221 research outputs found

    Problems in number theory from busy beaver competition

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    By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the following one: How can a small Turing machine manage to produce very big numbers? It provides the following answer: mostly by simulating Collatz-like functions, that are generalizations of the famous 3x+1 function. These functions, like the 3x+1 function, lead to new unsolved problems in number theory.Comment: 35 page

    Quantum, Stochastic, and Pseudo Stochastic Languages with Few States

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    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.Comment: A new version with new results. Previous version: Arseny M. Shur, Abuzer Yakaryilmaz: Quantum, Stochastic, and Pseudo Stochastic Languages with Few States. UCNC 2014: 327-33
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