Type Theory ציבורי
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In this episode I had the pleasure to have an in-depth conversation with ConalElliott about his life, his work, his philosophy and his many opinions aboutresearch and the current state of PL Research and how it lead him to come withthe concept of Denotational Design. Conal got his PhD at CMU in the 90s underFrank Pfenning working on Higher-Order Un…
 
In this episode I share my initial impressions -- very positive! -- of the Metamath system. Metamath allows one to develop theorems from axioms which you state. Typing or other syntactic requirements of axioms or theorems are also expressed axiomatically. The system exhibits an elegant coherent vision for how such a tool should work, and was super …
 
I discuss a book edited by Freek Wiedijk (pronounced "Frake Weedike"), which describes the solutions he received in response to a call for formalized proofs of the irrationality of the square root of 2. The book was published in 2006, and made an impression on me then. The provers we have discussed so far all have a solution in the book, except for…
 
in this episode we interview Jesper Cockx, one of the core developers on Agda.We talk about the philosophy behind Agda, his work on pattern matching, theUniqueness of Identity of Proofs, UIP for short, and why it is inconsistentwith Homotopy Type Theory. Links Jesper’s Website Jesper’s Twitter: @agdakx Jesper’s PhD Thesis Rewrite Theory paper Patte…
 
In this episode me, Eric and Nitin continues our conversation started in thelast episode. This time we move our attention to the cool projects happeningin Coq, in particular commenting through the projects mentioned in AndrewAppel’s keynote “Coq’s Vibrant Ecosystem for verification engineering” thattook place in CPP’22 which is colocated with POPL …
 
In this episode I gather with two good friends Eric and Nitin to randomly talk random subjects that pops up. Among them we talked about POPL, Scala, Isabelle, Parametricity, Dependent Object Types (DOT, for short) and more! Links Nitin Twitter @NitinJohnRaj2 Eric Twitter @EricBond10 Collection of links on logical relations Theorems for Free Reynold…
 
This episode is about the journey of a programmer that converted himself intoa Haskell developer after working with C/C++ for more than 10years. Here are a few questions that you’ll find the answer to in this episode: What does he find so compelling about Haskell? Why did it make him dive deeper into the Theoretical Computer Science? Why did it mak…
 
This is the start of Chapter 15, about interactive theorem provers (ITPs). In this episode, I talk about the difference between fully automatic and interactive provers, and my plan to discuss and compare several different ITPs, in future episodes of this chapter.על ידי Aaron Stump
 
In this episode, I outline the argument for why the proof-theoretic ordinal (in the sense of Rathjen, as presented last episode) is epsilon-0. My explanation has something of a hole, in explaining how one would go about deriving induction for ordinals strictly less than epsilon-0 in Peano Arithmetic. To help paper over this hole a little, I discuss…
 
Ordinal analysis seeks to determine the strength of a logical theory by assigning an ordinal to it. Which one? In this episode I describe a definition of the proof-theoretic ordinal of a logical theory from a paper by proof theorist Michael Rathjen. It is basically a measure of how strong an induction principle is derivable in the theory. (The firs…
 
Ordinal analysis is an important branch of proof theory, which seeks to compare, quantitatively, the strengths of different proof systems. The quantities in question are ordinals, which extend the ordering character of natural numbers into the infinite. In this episode, I discuss these ideas a bit further, and also review a little the ordinals up t…
 
Talia Ringer is an Assistant Professor at University of IllinoisUrbana-Champaign. She did her PhD at University of Washington with her thesison Proof Repair. She’s very active on twitter @taliaringer. And in this episode we will talkabout her transition from PhD to Professor, her work on diversity, her ADHD andhow it has affected her career so far,…
 
Cut elimination for sequent calculus is more involved that normalization of detours for natural deduction. There are more cases of cuts that must be transformed than correspond to detours (introductions followed by eliminations). In this episode, I explain why that is.על ידי Aaron Stump
 
In this episode we have talk with Alejandro Serrano Mena, heworks on 47 degrees and is a published author of two books about Haskell: TheBook of Monads and Practical Haskell. We talk about many interesting features behind functionalprogramming such as adts, pattern matching, impredicativity, monads, effects,hacking the ghc and how all this comes to…
 
We saw in the last few episodes that proofs in natural deduction can be simplified by removing detours, which occur when an introduction inference is immediately followed by an elimination inference on the introduced formula. What corresponds to this for sequent calculus proofs? The answer is cut elimination. This episode describes the cut rule and…
 
We talk about normalizing detours -- which are when an introduction inference is immediately followed by an elimination inference -- for the implication rules. Under Curry-Howard, this actually corresponds to beta-reduction, and could make the proof bigger (though less complex in a certain sense).על ידי Aaron Stump
 
Sequent calculus is a different style in which proof systems can be formulated, where for each connective, we have a left rule for introducing it (in the conclusion of the rule) in the left part of a sequent G => D (i.e., in G), and similarly a right rule for introducing it in the right part (D). The beauty of sequent calculus is disjunction is han…
 
We discuss the problem of the or-elimination rule in natural deduction, which does not have the correct form for natural deduction inferences. It is a research problem to fix this! Don't forget to get in touch with me if you want to do the mini-course over Zoom next month, on normalization.על ידי Aaron Stump
 
This episode begins the discussion of the style of proof known as Natural Deduction, invented by Gerhard Gentzen, a student of Hermann Weyl, himself a student of David Hilbert (sorry, I said incorrectly that Gentzen was Hilbert's own student). Each logical connective (like OR, AND, IMPLIES, etc.) has introduction rules that let you prove formulas b…
 
We continue our gradual entry into proof theory by talking about reflecting meta-logical reasoning into logical rules, and naming the three basic proof systems (Hilbert-style, natural deduction, and sequent calculus). Advertising for the October 3-session Zoom mini-course on normalization continues. Email me if you are interested! This is just for …
 
I highlight two basic points in this continuing warm-up to proof theory: there are different proof systems for the same logics, and it is customary to separate purely logical rules (dealing with propositional connectives and quantifiers, for example) from rules or axioms for some particular domain (like axioms about arithmetic, or whatever domain i…
 
This is the start of Season 3 of the podcast. We are beginning with a new chapter (Chapter 14) on Proof Theory. This episode gives some basic introduction to the history and broad concerns of proof theory. There is even music, composed and played by yours truly, chez nous...על ידי Aaron Stump
 
In this episode I discuss the paper "Modula-2 and Oberon" by Niklaus Wirth. Modula-2 introduced (it seems from the paper) the idea of having modules with explicit import and import lists -- something we saw at the start of our look at module systems with Haskell's module system. I note some interesting historical points raised by the paper.…
 
In this episode we host a discussion between Anupam Das and ThorstenAltenkirch on the role of constructivism in mathematics, logic and computerscience. Anupam is a lecturer in the University of Birmingham in the UK, and ThorstenAltenkirch is a CS Professor at the University of Nottingham. We discuss why constructive content in proofs matters, the l…
 
Analogously to the decomposition of a datatype into a functor (which can be built from other functors to assemble a bigger language from smaller pieces of languages) and a single expression datatype with a sole constructor that builds an Expr from an F Expr (where F is the functor) -- analogously, a recursion can be decomposed into algebras and a f…
 
Last episode we discussed how functors can describe a single level of a datatype. In this episode, we discuss how to put these functors back together into a datatype, using disjoint unions of functors and a fixed-point datatype. The latter expresses the idea that inductive data is built in any finite number of layers, where each layer is described …
 
This episode continues the discussion of Swierstra's paper "Datatypes à la Carte", explaining how we can decompose a datatype into the application of a fixed-point type constructor and then a functor. The functor itself can be assembled from other functors for pieces of the datatype. This makes modular datatypes possible.…
 
In a really wonderful paper of some few years back, Swierstra introduced the idea of modular datatypes using ideas from universal algebra. Modular datatypes allow one to assemble a bigger datatype from component datatypes, and combine functions written on the component datatypes in a modular way. In this episode I introduce the paper and the proble…
 
In a 2013 journal article titled "A Scalable Module System", Florian Rabe and Michael Kohlhase propose a module system called MMT (Modules for Mathematical Theories) for structuring mathematical knowledge. The paper has a very interesting general discussion of module systems, from programming languages but also other areas like algebraic specificat…
 
In this episode I interview Anupam Das we have a nice conversation on thehistorical perspective of how Logic and Proof Theory as we know today cameabout in the 30’s. The differences between Natural Deduction and SequentCalculus, Cut Elimination and much more. Links Anupam Das The Proof Theory Blog Stanford Encyclopedia of Philosophy Anupam’s Talk o…
 
I look back at the module systems we considered so far (Haskell's, Standard ML's, and Agda's), and consider a few points that are on my mind about these. I also mention a few other languages' module systems (Clojure, Java 9). I also ponder names and namespace management.על ידי Aaron Stump
 
In this episode I have a nice conversation with Chris Jenkins to talk aboutthe Cedille theorem prover, based on a very concise type theory called CDLE.The main selling point of Cedille is that the theory is so small that thetyping rules fit one page. And yet it is strong enough to do relevant theoremproving. This is probably the most technical epis…
 
SML has arguably the most complicated/powerful module system of any existing programming language. It is a luxury RV to Haskell's camper van. I take a high-level look, following a very nice paper by Xavier Leroy, titled "A modular module system".על ידי Aaron Stump
 
I start Chapter 13 (in Season 2) of the podcast, on module systems. Almost all programming languages I know include some kind of scheme for modules, packages, namespaces, or something like this. I discuss the high-level ideas of namespace management and type abstraction, two main use cases for module systems. Subsequent episodes will discuss module…
 
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