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 It was originally introduced by Klaus Weihrauch in an unpublished 1992 technical report.

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

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 $$f\le_{\mathrm{W}} g$$, if there are computable partial functions $$\Phi,\Psi:\subset \mathbb{N}^\mathbb{N} \to \mathbb{N}^\mathbb{N}$$ such that$$ (\forall G \vdash g )( \Psi \langle \mathrm{id}, G\Phi \rangle \vdash f ),$$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, $$f \le_\mathrm{W} g$$ if there are two computable maps $$\Phi, \Psi$$ such that the function $$p \mapsto \Psi(p, q) $$ is a realizer for $$f$$ whenever $$q$$ is a solution for $$g(\Phi(p))$$. The maps $$\Phi, \Psi$$ are often called forward and backward functional respectively.

We say that $$f$$ is strongly Weihrauch reducible to $$g$$, and write $$ f\le_{\mathrm{sW}} g$$, if the backward functional $$\Psi$$ does not have access to the original input. In symbols:$$ (\forall G \vdash g )( \Psi G\Phi \vdash f ).$$