Abstract
We investigate the minimal 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 non-unary case, we obtain the same result for double logarithmic space, which is also tight. When replacing the work tape with a few counters, we can still achieve similar results for unary linear-space two-counter automata, unary sublinear-space three-counter automata, and non-unary sublinear-space two-counter automata. We also show how to slightly improve the sublinear-space constructions by using more counters.
Highlights
When using uncountable transitions, bounded-error probabilistic and quantum models can recognize uncountably many languages [1, 8]
– Uncountably many unary languages can be defined by poly-time double logspace probabilistic Turing machines (PTMs) and linearithmic (O(n log n)) time log-space one-way PTMs
We investigate realtime probabilistic models that read the input in a streaming mode such that there is no pause on the input symbols. (This is referred as strict realtime.) On general alphabets, it is known that bounded-error one-way PTMs cannot recognize any nonregular language in space o(log log n) [5]
Summary
When using uncountable transitions, bounded-error probabilistic and quantum models can recognize uncountably many languages [1, 8]. – Uncountably many unary languages can be defined by poly-time double logspace probabilistic Turing machines (PTMs) and linearithmic (O(n log n)) time log-space one-way PTMs. (This is referred as strict realtime.) On general alphabets, it is known that bounded-error one-way PTMs cannot recognize any nonregular language in space o(log log n) [5]. We show that O(log log n)-space is enough for realtime PTMs to define uncountably many languages This bound is tight for general alphabets. We follow the same result for O(log n) space and we leave open whether realtime PTMs can recognize any unary nonregular languages in o(log n) space. We follow the same result for unary realtime probabilistic automata with counters and we show that two counters are sufficient. We first present the results for unary languages (Section 3.1), and for general alphabet languages (Section 3.2)
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