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WEIHRAUCH GOES BROUWERIAN

Part of: Computability and recursion theory Distributive lattices Proof theory and constructive mathematics Boolean algebras (Boolean rings) General logic

Published online by Cambridge University Press:  30 October 2020

VASCO BRATTKA  and
GUIDO GHERARDI
VASCO BRATTKA
Affiliation:
FACULTY OF COMPUTER SCIENCE UNIVERSITÄT DER BUNDESWEHR MÜNCHENNEUBIBERG, GERMANY DEPARTMENT OF MATHEMATICS & APPLIED MATHEMATICS UNIVERSITY OF CAPE TOWNCAPE TOWN, SOUTH AFRICAE-mail: [email protected]
GUIDO GHERARDI
Affiliation:
DIPARTIMENTO DI FILOSOFIA E COMUNICAZIONE UNIVERSITÀ DI BOLOGNABOLOGNA, ITALYE-mail: [email protected]
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Abstract

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of the respective types, where we allow any value outside of the original domain of the problem. This closure operator is of interest by itself, as it generates a total version of Weihrauch reducibility that is defined like the usual version of Weihrauch reducibility, but in terms of total realizers. From a logical perspective completion can be seen as a way to make problems independent of their premises. Alongside with the completion operator and total Weihrauch reducibility we need to study precomplete representations that are required to describe these concepts. In order to show that the parallelized total Weihrauch lattice forms a Brouwer algebra, we introduce a new multiplicative version of an implication. While the parallelized total Weihrauch lattice forms a Brouwer algebra with this implication, the total Weihrauch lattice fails to be a model of intuitionistic linear logic in two different ways. In order to pinpoint the algebraic reasons for this failure, we introduce the concept of a Weihrauch algebra that allows us to formulate the failure in precise and neat terms. Finally, we show that the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which also implies that the theory of our Brouwer algebra is Jankov logic.

Keywords

Weihrauch complexitycomputable analysisBrouwer algebraintuitionistic and linear logic

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03D30: Other degrees and reducibilities 03D78: Computation over the reals 03F52: Linear logic and other substructural logics 03F60: Constructive and recursive analysis 06D20: Heyting algebras
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1614 - 1653
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Copyright
© The Association for Symbolic Logic 2020

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