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Ackermannian Integer Compression and the Word Problem for Hydra Groups

Authors Will Dison, Eduard Einstein, Timothy R. Riley

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Will Dison
Eduard Einstein
Timothy R. Riley

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Will Dison, Eduard Einstein, and Timothy R. Riley. Ackermannian Integer Compression and the Word Problem for Hydra Groups. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.MFCS.2016.30

Abstract

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity.  The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions.  Here we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups.  Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by  strings of Ackermann functions.

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Keywords
  • Ackermann functions
  • hydra
  • word problem

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