Multiple-Length Division Revisited: A Tour of the Minefield

Long division of natural numbers plays a crucial role in Cobol arithmetic, cryptography, and primality testing. Only a handful of textbooks discuss the theory and practice of long division, and none of them do it satisfactorily. This tutorial attempts to fill this surprising gap in the literature on computer algorithms. We illustrate the subtleties of long division by examples, define the problem concisely, summarize the theory, and develop a complete Pascal algorithm using a consistent terminology

Law and Order in Algorithmics

An algorithm is the input-output effect of a computer program; mathematically, the notion of algorithm comes close to the notion of function. Just as arithmetic is the theory and practice of calculating with numbers, so is ALGORITHMICS the theory and practice of calculating with algorithms. Just as a law in arithmetic is an equation between numbers, like a(b+c) = ab + ac, so is a LAW in algorithmics an equation between algorithms. The goal of the research done is: (extending algorithmics by) the systematic detection and use of laws for algorithms. To this end category theory (a branch of mathematics) is used to formalise the notion of algorithm, and to formally prove theorems and laws about algorithms.\ud \ud The underlying motivation for the research is the conviction that algorithmics may be of help in the construction of computer programs, just as arithmetic is of help in solving numeric problems. In particular, algorithmics provides the means to derive computer programs by calculation, from a given specification of the input-output effect.\ud \ud In Chapter 2 the systematic detection and use of laws is applied to category theory itself. The result is a way to conduct and present proofs in category theory, that is an alternative to the conventional way (diagram chasing).\ud \ud In Chapter 3--4 several laws are formally derived in a systematic fashion. These laws facilitate to calculate with those algorithms that are defined by induction on their input, or on their output. Technically, initial algebras and terminal co-algebras play an crucial role here.\ud \ud In Chapter 5 a category theoretic formalisation of the notion of law itself is derived and investigated. This result provides a tool to formulate and prove theorems about laws-in-general, and, more specifically, about equationally specified datatypes.\ud \ud Finally, in Chapter 6 laws are derived for arbitrary recursive algorithms. Here the notion of ORDER plays a crucial role. The results are relevant for current functional programming languages

Interactive fuzzy numbers arithmetic in financial analyses

Fuzzy arithmetic seems to be a well-developed and well-formalized branch of fuzzy sets theory. Nevertheless, there are some problems with its practical implementation. In practice, one can very often encounter the question of how to calculate the value of an arithmetic expression when its arguments are presented as linked fuzzy numbers. This article presents the generalization of Zadeh’s extension principle for arithmetic operations on fuzzy linked variables. Typical kinds of interaction between financial analyses parameters in industrial enterprises are discussed later. These interactions make use of nonlinear programming methods necessary to calculate the values of selected financial indices expressed by the fuzzy numbers. The article presents the problem of nonlinear programming applied to the calculation of the values of these indices. Additionally, the paper proposes the use of computer simulations in arithmetic operations on fuzzy linked variables. Furthermore, the example of gross profit calculations for two metallurgical companies is presented. The calculations done show that the interaction between the arguments of the arithmetic expression influences the final result significantly.extension principle, fuzzy arithmetic, financial analyses

Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued functions. After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic. This formalization covers a large part of the theory which underlies the efficient algorithms implemented in practice in computer algebra. The success of this work paves the way for formal certification of these efficient methods.Comment: 40 pages, 4 figure

Numerical Analysis

Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud \ud In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud \ud With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright

Some observations on the logical foundations of inductive theorem proving

In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a goal. Based on this model, we then analyze the following aspects: the choice of a proof shape, the choice of an induction rule and the language of the induction formula. In particular, using model-theoretic techniques, we clarify the relationship between notions of inductiveness that have been considered in the literature on automated inductive theorem proving. This is a corrected version of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017

An Instantiation-Based Approach for Solving Quantified Linear Arithmetic

This paper presents a framework to derive instantiation-based decision procedures for satisfiability of quantified formulas in first-order theories, including its correctness, implementation, and evaluation. Using this framework we derive decision procedures for linear real arithmetic (LRA) and linear integer arithmetic (LIA) formulas with one quantifier alternation. Our procedure can be integrated into the solving architecture used by typical SMT solvers. Experimental results on standardized benchmarks from model checking, static analysis, and synthesis show that our implementation of the procedure in the SMT solver CVC4 outperforms existing tools for quantified linear arithmetic