Computability in Analysis and Physics

Abstract

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the first publication in the Perspectives in Logic series, Pour-El and Richards present the first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning. The book focuses on the computability or noncomputability of standard processes in analysis and physics. Topics include classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The work is self-contained, and although it is intended primarily for logicians and analysts, it should also be of interest to researchers and graduate students in physics and computer science.

Computability and complexity of Julia sets: a review

“…First, we introduce the classical notions of computability introduced by Turing [14], and computable real functions introduced by Pour-El [13]. Turing computability is defined by rather a physical model of human computation, called a Turing machine, which is an automaton which consisting of finite internal states and a head to read/write symbols on an external tape.…”

“…A definition of a computable real function is given by M. Pour-El in the context of computable analysis [13]. …”

“…An unbounded linear operator in a function space can take an input computable real function into uncomputable outputs [13].…”

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Abstract: Since A. M. Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1,10,13]. Some are of interest in nonlinear and statistical physics while others are extensions of the mathematical theory of computation.

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Domain theory and differential calculus (functions of one variable)

“…In Martin (2000), using some domain-theoretic ideas, a notion of informatic derivative is defined, which for a C 1 function evaluates at each point to the absolute value of the derivative at that point. In computable analysis, the relation between the computability of a function and its derivative has been investigated (Pour-El andRichards 1988). In Weihrauch (2000), a representation of a C 1 function is given, by brute force, such that the representation of the function and that of its derivative, using a type 2 machine, are both computable.…”