6,471 research outputs found
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
Complexity vs Energy: Theory of Computation and Theoretical Physics
This paper is a survey dedicated to the analogy between the notions of {\it
complexity} in theoretical computer science and {\it energy} in physics. This
analogy is not metaphorical: I describe three precise mathematical contexts,
suggested recently, in which mathematics related to (un)computability is
inspired by and to a degree reproduces formalisms of statistical physics and
quantum field theory.Comment: 23 pages. Talk at the satellite conference to ECM 2012, "QQQ Algebra,
Geometry, Information", Tallinn, July 9-12, 201
Weihrauch goes Brouwerian
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.Comment: 36 page
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