Abstract
Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all these numbers of states are optimal. After this, we completely characterize the class of languages recognized by 1-state GFAs, which is the only nontrivial class of languages recognized by 1-state automata. Finally, we consider the variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive cutpoint, and present some results on their expressive power.
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Whether Moore-Crutchfield quantum finite automata define more languages when the cutpoint is changed from 0 to a value from the open interval (0, 1).
MC stands for Moore and Crutchfield who introduced the model (Moore and Crutchfield 2000).
A GFA with \(v_0=0\) or \(f=0\) recognizes either \(\varnothing \) or \(\varSigma ^*\). The same effect can be achieved by setting all transition numbers to 0. Hence we assume w.l.o.g. \(v_0,\,f \ne 0\). We also use the standard convention that \(0^0=1\).
From the geometric point of view, a 1-state positive GFA defines a hyperplane in \({\mathbb {R}}^n\) and accepts exactly the words having the ends of their Parikh vectors on the prescribed side of this hyperplane.
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Acknowledgments
Arseny M. Shur was partially supported under the Agreement 02.A03.21.0006 of 27.08.2013 between the Ministry of Education and Science of the Russian Federation and Ural Federal University. Abuzer Yakaryılmaz was partially supported by CAPES with Grant 88881.030338/2013-01, ERC Advanced Grant MQC, and FP7 FET Projects QALGO.
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Shur, A.M., Yakaryılmaz, A. More on quantum, stochastic, and pseudo stochastic languages with few states. Nat Comput 15, 129–141 (2016). https://doi.org/10.1007/s11047-015-9511-8
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DOI: https://doi.org/10.1007/s11047-015-9511-8