Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Towards computability of elliptic boundary value problems in variational formulation

AbstractWe present computable versions of the Fréchet–Riesz Representation Theorem and the Lax–Milgram Theorem. The classical versions of these theorems play important roles in various problems of mathematical analysis, including boundary value problems of elliptic equations. We demonstrate how their computable versions yield computable solutions of the Neumann and Dirichlet boundary value problems for a simple non-symmetric elliptic differential equation in the one-dimensional case. For the discussion of these elementary boundary value problems, we also provide a computable version of the Theorem of Schauder, which shows that the adjoint of a computably compact operator on Hilbert spaces is computably compact again

Le direzioni della logica in Italia: la reverse mathematics e l'analisi computazionale

Nelle conversazioni tra matematici non \ue8 infrequente sentire affermazioni del tipo \u201ci teoremi \u3a6 e \u3a8 sono equivalenti\u201d, oppure \u201cil teorema \u3a6 \ue8 pi\uf9 forte del teorema \u3a8\u201d. Dato che \u3a6 e \u3a8 (essendo teoremi) sono entrambi dimostrabili, prendendo alla lettera le due affermazioni abbiamo che la prima \ue8 banalmente vera e la seconda banalmente falsa. Sappiamo tutti per\uf2 che queste affermazioni hanno un altro significato, molto meno banale, e c\u2019\ue8 quindi una ragione per cui vengono fatte. Negli ultimi decenni la logica matematica ha sviluppato alcuni strumenti in grado di rendere precise, e suscettibili di dimostrazione o refutazione, affermazioni come le precedenti. In particolare ci riferiamo alla reverse mathematics e all\u2019analisi computazionale. Questi sono due programmi di ricerca di origine diverse che nell\u2019ultimo decennio, anche grazie al contributo di alcuni ricercatori italiani, hanno trovato significativi punti di contatto. In questo lavoro presenteremo i due programmi, con particolare riferimento alle loro aree di contatto. Evidenzieremo in particolare i contributi dei ricercatori italiani attivi in queste aree, e concluderemo indicando alcune prospettive di sviluppo su cui anche in Italia si sta cercando di lavorare

'Sifting the significance from the data' - the impact of high-throughput genomic technologies on human genetics and health care.

This report is of a round-table discussion held in Cardiff in September 2009 for Cesagen, a research centre within the Genomics Network of the UK's Economic and Social Research Council. The meeting was arranged to explore ideas as to the likely future course of human genomics. The achievements of genomics research were reviewed, and the likely constraints on the pace of future progress were explored. New knowledge is transforming biology and our understanding of evolution and human disease. The difficulties we face now concern the interpretation rather than the generation of new sequence data. Our understanding of gene-environment interaction is held back by our current primitive tools for measuring environmental factors, and in addition, there may be fundamental constraints on what can be known about these complex interactions.RIGHTS : This article is licensed under the BioMed Central licence at http://www.biomedcentral.com/about/license which is similar to the 'Creative Commons Attribution Licence'. In brief you may : copy, distribute, and display the work; make derivative works; or make commercial use of the work - under the following conditions: the original author must be given credit; for any reuse or distribution, it must be made clear to others what the license terms of this work are

Aspects Topologiques des Représentations en Analyse Calculable

Computable analysis provides a formalization of algorithmic computations over infinite mathematical objects. The central notion of this theory is the symbolic representation of objects, which determines the computation power of the machine, and has a direct impact on the difficulty to solve any given problem. The friction between the discrete nature of computations and the continuous nature of mathematical objects is captured by topology, which expresses the idea of finite approximations of infinite objects.We thoroughly study the multiple interactions between computations and topology, analysing the information that can be algorithmically extracted from a representation. In particular, we focus on the comparison between two representations of a single family of objects, on the precise relationship between algorithmic and topological complexity of problems, and on the relationship between finite and infinite representations.L’analyse calculable permet de formaliser le traitement algorithmique d’objets mathématiques infinis. La théorie repose sur une représentation symbolique des objets, dont le choix détermine les capacités de calcul de la machine, notamment sa difficulté à résoudre chaque problème donné. La friction entre le caractère discret du calcul et la nature continue des objets est capturée par la topologie, qui exprime l’idée d’approximation finie d’objets infinis.Nous étudions en profondeur les multiples interactions entre calcul et topologie, cherchant à analyser l’information qui peut être extraite algorithmiquement d’une représentation. Je me penche plus particulièrement sur la comparaison entre deux représentations d’une même famille d’objets, sur les liens détaillés entre complexité algorithmique et topologique des problèmes, ainsi que sur les relations entre représentations finies et infinies

Topics in Programming Languages, a Philosophical Analysis through the case of Prolog

[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well. In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some: - the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog