- published: 08 Dec 2013
- views: 11839
In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type.
Type theory is closely related to (and in some cases overlaps with) type systems, which are a programming language feature used to reduce bugs. The types of type theory were created to avoid paradoxes in a variety of formal logics and rewrite systems and sometimes "type theory" is used to refer to this broader application.
Two well-known type theories that can serve as mathematical foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory.
The types of type theory were invented by Bertrand Russell in response to his discovery that Gottlob Frege's version of naive set theory was afflicted with Russell's paradox. This theory of types features prominently in Whitehead and Russell's Principia Mathematica. It avoids Russell's paradox by first creating a hierarchy of types, then assigning each mathematical (and possibly other) entity to a type. Objects of a given type are built exclusively from objects of preceding types (those lower in the hierarchy), thus preventing loops.
Robert Harper - Type Theory Foundations, Lecture 1, Oregon Programming Languages Summer School 2012, University of Oregon For more info about the summer school please visit http://www.cs.uoregon.edu/research/summerschool/summer12/
Traditionally, in Computer Science, sets are assumed to be the basis of a type theory, together with Boolean logic. In this version of type theory, we do not need sets or Boolean logic; intuitionism is enough ("no principle of excluded middle required"). The underlying math is Topos Theory, but you are not required to be even aware of its existence. The theory is described using diagrams, not the traditional (and tricky) deduction rules. The resulting theory turns out to have dependent types. A simple "real-life" example or two will illustrate all this. Help us caption & translate this video! http://amara.org/v/HcNK/
At the heart of intuitionistic type theory lies an intuitive semantics called the “meaning explanations." Crucially, when meaning explanations are taken as definitive for type theory, the core notion is no longer “proof” but “verification”. We’ll explore how type theories of this sort arise naturally as enrichments of logical theories with further judgements, and contrast this with modern proof-theoretic type theories which interpret the judgements and proofs of logics, not their propositions and verifications. Expect the following questions to be answered: What is the difference between a judgement and a proposition? What is a meaning explanation? What is the difference between a proof and a verification? The so-called semantical approach to type theory is, in the speaker's view, more imm...
The video for this talk http://www.meetup.com/Math-for-People/events/174966252/ slides https://docs.google.com/presentation/d/1IzRaCdIfisLWDF_8CL47NlO9NCJNCiR2L4Vbw6aDOqA/edit?usp=sharing code https://github.com/marklemay/introduction-to-type-theory/blob/master/introduction_to_type_theory.agda
Homotopy Type Theory (http://homotopytypetheory.org/) is a research program that brings together computer science (in the form of dependent type theory), logic, and algebraic topology in a single cohesive way. Its great insight is in systematizing the way in which these fields have all, in some sense been studying the “same” thing. From the standpoint of computer science, it consists of a new “geometric” interpretation of dependent type theory in which we think of types as topological spaces, and also the addition of a single axiom, univalence, which makes the initially confusing claim that “equivalence is equivalent to equality.” The implications of this work will take years before they affect the development of compilers and languages directly. However, the concepts and insights already ...
Talk intro runs till 1:37 min. Part 1 of Russell O'Connor talk at Intersections KW, Meetup http://www.meetup.com/Intersections-KW/events/219773810/ Jan. 20, 2015, in Waterloo, Ontario, Boltmade office. Part 1. The Logical Interpretation. http://youtu.be/Xm-sfrKMT4g Part 2. The Computational Interpretation. http://youtu.be/OUbxHGecO4M Part 3. The Mathematical Interpretation. http://youtu.be/zarQdwxcof0 Part 4. Dependent Types. http://youtu.be/1COvKfPZSnY Part 5. Proofs of Poincare Principle & 2 lemmas http://youtu.be/_8egAo1QSLc Part 6. Part 6. Interactive Proof Assistants. http://youtu.be/5iOq1qXPPa4 Recorded by @curelet with Canon PowerShot SX 520 HS superzoom point-and-shoot camera.
The first lecture on type theory in programming languages. Strap yourself in for a wild mathematical ride
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One of the key problems of Homotopy Type Theory is that it introduces axioms such as extensionality and univalence for which there is no known computational interpretation. We propose to overcome this by introducing a Type Theory where a heterogenous equality is defined recursively and equality for the universe just is univalence. This cubical type theory is inspired by Bernardy and Moulin's internal parametricity and by Coquand, Bezem and Huber's cubical set model. This is ongoing work with Ambrus Kaposi at Nottingham.
Did you REMEMBER to LIKE and Check the Links Below? Link to where Fairy Originates - https://en.wikipedia.org/wiki/Fairy Want to see my Pokemon Theories before anyone else? Donate over at Patreon - http://www.patreon.com/protomario This Link is only for a One Time Donation, link below- https://www.paypal.com/cgi-bin/webscr?cmd=_donations&business;=Protomario%40live%2ecom&lc;=US&item;_name=Protomario%27s%20Review¤ An explanation for what I use my Donations for can be found here - https://www.youtube.com/watch?v=fxUj7mQiph0 I stream at on TwitchTV Usually 4 PM EST. - http://www.twitch.tv/protomario Follow me and Tweet Me A Question and I'll answer https://twitter.com/Protomario Here is my fan Moderated page on Facebook - https://www.facebook.com/pages/Protomario-Theories/41155333901187...
A gentle introduction to type theory (well, not so gentle ...): my position is that one of the most important feature of a language, in order to help programmers to write correct code, is the type system. The talk aims at supporting that position and makes a brief excursus on the history of types.
What is the secret behind this mysterious Pokemon? A Pokemon Theory about TYPE: NULL! The chimera companion to Gladion, possible sibling of Lillie and Lusamine and ULTRA BEAST KILLER. Pokemon Reactions! ► http://goo.gl/1VHMmK More Theories! ► http://goo.gl/QznhWx STORY GAMES ► https://goo.gl/QIYVVV FUNNY GAMES ► https://goo.gl/wZpo8H SPOOKY GAMES ► https://goo.gl/RpXCyC GAY GAMES ► https://goo.gl/PdZ9gg ENJOY THE VIDEO? SUBSCRIBE! ► http://bit.ly/AnimaYES Instagram! ► https://www.instagram.com/animargh/ Twitter! ► https://twitter.com/AnimaDotTV Twitch! ► http://www.twitch.tv/animadottv ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬ Your support means the world! Every like, comment and share keeps us growing fast. I love you all :) - Animuh ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
Type:null along with along with Ultra beasts such as UB-01 were recently introduced. Not much is none about them, but from analyzing Type:Null It seems he is a combination of different creatures. This theory tries to explain how this may have a relation to eevee ------------------------------More Info------------------------------- PATREONS: Angelic Empyress Eternalsunrise Jordan Senpai Caleb Nidey ART Intro By: BvDesigns https://www.youtube.com/channel/UCK5w9wL7QQpFN921Jp0LDCA Channel Art and Character By : Draw With Rydi https://www.youtube.com/channel/UCaweqM5ARUl4VWTZRurLO_w Count Down Animations By: David G Burns https://www.youtube.com/user/ConCharacters CONTACT: Twitter:https://twitter.com/OviCuriosities Become My Patreon https://www.patreon.com/user?u=451428&ty;=h Thank You for ...
What's goin' on, Moxie Posse! Be sure to use a Black-Glasses-boosted Crunch on that LIKE button for Pokémon Sun and Moon! Welcome to a Pokémon Sun and Moon theory video! Today, I discuss the theory of Gladion creating Type: Null! What are your thoughts on this theory? Let me know in the comments! _________ Facebook: https://www.facebook.com/melissagamesyt Twitter: https://twitter.com/melissagamesyt Twitch: https://twitch.tv/melissagamesyt Music used includes: "Showdown in Kanto! [Pokémon Mix]" by GlitchxCity: https://www.youtube.com/watch?v=iOeMDS07_J8 Outro music is "Press Start" by MDK: https://www.youtube.com/watch?v=XoLou... http://www.facebook.com/MDKOfficial http://www.youtube.com/MDKOfficialYT
My theory on the connection between The Aether Foundation, Type: Null, and UB-01 in Pokemon Sun and Moon! SUBSCRIBE! http://www.youtube.com/subscription_center?add_user=danielinsomanywords FOLLOW ME ON TWITTER: http://www.twitter.com/supernerddaniel ========================================================== It's been a couple of days since the most recent substantial trailers dropped for Pokemon Sun and Moon, set to be released in about two months from now. And once again, we've been given quite a bit of information. From new Alolan Forms to newly discovered Pokemon entirely, the trailer had lots of interesting reveals -- including a mysterious new group, and two very strange and powerful creatures. The trailer first showed Type: Null, a horrific amalgum of different Pokemon body parts ...
Micheals vid: https://www.youtube.com/watch?v=M2vALXLfbzg Gnoggins vid: https://www.youtube.com/watch?v=1RSGN-REYj8
Oh man Pokemon Sun and Moon are spiralling everyone out of control here with the latest news of a new group, Pokemon and weird creature..let's talk about them!
Description coming soon. http://video.ias.edu/univalent/1213/0411-HomotopyGroup
Slides: https://docs.google.com/presentation/d/1D1JJQpR3GxVR6vw25SMhfCrcJ10iBrG8Iuk5_hKpGdY/edit?usp=sharing
On 9 Nov 2014, at Homotopy Type Theory Workshop (7-10 Nov 2014, Mathematical Institute, University of Oxford) Abstract: In this talk, I will describe work in progress (joint with Guillaume Brunerie) on a cubical syntax for type theory. The goal of the work is to provide a syntactic type theory where the computational aspects of the cubical sets model by Bezem, Coquand, and Huber can be expressed. I will describe a "boundaries-as-terms" cubical type theory, where the basic judgement is "the term u is an n-cube in the type A, together with its boundary", and the cubical operations (faces, degeneracies, symmetries, and diagonals) can be applied to any term. This syntax permits a clean formulation of the computation rules for Kan fillings in Pi, Sigma, and identity types. Moreover, diagonals ...
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But you know, every pretty wrapper, just holds,...
just holds another little lie.
Yeah, the road to hell winds through a gingerbread town.
And every ounce tells,
(Diptheria)
when every single pound kills,
(Diptheria)
and every loose tooth
(Diptheria)
Shows another wrong move
(Diptheria)
..of the diet...
(Diptheria)
..of denial.
(Denial)
Denial.
Denial.
Sell it uptown, Sell it downtown.
You're losing your pounds, just sleazing your bones.
Gotta,..get a shot from the man.
Got to...
Sell it uptown, Sell it downtown.
You're losing your pounds, just sleazing your bones.
Gotta,..get a shot from the man.
Got to...
Oh, yeah. And every ounce tells,
(Diptheria)
when every single pound kills,
(Diptheria)
and every loose tooth
(Diptheria)
Shows another wrong move
(Diptheria)
..of the diet...
(Diptheria)
..of denial.
(Denial)
Denial.
Denial.
AAAHHHHHH!
Yeah, the road to hell, is paved with chocolate sweets.
Yeah, and candy corn, that's sewn, that's sewn at the side of the streets.
The road to hell winds through a gingerbread town.
It looks real good, tastes real good,