On Randomized Computational Models and Complexity Classes: a Historical Overview

TL;DR

This paper provides a comprehensive historical overview of randomized computational models, clarifying their core features, differences, and relationships with counting models to enhance understanding of their development and computational power.

Contribution

It offers a detailed comparison of modern and original definitions of randomized models, clarifies terminology, and investigates links between probabilistic and counting complexity classes.

Findings

Clarified distinctions between randomized models and classes.

Compared modern definitions with original concepts.

Explored relationships between probabilistic and counting models.

Abstract

Since their appearance in the 1950s, computational models capable of performing probabilistic choices have received wide attention and are nowadays pervasive in almost every areas of computer science. Their development was also inextricably linked with inquiries about computation power and resource issues. Although most crucial notions in the field are well-known, the related terminology is sometimes imprecise or misleading. The present work aims to clarify the core features and main differences between machines and classes developed in relation to randomized computation. To do so, we compare the modern definitions with original ones, recalling the context in which they first appeared, and investigate the relations linking probabilistic and counting models.