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Many-one reductions and the category of multivalued functions

Published online by Cambridge University Press:  03 July 2015

ARNO PAULY
ARNO PAULY*
Affiliation:
Computer Laboratory, University of Cambridge Email: arno.pauly@cl.cam.ac.uk.
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Abstract

Multivalued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).

A more general understanding is possible, if the category-theoretic properties of multivalued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multivalued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch).

Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multivalued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multivalued functions.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Issue 3 , March 2017 , pp. 376 - 404
Copyright
Copyright © Cambridge University Press 2015 

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