# Representational Equality

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**Title:** Representational Equality

**Authors:** Sean Parent, Alex Stepanov

## 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