Complexity Classes of Equivalence Problems Revisited

TL;DR

This paper explores the complexity of equivalence problems, examining the relationships between polynomial-time decidability, canonical forms, and complete invariants, and extends prior results with new insights involving probabilistic and quantum computation.

Contribution

It extends previous work on the complexity of equivalence relations, establishing new connections with probabilistic and quantum computational models.

Findings

Polynomial-time decidability does not imply the existence of a canonical form or complete invariant.

The paper demonstrates that certain equivalence problems require non-relativizing techniques to resolve.

New links between equivalence problem complexity and quantum computation are established.

Abstract

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.