# Uncountable realtime probabilistic classes

## TL;DR

This paper explores the computational power of realtime probabilistic machines, demonstrating that minimal space resources suffice to define uncountably many languages, with specific results for unary and binary languages.

## Contribution

It establishes tight bounds on space requirements for realtime probabilistic machines to recognize uncountably many languages, extending known results to new memory configurations.

## Key findings

- Logarithmic space suffices for unary languages on realtime PTMs
- Double logarithmic space is enough for binary languages, which is tight
- Limited memory counters can also recognize uncountably many unary languages

## Abstract

We investigate the minimum cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On binary case, we follow the same result for double logarithmic space, which is tight. When replacing the worktape with some limited memories, we can follow uncountable results on unary languages for two counters.