# Difference between revisions of "Many-one reducibility"

A relation between sets of natural numbers. We say that $A$ is many-one reducible to $B$, $A \le_{\mathrm{m}} B$, if there is a general recursive function $\phi$ such that $x \in A$ if and only if $\phi(x) \in B$. This defines a pre-order on sets of natural numbers, and the equivalence classes are $\mathrm{m}$-degrees; they refine the Turing degrees.
Many-one reduction may also be regarded as a relation between functions on natural numbers: in this case, the characteristic functions of sets of natural numbers, with the pre-order $f \le_{\mathrm{m}} g$ if there is a recursive $\phi$ such that $f(x) = g(\phi(x))$ for all $x$.