Uncountable classical and quantum complexity classes

TL;DR

This paper explores how various restricted classical and quantum Turing machine models can recognize uncountably many languages within limited space bounds, extending prior results on unbounded models.

Contribution

It demonstrates that even highly restricted models like constant-space QTMs and logarithmic-space PTMs can recognize uncountably many languages, revealing new complexity boundaries.

Findings

Double logarithmic space suffices for unary PTMs in sweeping mode.

Logarithmic space enables unary quantum models with counters.

Small non-constant space is enough for binary PTMs with counters.

Abstract

Polynomial--time constant--space quantum Turing machines (QTMs) and logarithmic--space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error (Say and Yakary\i lmaz 2014, arXiv:1411.7647). In this paper, we investigate more restricted cases for both models to recognize uncountably many languages with bounded error. We show that double logarithmic space is enough for PTMs on unary languages in sweeping reading mode or logarithmic space for one-way head. On unary languages, for quantum models, we obtain middle logarithmic space for counter machines. For binary languages, arbitrary small non-constant space is enough for PTMs even using only counter as memory. For counter machines, when restricted to polynomial time, we can obtain the same result for linear space. For constant--space QTMs, we follow the result for a restricted sweeping head, known as restarting realtime.