Constructive Analysis

Towards a Physically Meaningful Definition of Computable Discontinuous and Multi-valued Functions (Constraints)

“…Intuitively, a real number is computable if we can compute it with any desired accuracy. In more precise terms, a real number x is called computable if there exists an algorithm that, given a natural number n, returns a rational number r n which is 2 −n -close to x: |x − r n | ≤ 2 −n ; [1,3]. Computable metric spaces: motivation.…”

“…Computable functions are continuous. The problem with the above definition is that all the functions computable according to this definition are continuous; see, e.g., [1,3]. Thus, we cannot use this definition to check how well we can compute a discontinuous function.…”

“…Indeed, let x be a computable element of X, and let the computable positive values ε < ε ′ be given. Then, according to a known result from [1], we can find a computable value ε 0 ∈ (ε, ε ′ ) for…”

“…⇒ Let us now prove that if R is a computable relation in the sense of Definition 2, then R is computable compact set. For that, we must show how, given a computable positive real number α > 0, we can generate an α-net for this set R. First, we use that fact that X is a computable compact, and generate an (α/2)-net {x 1 …”

From Global to Local Constraints: A Constructive Version of Bloch’s Principle

“…In this paper, we will use the usual definitions of computable numbers, functions, compact spaces, etc. ; see, e.g., [1,5]. Proof.…”