Iowa Type Theory Commute

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Natural deduction: or, the bad news! Sep 14, 2021

Show notes

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.

Implication rules for natural deduction Sep 14, 2021

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We discuss further inferences in natural deduction, in particular implication introduction and elimination.

Natural Deduction Sep 11, 2021

Show notes

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 built with that connective; and elimination rules that let you deduce consequences from a proven formula built with that connective.

Rules of proof, standard proof systems Sep 08, 2021

Show notes

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 ITTC listeners.
Finally, if you like the podcast and want to support it, would you consider a small ($5 or $10 is great) donation? This is not so much to cover the $12/month hosting fee at Buzzsprout but to get some kudos here at my institution, so people here know that I have a podcast [grin] and that listeners like it. To donate, go here, then choose "Colleges", then "College of Liberal Arts and Sciences", and then tick "Computer Science Development Fund". For "gift details", please leave "gift instructions" that say the gift is for Aaron Stump for the Iowa Type Theory Commute podcast. Then the money will get to the right place for me to use it for this. Thanks a lot in advance!

Different proof systems, distinguishing logical rules from domain axioms Sep 01, 2021

Show notes

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 is of interest).

Introduction to Proof Theory (Start of Season 3) Aug 31, 2021

Show notes

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...

Modula-2 Jul 27, 2021

Show notes

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.

Decomposing recursions using algebras Jul 12, 2021

Show notes

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 fold function that applies the algebra throughout the datatype.

Reassembling datatypes from functors using a fixed-point Jul 10, 2021

Show notes

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 by the functor for the datatype.

Decomposing datatypes into functors Jul 03, 2021

Show notes

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.

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