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Minsky machines and algorithmic problems

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 Added by Mark Sapir
 Publication date 2015
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and research's language is English
 Authors Mark Sapir




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This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.



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In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography.
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For every Turing machine, we construct an automaton group that simulates it. Precisely, starting from an initial configuration of the Turing machine, we explicitly construct an element of the group such that the Turing machine stops if, and only if, this element is of finite order.If the Turing machine is universal, the corresponding automaton group has an undecidable order problem. This solves a problem raised by Grigorchuk.The above group also has an undecidable Engel problem: there is no algorithm that, given g, h in the group, decides whether there exists an integer n such that the n-iterated commutator [...[[g,h],h],...,h]$ is the identity or not. This solves a problem raised by Bartholdi.
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