Type theory

In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general, type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation, a common one is Thierry Coquand's Calculus of Inductive Constructions. History of type theory Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic. Russell's paradox, which was discovered by Bertrand Russell, existed because a set could be defined using "all possible sets", which included itself. Between 1902 and 1908, Bertrand Russell proposed various "theories of type" to fix the problem. By 1908 Russell arrived at a "ramified" theory of types together with an "axiom of reducibility" both of which featured prominently in Whitehead and Russell's Principia Mathematica published between 1910 and 1913. This system avoided Russell's paradox by creating a hierarchy of types and then assigning each concrete mathematical entity to a type. Entities of a given type are built exclusively of subtypes of that type, thus preventing an entity from being defined using itself. Russell's theory of types ruled out the possibility of a set being a member of itself. Types were not always used in logic. There were other techniques to avoid Russell's paradox. Types did gain a hold when used with one particular logic, Alonzo Church's lambda calculus. The most famous early example is Church's simply typed lambda calculus. Church's theory of types helped the formal system avoid the Kleene–Rosser paradox that afflicted the original untyped lambda calculus. Church demonstrated that it could serve as a foundation of mathematics and it was referred to as a higher-order logic.

Ontology

Official source

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Related exercises (8)

A float in motion

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

The ramp

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 on a cylinder

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.

Electric field in a vertical wire

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

Hanging spider

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

Related notebooks (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"]