Weihrauch reducibility

In computable analysis Weihrauch reducibility is a notion of reducibility between multi-valued functions on represented spaces that roughly captures the uniform computational strength of computational problems. It was originally introduced by Karl Weihrauch in an unpublished 1992 technical report.

Definition

A represented space is a pair ${\textstyle (X,\delta )}$ of a set $X$ and a surjective partial function $\delta :\subseteq \mathbb {N} ^{\mathbb {N} }\rightarrow X$.

Let $(X,\delta _{X}),(Y,\delta _{Y})$ be represented spaces and let $f:\subset X\rightrightarrows Y$ be a partial multi-valued function. A realizer for $f$ is a (possibly partial) function $F:\subset \mathbb {N} ^{\mathbb {N} }\to \mathbb {N} ^{\mathbb {N} }$ such that, for every $p\in \mathrm {dom} f\circ \delta _{X}$, $\delta _{Y}\circ F(p)=f\circ \delta _{X}(p)$. Intuitively, a realizer $F$ for $f$ behaves "just like $f$" but it works on names. If $F$ is a realizer for $f$ we write $F\vdash f$.

Let $X,Y,Z,W$ be represented spaces and let $f:\subset X\rightrightarrows Y,g:\subset Z\rightrightarrows W$ be partial multi-valued functions. We say that $f$ is Weihrauch reducible to $g$, and write $f\leq _{\mathrm {W} }g$, if$$(\exists {\text{ computable }}\Phi ,\Psi :\subset \mathbb {N} ^{\mathbb {N} }\to \mathbb {N} ^{\mathbb {N} })(\forall G\vdash g)(\Psi \langle \mathrm {id} ,G\Phi \rangle \vdash f),$$where $\Psi \langle \mathrm {id} ,G\Phi \rangle :=\langle p,q\rangle \mapsto \Psi (\langle p,G\Phi (q)\rangle )$ and $\langle \cdot \rangle$ denotes the join in the Baire space. Very often, in the literature we find $\Psi$ written as a binary function, so to avoid the use of the join. In other words, $f\leq _{\mathrm {W} }g$ if there are two computable maps $\Phi ,\Psi$ such that the function $p\mapsto \Psi (p,q)$ is a realizer for $f$ whenever $G$ is a realizer for $g$. The maps $\Phi ,\Psi$ are often called forward and backward functional respectively.

We say that $f$ is strongly Weihrauch reducible to $g$, and write $f\leq _{\mathrm {sW} }g$, if the backward functional $\Psi$ does not have access to the original input. In symbols:$$(\exists {\text{ computable }}\Phi ,\Psi :\subset \mathbb {N} ^{\mathbb {N} }\to \mathbb {N} ^{\mathbb {N} })(\forall G\vdash g)(\Psi G\Phi \rangle \vdash f),$$