homotopy theory - Which properties of finite simplicial sets can be computed?

In the simply connected case, essentially everything is in principle computable, by some very early work of E.H. Brown:

bib{MR0083733}{article}{
   author={Brown, Edgar H., Jr.},
   title={Finite computability of Postnikov complexes},
   journal={Ann. of Math. (2)},
   volume={65},
   date={1957},
   pages={1--20},
   issn={0003-486X},
   review={MR{0083733 (18,753a)}},
}

In particular, if $X$ and $Y$ are finite simplicial complexes then $[Sigma^n X,Sigma^n Y]$ is computable for $ngeq 2$, and for large $n$ this gives the group of stable homotopy classes of maps. However, I do not think that there are practical algorithms for many such questions, although I am not up to date on this. Probably the simplest case that I do not know is as follows: is there a practical algorithm to compute the complex $K$-theory $K^0(X)$ for a finite simplicial complex $X$?