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Théorie des types

En mathématiques, logique et informatique, une théorie des types est une classe de systèmes formels, dont certains peuvent servir d'alternatives à la théorie des ensembles comme fondation des mathématiques. Ils ont été historiquement introduits pour résoudre le paradoxe d'un axiome de compréhension non restreint. En théorie des types, il existe des types de base et des constructeurs (comme celui des fonctions ou encore celui du produit cartésien) qui permettent de créer de nouveaux types à partir de types préexistant. Une différence notable avec la théorie des ensembles est que chaque terme possède un type, ce qui est un jugement subjectif et non pas une proposition . En d'autres termes la question de savoir si un terme appartient ou non à un type n'est pas réputée objective. Des types sont utilisés dans certains langages de programmation, ce qui permet d'éviter des bugs. Toutefois, la théorie des types fait plus souvent références à l'étude des systèmes de types qui servent de fondation alternative aux mathématiques. Le λ-calcul typé d'Alonzo Church et ses extensions permettent de faire de la logique à différents ordres ; ainsi elles servent à formaliser le système F. Dans la même ligne, la théorie des types intuitionniste de Per Martin-Löf ainsi que le calcul des constructions inductif offrent d'autres perspectives. Ceux-ci sont notamment à la base d' ou de Coq qui sont des assistants de preuve. D'un point de vue de la théorie mathématique : les types ont été pour la première fois théorisés par Bertrand Russell comme réponse à sa découverte du paradoxe ébranlant la théorie naïve des ensembles de Gottlob Frege. Cette théorie des types est développée dans l'ouvrage Principia Mathematica de Russell et Whitehead. Elle permet de contourner le paradoxe de Russell en introduisant tout d'abord une hiérarchie de types, puis en assignant un type à chaque entité mathématique. Les objets d'un certain type ne peuvent être construits qu'à partir d'objets leur préexistant (situés plus bas dans la hiérarchie), empêchant ainsi les boucles infinies et les paradoxes de surgir et de casser la théorie.

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Exercices associés (8)

Un flotteur en mouvement

xa0 The float of a fishing rod floats at the surface of water. This floater has a cylindrical shape with a radius r, a height h and with an homogeneous mass. The float is standing vertically in water and it moves from top to bottom while always staying partially submerged. In addition to its weight, the float is subject to the Archimedian buoyancy

\vec{P_A}

and at a viscous friction force F⃗u2004=u2004u2005−u2005kηv⃗. The volumic mass of the float has a value equal to two third of the one of water : $\rh

La rampe

An Inuit is sleighing down on the roof of his igloo. The igloo is represented as a semi sphere of radius R on a ground d. The position of the Inuit sleighs down without friction on the ice of the roof. You can name m the mass of the Inuit, represented by a material point, and g the acceleration of gravity. The Inuit sleighs down from the top of the igloo (his initial velocity can be neglected).

The Inuit takes off the roof when θu2004=u2004θ_(d). Find the expression of θ_(d).
Express the hor

Point sur un cylindre

A material point moves on an infinitely long cylinder of radius R. The material point is attracted to a point O on the cylinder’s axes by a force that’s proportional to the distance between the material point and O. There is neither gravity nor friction.

List the forces acting on the material point. Draw these forces on a diagram, as well as the frame of reference associated with the chosen coordinate system.
Write the equations of motion and project them on the frame of reference.

Champ électrique d un fil vertical

A wire of length L and of constant line charge density λ is vertically aligned between

z=-\frac{L}{2}

and

z=\frac{L}{2}

. Calculate the electric field

\bf E

at point Pu2004=u2004(r,0,0). Check the limits Lu2004≫u2004r and Lu2004≪u2004r. By the symmetry of the problem, the created field in P(r,0,0) is radial:

{\bf E}=E_r \,\hat{r}

. The contribution of an infinitesimal element dz of charge dq at point P(r,0,0) is: $$dE_{r}=\frac{1}{4 \pi \epsilon_{0}}\frac{dq}{{\mbox{$\resizebox{.16in}{.08in}{\includeg

Araignée suspendue

A spider of mass m is suspended by its thread to a tree. The thread has elastic constant k and rest length L and the tree is assumed to stand still. Gravity acts on the spider which oscillates vertically around its equilibrium position.

Draw a diagram of this problem and represent all forces acting on the spider. Choose the coordinates you will use to describe the spider’s motion. What is its equilibrium position?
Write the spider’s equation of motion.
The general solution for the equ

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Notebooks associés (15)

Exploring Quantization and Oversampling through One-Bit Music

One-Bit Music[1]¶ In this notebook we will talk about quantization and oversampling and we will do so by taking a trip down memory lane to revisit the early from monophonic square waves, introduce polyphony by way of pulse-width modulation and finish with the basics of sigma-delta quantization. pacman In [1]: import matplotlib import matplotlib.pyplot as plt import numpy as np import scipy.signal as sp from scipy.io import wavfile import IPython In [2]: import sys ! wget -q -O mus

Powder X-Ray Diffraction Analysis: Exploring Crystal Structures.

XRay Diffraction[1]¶ The purpose of this notebook is to study the powder XRay diffraction spectra of some systems. When the crystal is hit with an XRay wave, the wave is diffracted only for specific case where the Bragg condition is satisfied for one of the lattice planes in the material. For single crystal diffraction, the condition is satisfied only for specific orientation of the crystal, such that the incident ray has the correct

\theta

value for the plane. In powder XRay data, however, any plane ori

Introduction to Reciprocal Lattices with Periodic Boundary Conditions.

Real vs. Reciprocal Space in 2D[1]¶ Introduction[2]¶ In this short notebook, we will introduce the lattice with periodic(Born- von Karman) boundary condition. Following the boundary condition, there is a transformation from real lattice to the reciprocal lattice. In the code below, we define a1 and a2 through cartesian vectors. Hence, the reciprocal vectors will be the original rotated by 90 degrees multiplied by 2pi and divided by the area of the unit cell (|a1xa2|) In [1]: i

Voice Transformation Techniques Using Digital Signal Processing

A Gallery of Voice Transformers[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will explore various ways to implement a system that changes the voice of a speaker. Typical applications for these techniques, strictly sorted by plausibility, include international spy missions and prank telephone calls, and devices such as the one displayed here can be easily bought on the web... or in a novelty shop. No description has been provided for t

Exploring the Karplus-Strong Algorithm for Sound Synthesis

The Karplus-Strong Algorithm[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

The Karplus-Strong algorithm is a simple method to generate synthetic sounds with arbitrary pitch. In this notebook we will explore a few variations of the method to illustrate some basic notions of discrete-time signal processing. In [1]: %matplotlib inline import matplotlib import matplotlib.pyplot as plt import numpy as np import IPython In [2]: plt.rcParams["figure.figsize"]

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Exercices associés (8)

Un flotteur en mouvement

\vec{P_A}

and at a viscous friction force F⃗u2004=u2004u2005−u2005kηv⃗. The volumic mass of the float has a value equal to two third of the one of water : $\rh

La rampe

The Inuit takes off the roof when θu2004=u2004θ_(d). Find the expression of θ_(d).
Express the hor

Point sur un cylindre

List the forces acting on the material point. Draw these forces on a diagram, as well as the frame of reference associated with the chosen coordinate system.
Write the equations of motion and project them on the frame of reference.

Champ électrique d un fil vertical

A wire of length L and of constant line charge density λ is vertically aligned between

z=-\frac{L}{2}

and

z=\frac{L}{2}

. Calculate the electric field

\bf E

at point Pu2004=u2004(r,0,0). Check the limits Lu2004≫u2004r and Lu2004≪u2004r. By the symmetry of the problem, the created field in P(r,0,0) is radial:

{\bf E}=E_r \,\hat{r}

. The contribution of an infinitesimal element dz of charge dq at point P(r,0,0) is: $$dE_{r}=\frac{1}{4 \pi \epsilon_{0}}\frac{dq}{{\mbox{$\resizebox{.16in}{.08in}{\includeg

Araignée suspendue

Draw a diagram of this problem and represent all forces acting on the spider. Choose the coordinates you will use to describe the spider’s motion. What is its equilibrium position?
Write the spider’s equation of motion.
The general solution for the equ

Afficher plus

Notebooks associés (15)

Exploring Quantization and Oversampling through One-Bit Music

Powder X-Ray Diffraction Analysis: Exploring Crystal Structures.

\theta

value for the plane. In powder XRay data, however, any plane ori

Introduction to Reciprocal Lattices with Periodic Boundary Conditions.

Voice Transformation Techniques Using Digital Signal Processing

A Gallery of Voice Transformers[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

Exploring the Karplus-Strong Algorithm for Sound Synthesis

The Karplus-Strong Algorithm[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

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Théorie des types

Chattez avec Graph Search

Modular Substructural Constraints for Embedded DSLs

Interactive and Automated Reasoning with Orthologic

Precise Reasoning about Container-Internal Pointers with Logical Pinning