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COMPLEXITY OF EQUIVALENCE RELATIONS AND PREORDERS FROM COMPUTABILITY THEORY

  • EGOR IANOVSKI (a1), RUSSELL MILLER (a2), KENG MENG NG (a3) and ANDRÉ NIES (a4)

Abstract

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

RS ⇔ ∃fx, y [x R yf (x) S f (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 ${\rm{\Pi }}_1^0$ -complete equivalence relation, but no ${\rm{\Pi }}_k^0$ -complete for k ≥ 2. We show that ${\rm{\Sigma }}_k^0$ preorders arising naturally in the above-mentioned areas are ${\rm{\Sigma }}_k^0$ -complete. This includes polynomial time m-reducibility on exponential time sets, which is ${\rm{\Sigma }}_2^0$ , almost inclusion on r.e. sets, which is ${\rm{\Sigma }}_3^0$ , and Turing reducibility on r.e. sets, which is ${\rm{\Sigma }}_4^0$ .

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