TITLE: Which properties of finite simplicial sets can be computed?
QUESTION [11 upvotes]: A simplicial set $X$ is a a combinatorial model for a topological space $|X|$, its realization, and conversely every topological space is weakly equivalent to such a realization of a simplicial set. I am wondering, which properties of finite simplicial sets can be effectively (or even theoretically) computed using a computer program. Maybe there are some implementations and I am not aware of their existence, so I would be very pleased about any information.
The first problem is how to input the simplicial set (maybe that's not really a problem).
Does one know terminating algorithms for the following problems then? Are there other problems for which anything is known in this direction?

Is a given finite simplicial set a Kan complex?
Given two finite simplicial sets $X$ and $Y$, are $|X|$ and $|Y|$ homeomorphic?  
Given two finite simplicial sets $X$ and $Y$, are $|X|$ and $|Y|$ homotopy equivalent?
What are the homotopy/homology groups of a given Kan complex? 
Given a finite simplicial set $X$, what are the homotopy/homology groups of $|X|$?

etc.

REPLY [5 votes]: 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 $n\geq 2$, 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 $X$?