We can think of Kolmogorov complexity of a string $x$ as the length of the shortest program $P$ and input $y$ such that $x = P(y)$. Usually these programs are drawn from some Turing-complete set (like $P$ might be the description of a Turing machine, or it could be a program in LISP or C). Even when we look at resource-bounded Kolmogorov complexity, we still look at Turing machines but with some bounds on their runtime or space usage. One of the consequences of this, is that the complexity of a string is undecidable. This seems like an awkward feature.
What happens if we use non-Turing complete models of computation to define Kolmogorov complexity?
If we pick a restrictive enough model (say our model can only implement the identity), then the complexity of a string becomes decidable, although we also lose the invariance theorem. Is it possible to have a model strong enough to have complexity equal (upto a constant offset, or even a multiplicative factor) to the Turing-complete model, but weak enough to still allow the complexity of a string to be decidable? Is there a standard name for Kolmogorov complexity with non-Turing complete models of computation? Where could I read more about this?