Concept and Definition of Complexity

TL;DR

This paper explores the dual notions of complexity as both a qualitative measure of understanding and a quantitative measure of difficulty, analyzing their formalization in the 20th century.

Contribution

It provides a comprehensive analysis of the historical development and formal definitions of complexity as both a qualitative and quantitative concept.

Findings

Complexity has dual meanings: qualitative and quantitative.

Formalizations of complexity emerged mainly in the 20th century.

Understanding of complexity has evolved with formal theories and definitions.

Abstract

The term {\em complexity} is used informally both as a quality and as a quantity. As a quality, complexity has something to do with our ability to understand a system or object -- we understand simple systems, but not complex ones. On another level, {\em complexity} is used as a quantity, when we talk about something being more complicated than another. In this chapter, we explore the formalisation of both meanings of complexity, which happened during the latter half of the twentieth century.