# The minimal probabilistic and quantum finite automata recognizing uncountably many languages with fixed cutpoints

## TL;DR

This paper demonstrates that minimal probabilistic and quantum finite automata with fixed cutpoints can recognize uncountably many languages, using new constructions that associate each language with a specific automaton.

## Contribution

It introduces novel constructions showing that fixed cutpoint automata can recognize uncountably many languages, extending prior results that used varying cutpoints.

## Key findings

- Probabilistic automata with 2 states recognize uncountably many languages.
- Quantum automata with 2 states recognize uncountably many languages.
- New constructions for fixed cutpoint recognition are provided.

## Abstract

It is known that 2-state binary and 3-state unary probabilistic finite automata and 2-state unary quantum finite automata recognize uncountably many languages with cutpoints. These results have been obtained by associating each recognized language with a cutpoint and then by using the fact that there are uncountably many cutpoints. In this note, we prove the same results for fixed cutpoints: each recognized language is associated with an automaton (i.e., algorithm), and the proofs use the fact that there are uncountably many automata. For each case, we present a new construction.