Linear speed-up, information vicinity, and finite-state machines

Connections are shown between two properties of a machine model: linear speed-up and polynomial vicinity. In the context of the author's Block Move (BM) model, these relate to: `How long does it take to simulate a finite transducer S on a given input z?' This question is related to the century-old problem of finding economical representations for finite groups. Under some cost measures for computing S(z), the BM enjoys the linear speed-up property, but under more-realistic measures, and subject to a reasonable but unproved hypothesis, it has the antithetical property of a constant-factor time hierarchy.