Uncountable classical and quantum complexity classes

Abstract

Polynomial-time constant-space quantum Turing machines (QTMs) and logarithmic-space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error Yakaryılmaz 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.1 As another "probabilistic" but unconventional model, ultrametric automata can also define uncountably many languages with 2 states [6].

Uncountable realtime probabilistic classes

“…Due to Fact 1 the membership of k ∈ I for LOGLOG(I) will be computed with probability at least 3 4 . Therefore language LOGLOG(I) is recognized correctly with probability at least (1 − ǫ) • 3 4 , which can be arbitrarily close to 3 4 by picking a suitable c. The space used on the work tape is linear in the length of the counter for |bin(i)|. The value of bin(i) is logarithmic to the length of input word, and so the length of the counter is double logarithmic to the input length.…”

“…It is interesting to identify the minimum resources that are sufficient to follow this result. Some of the known results [8,3] are as follows:…”

Probabilistic verification of all languages

“…When using arbitary real-valued transitions, quantum and probabilistic machines can recognize uncountably many languages with bounded error [1,6,7,13] because it is possible to encode an infinite sequence as a transition value and then determine its i-th bit by using some probabilistic or quantum experiments.…”

“…But, we obtain some other strong results for constant-space IPSs such that the binary (unary) languages having constantspace linear-time (quadratic-time) IPSs form an uncountable set. Remark that it is also open whether constant-space probabilistic machines can recognize uncountably many languages or not [6].…”