  To determine if two lists of numbers are the same set, we sort both lists and
see if we get the same result. The sorted list is a canonical form for the
equivalence relation of set equality. Other canonical forms arise in graph
isomorphism algorithms, and the equality of permutation groups given by
generators. To determine if two graphs are cospectral (have the same
eigenvalues), however, we compute their characteristic polynomials and see if
they are the same; the characteristic polynomial is a complete invariant for
the equivalence relation of cospectrality. This is weaker than a canonical
form, and it is not known whether a polynomial-time canonical form for
cospectrality exists. Note that it is a priori possible for an equivalence
relation to be decidable in polynomial time without either a complete invariant
or canonical form.
  Blass and Gurevich (SIAM J. Comput., 1984) ask whether these conditions on
equivalence relations -- having an FP canonical form, having an FP complete
invariant, and simply being in P -- are in fact different. They showed that
this question requires non-relativizing techniques to resolve. Here we extend
their results, and give new connections to probabilistic and quantum
computation.
