On Randomized Computational Models and Complexity Classes: a Historical Overview

TL;DR

The paper addresses historical definitions and terminology of randomized computation, clarifying core features and differences among probabilistic and counting models, and relating machine models to complexity classes. It surveys early probabilistic machines, probabilistic Turing machines, threshold and counting devices, and the resulting complexity classes, highlighting how multiple definitions can coexist and link to familiar classes. It highlights nuanced distinctions between Santos and Gill PTMs, as well as between threshold and counting machines, and shows how modern classes such as PP, BPP, ZPP, and #P connect to these historical models. The work culminates in a clarified historical account that reduces terminological confusion and aids interpretation of randomized computation across computer science.

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.