It just dawned on me, we can express an analog of the Kolmogorov Complexity that measures the runtime complexity of an algorithm. Specifically, let and be equivalent functions run on the same UTM , in that , for all . During the operation of the functions, the tape of the UTM will change. Simply count the number of changes to the tape, which has units bits, which will allow us to compare the runtimes of the functions in the same units as the Kolmogorov Complexity. As a general matter, we can define a measure of runtime complexity, , given by the number of bits changed during the runtime of as applied to .

Interestingly, my model of physics implies an equivalence between energy and information, and so a change in the information content of a system must be the result of acceleration. See Equation 10 of, A Computational Model of Time-Dilation. This connects computation with forces applied to systems, which are obviously related, since you need to apply a force to change the state of a tape in a UTM. Note there’s a bit of pedantry to this, since a changed bit is not necessarily the same as the units of bits themselves, but in any case, it nonetheless measures runtime complexity in some form of bits.