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• W115672173 abstract "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 NP , integers can be factored in probabilistic polynomial time, and deterministic collision-free hash functions do not exist;" @default.
• W115672173 created "2016-06-24" @default.
• W115672173 creator A5062065155 @default.
• W115672173 date "2008-01-01" @default.
• W115672173 modified "2023-09-27" @default.
• W115672173 title "The Complexity of Equivalence Relations" @default.
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• W115672173 hasPublicationYear "2008" @default.
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