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An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
In this paper we develop an axiomatic setup for algorithmic homological
algebra of Abelian categories. This is done by exhibiting all existential
quantifiers entering the definition of an Abelian category, which for the sake
of computability need to be turned into constructive ones. We do this
explicitly for the often-studied example Abelian category of finitely presented
modules over a so-called computable ring , i.e., a ring with an explicit
algorithm to solve one-sided (in)homogeneous linear systems over . For a
finitely generated maximal ideal in a commutative ring we
show how solving (in)homogeneous linear systems over can be
reduced to solving associated systems over . Hence, the computability of
implies that of . As a corollary we obtain the computability
of the category of finitely presented -modules as an Abelian
category, without the need of a Mora-like algorithm. The reduction also yields,
as a by-product, a complexity estimation for the ideal membership problem over
local polynomial rings. Finally, in the case of localized polynomial rings we
demonstrate the computational advantage of our homologically motivated
alternative approach in comparison to an existing implementation of Mora's
algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
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