72 research outputs found
An algorithmic approach to the existence of ideal objects in commutative algebra
The existence of ideal objects, such as maximal ideals in nonzero rings,
plays a crucial role in commutative algebra. These are typically justified
using Zorn's lemma, and thus pose a challenge from a computational point of
view. Giving a constructive meaning to ideal objects is a problem which dates
back to Hilbert's program, and today is still a central theme in the area of
dynamical algebra, which focuses on the elimination of ideal objects via
syntactic methods. In this paper, we take an alternative approach based on
Kreisel's no counterexample interpretation and sequential algorithms. We first
give a computational interpretation to an abstract maximality principle in the
countable setting via an intuitive, state based algorithm. We then carry out a
concrete case study, in which we give an algorithmic account of the result that
in any commutative ring, the intersection of all prime ideals is contained in
its nilradical
Non-principal ultrafilters, program extraction and higher order reverse mathematics
We investigate the strength of the existence of a non-principal ultrafilter
over fragments of higher order arithmetic.
Let U be the statement that a non-principal ultrafilter exists and let
ACA_0^{\omega} be the higher order extension of ACA_0. We show that
ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that
ACA_0^{\omega}+\U is conservative over PA.
Moreover, we provide a program extraction method and show that from a proof
of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U
a realizing term in G\"odel's system T can be extracted. This means that one
can extract a term t, such that A(f,t(f))
From Nonstandard Analysis to various flavours of Computability Theory
As suggested by the title, it has recently become clear that theorems of
Nonstandard Analysis (NSA) give rise to theorems in computability theory (no
longer involving NSA). Now, the aforementioned discipline divides into
classical and higher-order computability theory, where the former (resp. the
latter) sub-discipline deals with objects of type zero and one (resp. of all
types). The aforementioned results regarding NSA deal exclusively with the
higher-order case; we show in this paper that theorems of NSA also give rise to
theorems in classical computability theory by considering so-called textbook
proofs.Comment: To appear in the proceedings of TAMC2017 (http://tamc2017.unibe.ch/
Computational interpretations of analysis via products of selection functions
We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions
Quantitative Coding and Complexity Theory of Compact Metric Spaces
Specifying a computational problem requires fixing encodings for input and
output: encoding graphs as adjacency matrices, characters as integers, integers
as bit strings, and vice versa. For such discrete data, the actual encoding is
usually straightforward and/or complexity-theoretically inessential (up to
polynomial time, say); but concerning continuous data, already real numbers
naturally suggest various encodings with very different computational
properties. With respect to qualitative computability, Kreitz and Weihrauch
(1985) had identified ADMISSIBILITY as crucial property for 'reasonable'
encodings over the Cantor space of infinite binary sequences, so-called
representations [doi:10.1007/11780342_48]: For (precisely) these does the
sometimes so-called MAIN THEOREM apply, characterizing continuity of functions
in terms of continuous realizers.
We rephrase qualitative admissibility as continuity of both the
representation and its multivalued inverse, adopting from
[doi:10.4115/jla.2013.5.7] a notion of sequential continuity for
multifunctions. This suggests its quantitative refinement as criterion for
representations suitable for complexity investigations. Higher-type complexity
is captured by replacing Cantor's as ground space with Baire or any other
(compact) ULTRAmetric space: a quantitative counterpart to equilogical spaces
in computability [doi:10.1016/j.tcs.2003.11.012]
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