On the Complexity of Finite Sorted Algebras

Abstract

The general problem of testing the isomorphism of two given finite algebras is known to be isomorphism complete, i.e. polynomially equivalent to the graph isomorphism problem (GI). It is easy to see that this fact still holds when sorts are introduced. However, this isomorphism problem is relevant only for algebras (or interpretations) of a fixed signature, and in some cases, according to the signature, is much simpler than the general problem. We therefore establish exactly for which signatures is the associated isomorphism problem simpler than GI, and for which is it isomorphism complete. It turns out that for non-monadic signatures, this problem is isomorphism complete just as is the case without sorts, while the classification of monadic signatures is more complex and interesting in the presence of sorts.