A Two-Layer Approach to the Computability and Complexity of Real Numbers

Computational Complexity of Real Functions

We devise imperative programming languages for verified real number computation where real numbers are provided as abstract data types such that the users of the languages can express real number computation by considering real numbers as abstract mathematical entities. Unlike other common approaches toward real number computation, based on an algebraic model that lacks implementability or transcendental computation, or finite-precision approximation such as using double precision computation that lacks a formal foundation, our languages are devised based on computable analysis, a foundation of rigorous computation over continuous data. Consequently, the users of the language can easily program real number computation and reason about the behaviours of their programs, relying on their mathematical knowledge of real numbers without worrying about artificial roundoff errors. As the languages are imperative, we adopt precondition–postcondition-style program specification and Hoare-style program verification methodologies. Consequently, the users of the language can easily program a computation over real numbers, specify the expected behaviour of the program, including termination, and prove the correctness of the specification. Furthermore, we suggest extending the languages with other interesting continuous data, such as matrices, continuous real functions, et cetera.Abstract taken directly from the thesis.E-mail: sewonpark17@gmail.comURL: https://sewonpark.com/thesis

We call a sequence [Formula: see text] of elements of a metric space nearly computably Cauchy if for every increasing computable function [Formula: see text] the sequence [Formula: see text] converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence [Formula: see text] of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte ( Theoretical Computer Science 284 ( 2002 ) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup ( Theory of Computing Systems 60 ( 2017 ) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings ( 2005 ) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte ( Theoretical Computer Science 284 ( 2002 ) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.

On the hierarchy and extension of monotonically computable real numbers

Guest editors’ introduction: Special issue on practical development of exact real number computation

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.

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.

Computational complexity of real functions

Type Two Theory of Effectivity and Real PCF

A Banach–Mazur computable but not Markov computable function on the computable real numbers

Reducibilities on real numbers

Turing machines, in particular, finite Mill automata can be considered as a means for giving corresponding subclasses of the class of real functions. From this point of view, finite Mili automata and generalized sequences of machines were considered in [I]. Here a real function f is defined by a finite automaton A in the following way. To determine the value f(x) of the automaton A one successively transforms the input object, the expansion of the real number x in the binary (decimal) system, into the output object, which is the binary (decimal) expansion of the number y, where y = f(x). Analogously for determining real functions one can use Turing machines. The subclass of the class of real functions arising here is actually a proper subclass of the class of C-computable functions, defined below. The latter is strictly included in the class of R-functions, or real functions, and that, in its own right, is contained in the class of V-functions. Actually, each continuous real functions turns out to be a real function. V-functions are given by indeterminate R-transformers, transforming binary representations of real numbers. R-transformers with finite memory, defining real functions are said to be finite reals. The latter are richer than Mill automata, defining real functions according to [I], in two respects: they can be asynchronous and can define functions which are not strictly real. For example, the function f(x) = 3x is defined by a finitely real, but is not strictly real. For finite reals, on the basis of [2] the solvability of the equivalence problem and of some other algorithmic questions is proved. This generalizes the corresponding results found in [i] for finite Mili automata and generalized succcession machines preserving length. The solvability is also established of the equivalence problem for single-valued indeterminate finite reals with finitely turning accumulators. An algebraic characterization is given of the class of partial C-computable functions. We note that in the papers of Red'ko [3, 4] the problem of describing the algebra of computable (partially computable) functions of a real variable was formulated as open. Bui effected the construction of the algebra of computable operations on the set of rational numbers [5].

In this chapter we introduce the formal definitions of computational complexity of real functions. We first review the definitions of computable real numbers and computable real functions and their basic properties. Then, the oracle Turing machine is introduced as the formal model for computing real functions. This model allows us to define the complexity measures for computing real functions in a natural way. The class of polynomial time computable real functions is then defined, and several characterizations of this class will be given. Other complexity classes of real numbers and real functions, such as NP real functions and log-space computable real functions, will be defined in later chapters.

In scientific computation and engineering real numbers are typically approximated by rational numbers which approximate, in principle, the real numbers up to any given precision. This means that they are treated as the limits of computable sequences of rational numbers where an effective error estimation is often expected. Although such approximations exist for many constants, they do not exist for all constants. More precisely, approximations with an effective control over the approximation error exist only for the computable real numbers. As long as we are interested in approximations of real numbers, the weakest condition we can ask for is that a real number can be approximated by a computable sequence of rational numbers without any information about the approximation error. These real numbers are called computably approximable. By relaxing and varying conditions on the knowledge one has about the approximation, one gets several natural classes between these two classes of real numbers. In this paper we review the most natural classes which have been investigated over the past decades with respect to this viewpoint. Most of these classes can be characterized in different ways, partly by purely mathematical properties and partly by their computational properties.

The maximum value problem and NP real numbers