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.