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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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