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We show in this article that uncomputability is also a relative property ofsubrecursive classes built on a recursive relative incompressible function,which acts as a higher-order "yardstick" of irreducible information for therespective subrecursive class. We define the concept of a Turing submachine,and a recursive relative version for the Busy Beaver function and for thehalting probability (or Chaitin's constant) Omega; respectively the Busy BeaverPlus (BBP) function and a time-bounded halting probability. Therefore, we provethat the computable BBP function defined on any Turing submachine is neithercomputable nor compressible by any program running on this submachine. Inaddition, we build a Turing submachine that can use lower approximations to itsown time-bounded halting probability to calculate the values of its Busy BeaverPlus function, in the "same" manner that universal Turing machines useapproximations to Omega to calculate Busy Beaver values. Thus, the algorithmicinformation carried by the BBP function is relatively incompressible (anduncomputable) at the same time that it still is occasionally reached bysubmachines. We point that this phenomenon enriches the research on therelativization and simulation of uncomputability and irreducible information.

Relativizing an incompressible number and an incompressible function through subrecursive extensions of Turing machines