Weihrauch reducibility

A represented space is a pair ${\textstyle (X,\delta )}$ of a set ${\displaystyle X}$ and a surjective partial function ${\displaystyle \delta$ :\subset \mathbb {N} ^{\mathbb {N} }\rightarrow X} .[1]

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

Let ${\displaystyle X,Y,Z,W}$ be represented spaces and let ${\displaystyle f:\subset X\rightrightarrows Y,g:\subset Z\rightrightarrows W}$ be partial multi-valued functions. We say that ${\displaystyle f}$ is Weihrauch reducible to ${\displaystyle g}$, and write ${\displaystyle f\leq _{\mathrm {W} }g}$, if there are computable partial functions ${\displaystyle \Phi ,\Psi$ :\subset \mathbb {N} ^{\mathbb {N} }\to \mathbb {N} ^{\mathbb {N} }} such that

$${\displaystyle (\forall G\vdash g)(\Psi \langle \mathrm {id} ,G\Phi \rangle \vdash f),}$$

where ${\displaystyle \Psi \langle \mathrm {id} ,G\Phi \rangle$ :=\langle p,q\rangle \mapsto \Psi (\langle p,G\Phi (q)\rangle )} and ${\displaystyle \langle \cdot \rangle }$ denotes the join in the Baire space. Very often, in the literature we find ${\displaystyle \Psi }$ written as a binary function, so to avoid the use of the join. In other words, ${\displaystyle f\leq _{\mathrm {W} }g}$ if there are two computable maps ${\displaystyle \Phi ,\Psi }$ such that the function ${\displaystyle p\mapsto \Psi (p,q)}$ is a realizer for ${\displaystyle f}$ whenever ${\displaystyle q}$ is a solution for ${\displaystyle g(\Phi (p))}$. The maps ${\displaystyle \Phi ,\Psi }$ are often called forward and backward functional respectively. We say that ${\displaystyle f}$ is strongly Weihrauch reducible to ${\displaystyle g}$, and write ${\displaystyle f\leq _{\mathrm {sW} }g}$, if the backward functional ${\displaystyle \Psi }$ does not have access to the original input. In symbols:

$${\displaystyle (\forall G\vdash g)(\Psi G\Phi \vdash f).}$$

• Wadge reducibility