Representational Equality

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**Title:** Representational Equality

**Authors:** Sean Parent, Alex Stepanov

#### Contents

## Abstract

## Introduction

## Body

Definitions -

- identical
- I(a, b) - same instance, &a == &b representationally equal: R(a, b) - representation is equal logically equal: L(a, b) - all corresponding parts are equal
- locational less than
- std::less<T*>(&a, &b) representationally less than:

I ==> R ==> L

- I - complexity
- small, constant time (pointer comparison)
- R - complexity
- average small constant time, worst case O(area)
- L - complexity
- may be significant - time to find corresponding parts
- identical
- &a == &b
- representationally equal
- R(a, b) ==>

Desired attributes for logical less symmetric with other operators - esp. + and *

Examples of types where < and/or == is problematic:

float and double - reason is NaN complex - operator < not defined because of symmetry with operator*() solution is provide representational_less priority_queue - operator == and operator < not defined because of complexity and resource consumption