Equivalence relations, invariants, and normal forms, II

Abstract

For an equivalence relation E on the words over some finite alphabet, we consider the following four problems, listed in order of increasing difficulty. Recognition: Decide whether two words are equivalent. Invariant: Calculate a function constant on precisely the equivalence classes. Normal form: Calculate a particular member of an equivalence class, given an arbitrary member. First member: Calculate the first member of an equivalence class, given an arbitrary member. We consider the questions whether p solutions for the easier problems yield NP solutions for the harder ones, or vice versa. We show that affirmative answers to several of these questions are equivalent to natural principles like NP=co-NP, NP ∩co-NP=P, and the shrinking principle for NP sets. We supplement known oracles with enough new ones to show that all the questions considered have negative answers relative to some oracles. In other words, these questions cannot be answered affirmatively by means of relativizable polynomial-time Turing reductions. Finally, we show that the analogous questions with "p" replaced by "Borel" have negative answers.