Adjunctions for exceptions

An algebraic method is used to study the semantics of exceptions in computer languages. The exceptions form a computational effect, in the sense that there is an apparent mismatch between the syntax of exceptions and their intended semantics. We solve this apparent contradiction by efining a logic for exceptions with a proof system which is close to their syntax and where their intended semantics can be seen as a model. This requires a robust framework for logics and their morphisms, which is provided by categorical tools relying on adjunctions, fractions and limit sketches.Comment: In this Version 2, minor improvements are made to Version

Multiple-Precision Correctly Rounded Gauss-Legendre Quadrature

Numerical integration is an operation that is frequently available in multiple precision numerical software packages. The different quadrature schemes used are considered well studied but the roundoff errors that result from the computation are often neglected, and the actual accuracy of the results are therefore seldom rigorously proven. We focus on the Gauss-Legendre quadrature scheme and describe the algorithms needed in our implementation. A thorough error analysis is given as well as experimental error measurements and timings

A duality between exceptions and states

In this short note we study the semantics of two basic computational effects, exceptions and states, from a new point of view. In the handling of exceptions we dissociate the control from the elementary operation which recovers from the exception. In this way it becomes apparent that there is a duality, in the categorical sense, between exceptions and states

States and exceptions considered as dual effects

In this paper we consider the two major computational effects of states and exceptions, from the point of view of diagrammatic logics. We get a surprising result: there exists a symmetry between these two effects, based on the well-known categorical duality between products and coproducts. More precisely, the lookup and update operations for states are respectively dual to the throw and catch operations for exceptions. This symmetry is deeply hidden in the programming languages; in order to unveil it, we start from the monoidal equational logic and we add progressively the logical features which are necessary for dealing with either effect. This approach gives rise to a new point of view on states and exceptions, which bypasses the problems due to the non-algebraicity of handling exceptions

Decorated proofs for computational effects: Exceptions

We define a proof system for exceptions which is close to the syntax for exceptions, in the sense that the exceptions do not appear explicitly in the type of any expression. This proof system is sound with respect to the intended denotational semantics of exceptions. With this inference system we prove several properties of exceptions.Comment: 11 page

Correctly Rounded Newton-Cotes Quadrature

Numerical integration is an important operation for scientific computations. Although the different quadrature methods have been well studied from a mathematical point of view, the analysis of the actual error when performing the quadrature on a computer is often neglected. This step is however required for certified arithmetics. We study the Newton-Cotes quadrature scheme and give enough details on the algorithms and the error bounds to enable software developers to write a correctly-rounded Newton-Cotes quadrature

Compressed Modular Matrix Multiplication

We propose to store several integers modulo a small prime into a single machine word. Modular addition is performed by addition and possibly subtraction of a word containing several times the modulo. Modular Multiplication is not directly accessible but modular dot product can be performed by an integer multiplication by the reverse integer. Modular multiplication by a word containing a single residue is a also possible. Therefore matrix multiplication can be performed on such a compressed storage. We here give bounds on the sizes of primes and matrices for which such a compression is possible. We also explicit the details of the required compressed arithmetic routines.Comment: Published in: MICA'2008 : Milestones in Computer Algebra, Tobago : Trinit\'e-et-Tobago (2008

Decorated proofs for computational effects: States

The syntax of an imperative language does not mention explicitly the state, while its denotational semantics has to mention it. In this paper we show that the equational proofs about an imperative language may hide the state, in the same way as the syntax does.Comment: In Proceedings ACCAT 2012, arXiv:1208.430

MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding

This paper presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend ideas from the IEEE-754 standard to arbitrary precision, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved | with no signicant slowdown with respect to other tools | and discuss a few applications where such a library can be useful

Computational Complexity of Iterated Maps on the Interval (Extended Abstract)

The exact computation of orbits of discrete dynamical systems on the interval is considered. Therefore, a multiple-precision floating point approach based on error analysis is chosen and a general algorithm is presented. The correctness of the algorithm is shown and the computational complexity is analyzed. As a main result, the computational complexity measure considered here is related to the Ljapunow exponent of the dynamical system under consideration