Computational Aspects of Σ-Definability over the Real Numbers without the Equality Test

2003 ◽  
pp. 330-344 ◽  
Author(s):  
Margarita Korovina
Keyword(s):  
Real Numbers ◽  
The Real ◽  
Computational Aspects ◽  
Equality Test

scholarly journals Recent Advances in Σ-definability over Continuous Data Types

2003 ◽  
Vol 10 (23) ◽  
Author(s):  
Margarita Korovina
Keyword(s):  
Expressive Power ◽  
Continuous Data ◽  
Data Types ◽  
Real Numbers ◽  
Recursively Enumerable Set ◽  
The Real ◽  
Recursively Enumerable ◽  
Definable Relations ◽  
Computational Aspects ◽  
Equality Test

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas.

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

On expressible sets and p-adic numbers

2011 ◽  
Vol 54 (2) ◽  
pp. 411-422
Author(s):  
Jaroslav Hančl ◽  
Radhakrishnan Nair ◽  
Simona Pulcerova ◽  
Jan Šustek
Keyword(s):  
Haar Measure ◽  
Real Numbers ◽  
The Real ◽  
Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

Models of the Real Numbers

2020 ◽  
pp. 203-221
Author(s):  
Lorenz Halbeisen ◽  
Regula Krapf
Keyword(s):  
Real Numbers ◽  
The Real

The Real Numbers

2020 ◽  
pp. 29-54
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Real Numbers ◽  
The Real

scholarly journals On Certain Approximation Problem Connected with the Sums of Subseries

2013 ◽  
Vol 55 (1) ◽  
pp. 37-45
Author(s):  
Roman Wituła ◽  
Konrad Kaczmarek ◽  
Edyta Hetmaniok ◽  
Damian Słota
Keyword(s):  
Approximation Problem ◽  
Real Numbers ◽  
The Real ◽  
The Given

Abstract In this paper a problem of approximating the real numbers by using the series of real numbers is considered. It is proven that if the given family of sequences of real numbers satisfies some conditions of set-theoretical nature, like being closed under initial subsequences and (additionally) possessing properties of adding and removing elements, then it automatically possesses some approximating properties, like, for example, reaching supremum of the set of sums of subseries.

Sign in / Sign up

Export Citation Format

Share Document