The computational power of concurrent data types has been the focus of much recent research. Herlihy showed that such power may be measured by examining the type's ability to implement wait-free consensus. Jayanti argued that this "ability" could be measured in different ways, depending, for example, on whether or not read/write registers could be used in an implementation. He demonstrated the significance of this distinction by exhibiting a nondeterministic type whose ability to implement consensus was increased with the availability of registers. We show that registers cannot increase the computational power (to implement consensus) of any deterministic type or of any type that can implement 2-process consensus. These results significantly impact upon the study of the wait-free hierarchies of concurrent data types. In particular, the combination of these results with other recent works shows that Jayanti's hm hierarchy is robust for deterministic types.