RAIRO-Theor. Inf. Appl.
Volume 52, Number 2-3-4, April–December 2018
8th Workshop on Non-classical Models of Automata and Applications (NCMA 2016)
|Page(s)||111 - 126|
|Published online||07 February 2019|
Uncountable classical and quantum complexity classes★
Faculty of Computing, University of Latvia,
* Corresponding author: firstname.lastname@example.org
Accepted: 21 November 2018
It is known that poly-time constant-space quantum Turing machines (QTMs) and logarithmic-space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error (A.C. Cem Say and A. Yakaryılmaz, Magic coins are useful for small-space quantum machines. Quant. Inf. Comput. 17 (2017) 1027–1043). 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 obtain the result for a restricted sweeping head, known as restarting realtime.
Mathematics Subject Classification: 68Q05 / 68Q15 / 68Q75
Key words: Probabilistic and quantum computation / small-space bounds / unary languages / uncountable classes / counter machines
A preliminary version appeared as “Maksims Dimitrijevs, Abuzer Yakaryılmaz: Uncountable classical and quantum complexity classes. NCMA 2016: 131-146” . The arXiv number is 1608.00417.
© EDP Sciences, 2019
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