Mathematics and Computation

A blog about mathematics for computers

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The preprint version of the paper A general definition of dependent type theories has finally appeared on the arXiv! Over three years ago Peter Lumsdaine invited me to work on the topic, which I gladly accepted, and dragged my student Philipp Haselwarter into it. We set out to give an answer to the queation:

What is type theory, precisely?

At least for me the motivation to work on such a thankless topic came from Vladimir Voevodsky, who would ask the question to type-theoretic audiences. Some took him to be a troll and others a newcomer who just had to learn more type theory. I was among the latter, but eventually the question got through to me – I could point to any number of specific examples of type theories, but not a comprehensive and mathematically precise definition of the general concept.

It is too easy to dismiss the question by claiming that type theory is an open-ended concept which therefore cannot be completely captured by any mathematical definition. Of course it is open-ended, but it does not follow at all that we should not even attempt to define it. If geometers were equally fatalistic about the open-ended notion of space we would never have had modern geometry, topology, sheaves – heck, half of 20th century mathematics would not be there!

Of course, we are neither the first nor the last to give a definition of type theory, and that is how things should be. We claim no priority or supremacy over other definitions and views of type theory. Our approach could perhaps be described as "concrete" and "proof-theoretic":

  1. We wanted to stay close to traditional syntax.
  2. We gave a complete and precise definition.
  3. We aimed for a level of generality that allows useful meta-theory of a wide range of type theories.

One can argue each of the above points, and we have done so among ourselves many times. Nevertheless, I feel that we have accomplished something worthwhile – but the ultimate judges will be our readers, or lack of them. You are kindly invited to take a look at the paper.

Download PDF: arxiv.org/pdf/2009.05539.pdf

I should not forget to mention that Peter, with modest help from Philipp and me, formalized almost the entire paper in Coq! See the repository general-type-theories at GitHub.

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I am happy to announce that we are restarting the "Every proof assistants" series of talks with Anders Mörtberg who will talk about Cubical Agda. Note that we are moving the seminar time to a more reasonable hour, at least as far as the working people in Europe are concerned.

Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types

Time: Thursday, September 17, 2020 from 15:00 to 16:00 (Central European Summer Time, UTC+2)
Location: online at Zoom ID 989 0478 8985
Speaker: Anders Mörtberg (Stockholm University)
Proof assistant: Cubical Agda

Abstract: The dependently typed programming language Agda has recently been extended with a cubical mode which provides extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically to proof assistants based on dependent type theory which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types. In the talk I will discuss how Agda was extended to a full-blown proof assistant with native support for univalence and a general schema of higher inductive types. I will also show a variety of examples of how to use Cubical Agda in practice to reason about mathematics and computer science.

The talk video recording is available.

We have more talks in store, but we will space them out a bit to give slots to our local seminar.

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We shall finish the semester with a "Every proof assistant" talk by William Bowman. Note that we start an hour later than usual, at 17:00 UTC+2.

Cur: Designing a less devious proof assistant

Time: Thursday, June 25, 2020 from 17:00 to 18:00 (Central European Summer Time, UTC+2)
Location: online at Zoom ID 989 0478 8985
Speaker: William J. Bowman (University of British Columbia)
Proof assistant: Cur

Abstract:

Dijkstra said that our tools can have a profound and devious influence on our thinking. I find this especially true of modern proof assistants, with "devious" out-weighing "profound". Cur is an experiment in design that aims to be less devious. The design emphasizes language extension, syntax manipulation, and DSL construction and integration. This enables the user to be in charge of how they think, rather than requiring the user to contort their thinking to that of the proof assistant. In this talk, my goal is to convince you that you want similar capabilities in a proof assistant, and explain and demonstrate Cur's attempt at solving the problem.

The talk video recording and slides with notes and demo code are available.

Upcoming talks: Anders Mörtberg's talk on Cubical Agda will take place in September 2020.

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This week shall witness a performance by Conor McBride.

Epigram 2: Autopsy, Obituary, Apology

Time: Thursday, June 11, 2020 from 16:00 to 17:00 (Central European Summer Time, UTC+2)
Location: online at Zoom ID 989 0478 8985
Speaker: Conor McBride (University of Strathclyde)
Proof assistant: Epigram 2

Abstract: "A good pilot is one with the same number of take-offs and landings." runs the old joke, which makes me a very bad pilot indeed. The Epigram 2 project was repeatedly restarted several times in the late 2000s and never even reached cruising altitude. This talk is absolutely not an attempt to persuade you to start using it. Rather, it is an exploration of the ideas which drove it: proof irrelevant observational equality, first class datatype descriptions, nontrivial equational theories for neutral terms. We may yet live to see such things. Although the programming language elaborator never happened, the underlying proof engine was accessible via an imperative interface called "Cochon": we did manage some interesting constructions, at least one of which I can walk through. I'll also explore the reasons, human and technological, why the thing did not survive the long dark.

The video recording of the talk.

Upcoming talks:

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This week the speaker will be Jon Sterling, and we are getting two proof assistants for the price of one!

redtt and the future of Cartesian cubical type theory

Time: Thursday, June 4, 2020 from 16:00 to 17:00 (Central European Summer Time, UTC+2)
Location: online at Zoom ID 989 0478 8985
Speaker: Jon Sterling (Carnegie Mellon University)
Proof assistant: redtt and cooltt

Abstract: redtt is an interactive proof assistant for Cartesian cubical type theory, a version of Martin-Löf type theory featuring computational versions of function extensionality, higher inductive types, and univalence. Building on ideas from Epigram, Agda, and Idris, redtt introduces a new cubical take on interactive proof development with holes. We will first introduce the basics of cubical type theory and then dive into an interactive demonstration of redtt’s features and its mathematical library.

After this we will catch a first public glimpse of the future of redtt, a new prototype that our team is building currently code-named “cooltt”: cooltt introduces syntax to split on disjunctions of cofibrations in arbitrary positions, implementing the full definitional eta law for disjunction. While cooltt is still in the early stages, it already has full support for univalence and cubical interactive proof development.

The video recording of the talk.

Upcoming talks:

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We are marching on with the Every proof assistant series!

Mechanizing Meta-Theory in Beluga

Time: Thursday, May 28, 2020 from 16:00 to 17:00 (Central European Summer Time, UTC+2)
Location: online at Zoom ID 989 0478 8985
Speaker: Brigitte Pientka (McGill University)
Proof assistant: Beluga

Abstract: Mechanizing formal systems, given via axioms and inference rules, together with proofs about them plays an important role in establishing trust in formal developments. In this talk, I will survey the proof environment Beluga. To specify formal systems and represent derivations within them, Beluga relies on the logical framework LF; to reason about formal systems, Beluga provides a dependently typed functional language for implementing (co)inductive proofs about derivation trees as (co)recursive functions following the Curry-Howard isomorphism. Key to this approach is the ability to model derivation trees that depend on a context of assumptions using a generalization of the logical framework LF, i.e. contextual LF which supports first-class contexts and simultaneous substitutions.

Our experience demonstrated that Beluga enables direct and compact mechanizations of the meta-theory of formal systems, in particular programming languages and logics.

The video recording of the talk.

Upcoming talks:

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I am happy to announce the next seminar in the "Every proof assistant" series.

MMT: A Foundation-Independent Logical System

Time: Thursday, May 21, 2020 from 16:00 to 17:00 (Central European Summer Time, UTC+2)
Location: online at Zoom ID 989 0478 8985
Speaker: Florian Rabe (University of Erlangen)
Proof assistant: The MMT Language and System

Abstract: Logical frameworks are meta-logics for defining other logics. MMT follows this approach but abstracts even further: it avoids committing to any foundational features like function types or propositions. All MMT algorithms are parametric in a set of rules, which are self-contained objects plugged in by the language designer. That results in a framework general enough to develop many formal systems including other logical frameworks in it, enabling the rapidly prototyping of new language features.

Despite this high level of generality, it is possible to develop sophisticated results in MMT. The current release includes, e.g., parsing, type reconstruction, module system, IDE-style editor, and interactive library browser. MMT is systematically designed to be extensible, providing multiple APIs and plugin interfaces, and thus provides a versatile infrastructure for system development and integration.

This talk gives an overview of the current state of MMT and its future challenges. Examples are drawn from the LATIN project, a long-running project of building a modular, highly inter-related suite of formalizations of logics and related formal systems.

The video recording of the talk.

The spring schedule of talks is planned as follows:

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For a while now I have been contemplating a series of seminars titled "Every proof assistant" that would be devoted to all the different proof assistants out there. Apart from the established ones (Isabelle/HOL, Coq, Agda, Lean), there are other interesting experimental proof assistants, and some that are still under development, or just proofs of concept. I would like to know more about them, and I suspect I am not the only one.

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I forgot to record the fact that already two years ago I wrote a paper on Lawvere's fixed-point theorem in synthetic computability:

Andrej Bauer: On fixed-point theorems in synthetic computability. Tbilisi Mathematical Journal, Volume 10: Issue 3, pp. 167–181.

It was a special issue in honor of Professors Peter J. Freyd and F. William Lawvere on the occasion of their 80th birthdays.

Lawvere's paper "Diagonal arguments and cartesian closed categories proves a beautifully simple fixed point theorem.

Theorem: (Lawvere) If $e : A \to B^A$ is a surjection then every $f : B \to B$ has a fixed point.

Proof. Because $e$ is a surjection, there is $a \in A$ such that $e(a) = \lambda x : A \,.\, f(e(x)(x))$, but then $e(a)(a) = f(e(a)(a)$. $\Box$

Lawvere's original version is a bit more general, but the one given here makes is very clear that Lawvere's fixed point theorem is the diagonal argument in crystallized form. Indeed, the contrapositive form of the theorem, namely

Corollary: If $f : B \to B$ has no fixed point then there is no surjection $e : A \to B^A$.

immediately implies a number of famous theorems that rely on the diagonal argument. For example, there can be no surjection $A \to \lbrace 0, 1\rbrace^A$ because the map $x \mapsto 1 - x$ has no fixed point in $\lbrace 0, 1\rbrace$ -- and that is Cantors' theorem.

It not easy to find non-trivial instances to which Lawvere's theorem applies. Indeed, if excluded middle holds, then having a surjection $e : A \to B^A$ implies that $B$ is the singleton. We should look for interesting instances in categories other than classical sets. In my paper I do so: I show that countably based $\omega$-cpos in the effective topos are countable and closed under countable products, which gives us a rich supply of objects $B$ such that there is a surjection $\mathbb{N} \to B^\mathbb{N}$.

Enjoy the paper!

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It has been almost a decade since Matija Pretnar and I posted the first blog posts about programming with algebraic effects and handlers and the programming language Eff. Since then handlers have become a well-known control mechanism in programming languages.

Handlers and monads excel at simulating effects, either in terms of other effects or as pure computations. For example, the familiar state monad implements mutable state with (pure) state-passing functions, and there are many more examples. But I have always felt that handlers and monads are not very good at explaining how a program interacts with its external environment and how it gets to perform real-world effects.

Danel Ahman and I have worked for a while on attacking the question on how to better model external resources and what programming constructs are appropriate for working with them. The time is right for us to show what we have done so far. The theoretical side of things is explained in our paper Runners in action, Danel implemented a Haskell library Haskell-Coop to go with the paper, and I implemented a programming language Coop.

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