Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.

What algorithms could not be

Abstract: This dissertation addresses a variety of foundational issues pertaining to the notion of algorithm employed in mathematics and computer science. In these settings, an algorithm is taken to be an effective mathematical procedure for solving a previously stated mathematical problem. Procedures of this sort comprise the notional subject matter of the subfield of computer science known as algorithmic analysis. In this context, algorithms are referred to via proper names (e.g. Mergesort) of which computational properties are directly predicated (e.g. Mergesort has running time O(n log(n))). Moreover, many formal results are traditionally stated by explicitly quantifying over algorithms (e.g. there is a polynomial time primality algorithm; there is no linear time comparison sorting algorithm). These observations motivate the view that algorithms are themselves abstract mathematical objects – a view I refer to as algorithmic realism. The status of this view is clearly related to that of Church's Thesis – i.e. the claim that a function is computable by an algorithm just in case it is partial recursive. But whereas Church's Thesis is widely accepted on the basis of several well-known mathematical analyses of algorithmic computability, there is presently no consensus on how (or if) we can uniformly identify individual algorithms with mathematical objects. In the first chapter of this work, I attempt to illustrate the theoretical significance of algorithmic realism. I suggest that this view is not only of foundational significance to computer science, but also worth highlighting due to the role algorithms play in the fixation of mathematical knowledge and their relationship to intensional entities such as propositions and properties. Chapter Two presents a formal framework for reducing computational discourse to mathematical discourse modeled on contemporary discussion of mathematical nominalism. Chapter Three is centered on a case study intended to illustrate the technical exigencies involved with defending algorithmic realism. Chapter Four explores various ways in which algorithms might be identified with models of computation. And finally, Chapter Five argues that no such identification can uniformly satisfy all statements of algorithmic identity and non-identity affirmed by computational practice. I suggest that the technical crux of algorithmic realism lies in the necessity of reducing recursive specifications of algorithms to iterative models of computation in a manner which preserves algorithmic identity.%%%%

Semantic Characterisations of Second-Order Computability over the Real Numbers

Abstract: We propose semantic characterisations of second-order computability over the reals based on Σ-definability theory. Notions of computability for operators and real-valued functionals defined on the class of continuous functions are introduced via domain theory. We consider the reals with and without equality and prove theorems which connect computable operators and real-valued functionals with validity of finite Σ-formulas.

${\cal H}_{\infty}$ Filtering for Nonlinear Singular Systems

Abstract: In this paper, we consider the H∞-filtering problem for affine nonlinear singular (or descriptor systems). Two types of filters are discussed, namely, 1) singular and 2) normal, and sufficient conditions for the solvability of the problem in terms of Hamilton-Jacobi-Isaacs equations (HJIEs) are presented. The results are also specialized to linear systems in which case the HJIEs reduce to a system of bilinear-matrix-inequalities (BLMIs) which can still be solved efficiently. Some simple examples are also given to illustrate the approach.

Computable analysis of the abstract Cauchy problem in a Banach space and its applications I

Abstract: We study computability of the abstract linear Cauchy problem ((1)) where A is a linear operator, possibly unbounded, on a Banach space X. We give necessary and sufficient conditions for A such that the solution operator K: x ↦ u of the problem (1) is computable. For studying computability we use the representation approach to computable analysis developed by Weihrauch and others. This approach is consistent with the model used by Pour-El/Richards. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A geometric approach to output regulation for discrete-time switched linear systems

Abstract: This paper considers the problem of asymptotic output regulation by dynamic feedback for discrete-time switched linear systems, with the requirement of asymptotic stability of the regulation loop. Using the methods of the geometric approach, sufficient conditions for solvability of the problem, under suitable assumptions, are shown. A synthesis procedure is outlined.