable nonexpansive f : K → K such that Fix(f ) = A, at least in the case where K is computably compact, and so in particular in the finite-dimensional case (if K is not computably compact there is no uniform majorant on the names of the points in K, so one would have to work in the framework of second-order complexity [39]). This allows us to characterise the computational complexity of fixed points of Lipschitz-continuous polynomial-time computable functions according to their Lipschitz constant.…”

Function Spaces for Second-Order Polynomial Time

“…This defines the class of (polynomial-time) computable functions from Reg to Reg. We can suitably define some other complexity classes related to nondeterminism or space complexity, as well as the notions of reduction and hardness [15].…”

“…[19]). This was due to the absence of a sufficiently general theory of secondorder polynomial-time computability -a gap which was filled by Cook and the first author in [15]. This theory can be considered as a refinement of the computability theory.…”