从可计算性理论看等价关系和前序的复杂性
Complexity of equivalence relations and preorders from computability theory
摘要：
从可计算性理论和复杂性理论出发，研究了等价关系和前序的相对复杂性。给定二元关系$R, S$，按分量约简由$ R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. $定义。对我们来说，关系将是自然数，$f$必须是可计算的。证明了对于$k \ge 2$，存在$Π_1$ -完全等价关系，但不存在$Πk$ -完全等价关系.我们表明，自然产生的$Σk$前序在上述领域是$Σk$ -完全的。这包括多项式时间$m$ -指数时间集上的约简，即$Σ2$，几乎包含在r. e上。集合，也就是$Σ3$，以及r.e.上的图灵归约。\集合，即$Σ4$。
英文摘要：
We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by $ R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. $ Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $Π_1$-complete equivalence relation, but no $Πk$-complete for $k \ge 2$. We show that $Σk$ preorders arising naturally in the above-mentioned areas are $Σk$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $Σ2$, almost inclusion on r.e.\ sets, which is $Σ3$, and Turing reducibility on r.e.\ sets, which is $Σ4$.
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