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In this paper we introduce a way to estimate a level of closeness of Cayleyautomatic groups to the class of automatic groups using a certain numericalcharacteristic. We characterize Cayley automatic groups which are not automaticin terms of this numerical characteristic and then study it for the lamplightergroup, the Baumslag--Solitar groups and the Heisenberg group.

Measuring closeness between Cayley automatic groups and automatic groups