The Complexity of Equivalence Relations

To determine if two given lists of numbers are the same set, we would 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 for equivalences arise in graph isomorphism and its variants, and the equality of permutation groups given by generators. To determine if two given graphs are cospectral, 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 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 (“Equivalence relations, invariants, and normal forms, I and II”, 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 using generic oracles, and give new connections to probabilistic and quantum computation. We denote the class of equivalence problems in P by PEq, the class of problems with complete FP invariants Ker, and the class with FP canonical forms CF; CF ⊆ Ker ⊆ PEq, and we ask whether these inclusions are proper. If x ∼ y implies |y| ≤ poly(|x|), we say that ∼ is polynomially bounded; we denote the corresponding classes of equivalence relation CFp, Kerp, and PEqp. Our main results are: • If CF = PEq then NP = UP = RP and thus PH = BPP; • If CF = Ker then NP = UP, PH = ZPP, integers can be factored in probabilistic polynomial time, and deterministic collision-free hash functions do not exist; • If Ker = PEq then UP ⊆ BQP; • There is an oracle relative to which CF 6= Ker 6= PEq; and • There is an oracle relative to which CFp = Kerp and Ker 6= PEq. Attempting to generalize the third result above from UP to NP leads to a new open question about the structure of witness sets for NP problems (roughly: can the witness sets for an NPcomplete problem form an Abelian group?). We also introduce a natural notion of reduction between equivalence problems, and present several open questions about generalizations of these concepts to the polynomial hierarchy, to logarithmic space, and to counting problems. Many of the new results in this thesis were obtained in collaboration with Lance Fortnow, and have been submitted for conference presentation.