formalisms of computability

an orthogonal approach to the fuzzification of both multisets and hybridsets is presented. in particular, we introduce $l$-multi-fuzzy and$l$-fuzzy hybrid sets, which are general enough and in spirit with thebasic concepts of fuzzy set theory. in addition, we study the properties ofthese structures. also, the usefulness of these structures is examined inthe framework of mechanical multiset proc...

Let h : N → Q be a computable function. A real number x is h-monotonically computable (h-mc, for short) if there is a computable sequence (xs) of rational numbers which converges to x in such a way that the ratios of the approximation errors are bounded by h. In this paper we discuss the h-monotonic computability of semi-computable real numbers which are limits of monotone computable sequences ...

We begin by revisiting partiality in univalent type theory via the notion of dominance. We then perform first steps in constructive computability theory, discussing the consequences of working with computability as property or structure, without assuming countable choice or Markov’s principle. A guiding question is what, if any, notion of partial function allows the proposition “all partial fun...

We are to discuss how to view notions of computability for discontinuous functions. We con ne ourselves to real-valued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk about computable functions and other mathematical objects just as one talks about continuous functions, integrable functions, etc. I...

Peter Wegner’s definition of computability differs markedly from the classical term as established by Church, Kleene, Markov, Post, Turing et al.. Wegner identifies interaction as the main feature of today’s systems lacking in the classical treatment of computability. We compare the different approaches and argue whether or not Wegner’s criticism is appropriate.

Abstract: Adding versatile interactions to imperative programming – C, Java and Android – is an essential task. Unfortunately, existing languages provide little constructs for user interaction. We propose a computability-logical approach [1] to user interaction. We illustrate our idea via C, an extension of the core C with a new choice statement. keywords: user interaction, choose, computabilit...

Computability theory formalizes the intuitive concepts of computation and information content. Pure computability studies these notions and how they relate to definability in arithmetic and set theory. Applied computability examines other areas of mathematics through the lens of effective processes. Tools from computability can be used to give precise definitions of concepts such as randomness....