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. ^[1] It was originally introduced by Klaus Weihrauch in an unpublished 1992 technical report. ^[2]

Definition

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 ${\displaystyle (X,\delta _{X})}$ and ${\displaystyle (Y,\delta _{Y})}$ 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.^{[ citation needed ]} 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:$${\displaystyle (\forall G\vdash g)(\Psi G\Phi \vdash f).}$$

See also

• Wadge reducibility

References

1. Brattka, Vasco; Gherardi, Guido; Pauly, Arno (2021), Brattka, Vasco; Hertling, Peter (eds.), "Weihrauch Complexity in Computable Analysis", Handbook of Computability and Complexity in Analysis, Cham: Springer International Publishing, pp. 367–417, arXiv: 1707.03202 , doi:10.1007/978-3-030-59234-9_11, ISBN   978-3-030-59233-2, S2CID   7903709 , retrieved 2022-06-29
2. Weihrauch, Klaus (1992). The Degrees of Discontinuity of some Translators between Representations of the Real Numbers (Report). Informatik-Berichte. Vol. 129. FernUniversität in Hagen.